Compute acos with less cancellation near x=1

decimal.js

@@ -744,14 +744,13 @@
 
     Ctor.precision = pr + 6;
     Ctor.rounding = 1;
-
-    x = x.asin();
-    halfPi = getPi(Ctor, pr + 4, rm).times(0.5);
+
+    x = (new Ctor(1)).minus(x).div(x.plus(1)).sqrt().atan();
 
     Ctor.precision = pr;
     Ctor.rounding = rm;
 
-    return halfPi.minus(x);
+    return x.times(2);
   };
 
 

decimal.mjs

@@ -741,13 +741,12 @@ P.inverseCosine = P.acos = function () {
   Ctor.precision = pr + 6;
   Ctor.rounding = 1;
 
-  x = x.asin();
-  halfPi = getPi(Ctor, pr + 4, rm).times(0.5);
+  x = (new Ctor(1)).minus(x).div(x.plus(1)).sqrt().atan();
 
   Ctor.precision = pr;
   Ctor.rounding = rm;
 
-  return halfPi.minus(x);
+  return x.times(2);
 };
 
 

test/hypothesis/.gitignore

@@ -0,0 +1,3 @@
+__pycache__
+.hypothesis
+.ipynb_checkpoints

test/hypothesis/error_hunt.py

@@ -0,0 +1,99 @@
+from decimal import Decimal, getcontext
+import json
+import subprocess
+import hypothesis
+from hypothesis import given, settings
+from hypothesis.strategies import decimals, integers, tuples
+from mpmath import mp
+import pytest
+
+mp.prec = 500
+getcontext().prec = 14
+
+node = subprocess.Popen(
+    ["node", "evaluate.mjs"], stdout=subprocess.PIPE, stdin=subprocess.PIPE
+)
+
+
+def get_decimal(func: str, args: list, config: dict):
+    arg = json.dumps({"func": func, "args": args, "config": config}).encode() + b"\r"
+    node.stdin.write(arg)
+    node.stdin.flush()
+    return Decimal(node.stdout.readline().strip().decode())
+
+
+def assert_matches(x, mpfunc, jsfunc=None):
+    if jsfunc is None:
+        jsfunc = mpfunc
+    y = Decimal(str(getattr(mp, mpfunc)(x))) * Decimal("1.0")
+    z = get_decimal(jsfunc, [str(x)], {"precision": 14})
+    assert y == z
+
+
+@pytest.mark.parametrize("fn", "sin cos tan atan asinh".split())
+@given(
+    x=tuples(
+        decimals(
+            allow_nan=False, allow_infinity=False, min_value=-1, max_value=1, places=14
+        ),
+        integers(min_value=-99, max_value=99),
+    ).map(lambda tup: tup[0] * Decimal(10) ** tup[1])
+)
+@settings(max_examples=100_000)
+def test_matches(x, fn):
+    assert_matches(x, fn)
+
+
+@pytest.mark.parametrize("fn", "ln log10 sqrt".split())
+@given(
+    x=tuples(
+        decimals(
+            allow_nan=False,
+            allow_infinity=False,
+            min_value=1e-13,
+            max_value=1,
+            places=14,
+        ),
+        integers(min_value=-99, max_value=99),
+    ).map(lambda tup: tup[0] * Decimal(10) ** tup[1])
+)
+@settings(max_examples=100_000)
+def test_positive_domain(x, fn):
+    assert_matches(x, fn)
+
+
+@pytest.mark.parametrize("fn", "asin acos atanh".split())
+@given(
+    x=decimals(
+        allow_nan=False, allow_infinity=False, min_value=-1, max_value=1, places=14
+    )
+)
+@settings(max_examples=100_000)
+def test_inverse_trig(x, fn):
+    assert_matches(x, fn)
+
+
+@pytest.mark.parametrize("fn", "sinh cosh tanh exp".split())
+@given(
+    x=tuples(
+        decimals(
+            allow_nan=False, allow_infinity=False, min_value=-1, max_value=1, places=14
+        ),
+        integers(min_value=-99, max_value=3),
+    ).map(lambda tup: tup[0] * Decimal(10) ** tup[1])
+)
+@settings(max_examples=100_000)
+def test_small_domain(x, fn):
+    assert_matches(x, fn)
+
+@given(
+    x=tuples(
+        decimals(
+            allow_nan=False, allow_infinity=False, min_value=1, max_value=10, places=14
+        ),
+        integers(min_value=0, max_value=99),
+    ).map(lambda tup: tup[0] * Decimal(10) ** tup[1])
+)
+@settings(max_examples=100_000)
+def test_acosh(x):
+    assert_matches(x, 'acosh')

test/hypothesis/evaluate.mjs

@@ -0,0 +1,26 @@
+// listen for test cases, and provide the results.
+// give JSON of the test case, receive the result
+// Example: 
+// > {"func":"tan", "args":["12.5"], "config":{"precision":8}}
+// -0.066468242
+
+
+import {Decimal} from '../../decimal.mjs';
+import {createInterface} from 'readline';
+
+const readline = createInterface({
+	input: process.stdin,
+	output: process.stdout
+});
+
+readline.on("close", () => {console.log('\n'); process.exit(0);});
+
+readline.on("line", (line) => {
+    if (line) {
+        const {func, args, config} = JSON.parse(line);
+        config.defaults = true;
+        Decimal.set(config);
+        const result = Decimal[func](...args);
+        console.log(result);
+    }
+});

test/hypothesis/requirements.txt

@@ -0,0 +1,3 @@
+hypothesis
+mpmath
+node

test/modules/acos.js

@@ -68,6 +68,10 @@ T('acos', function () {
   t('0.41923186648524998285699814886092351368793359300547574', 42, 3, '1.13819724675000902666504291062053681280681');
   t('-0.19508761025300975791021816036', 27, 1, '1.76714310275532020878366926');
   t('0.0623252416', 19, 0, '1.508430664767542249');
+  t('0.9999999297625', 14, 4, '0.00037479994883195');
+  t('0.99999999467518', 14, 4, '0.0001031970930281');
+  t('0.9999999999999999995', 25, 4, '0.000000001000000000000000000041667');
+  t('0.99999999999999999999995', 30, 4, '0.0000000000100000000000000000000000416667');
 
   /*
   t('0.95', 6, 5, '0.31756');

test/modules/sin.js

@@ -119,4 +119,5 @@ T('sin', function () {
   t('22011131111011111.111111611113111111111151111111', 143, 2, '0.82582504036277799386306063085803210583251158969990606609364360685569588545519071481543672724620118406694191888115120286393881609546697317692404');
   t('996270725099515169352424536636186062915113219400094989.8763797268889422850038402633796294758036260533902551191769915343780424028900449342752548782035', 46, 2, '0.6613706114081017074779805460666900787572253475');
   t('0.780360750628373', 37, 5, '0.7035358359376557390803090830882458906');
+  t('5900', 14, 4, '0.088879123681079');
 });