Rust f64::asinh

Percentage Accurate: 29.6% → 99.9%

Time: 2.1s

Alternatives: 2

Speedup: N/A×

Specification

Math

\[\begin{array}{l} \\ \sinh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (asinh x))

C

double code(double x) {
	return asinh(x);
}

Python

def code(x):
	return math.asinh(x)

Julia

function code(x)
	return asinh(x)
end

MATLAB

function tmp = code(x)
	tmp = asinh(x);
end

Wolfram

code[x_] := N[ArcSinh[x], $MachinePrecision]

Initial Program: 29.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))

C

double code(double x) {
	return copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
}

Java

public static double code(double x) {
	return Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
}

Python

def code(x):
	return math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)

Julia

function code(x)
	return copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
end

Wolfram

code[x_] := N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

Alternative 1: 99.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\sinh^{-1} x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (copysign (asinh x) x))

C

double code(double x) {
	return copysign(asinh(x), x);
}

Python

def code(x):
	return math.copysign(math.asinh(x), x)

Julia

function code(x)
	return copysign(asinh(x), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(asinh(x));
end

Wolfram

code[x_] := N[With[{TMP1 = Abs[N[ArcSinh[x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

Derivation

1. Initial program 31.8%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}, x\right) \]

   2. lift-+.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}, x\right) \]

   3. lift-fabs.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left|x\right|} + \sqrt{x \cdot x + 1}\right), x\right) \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\sqrt{x \cdot x + 1}}\right), x\right) \]

   5. lift-+.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{x \cdot x + 1}}\right), x\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{x \cdot x} + 1}\right), x\right) \]

   7. pow2 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{{x}^{2}} + 1}\right), x\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + {x}^{2}}}\right), x\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{{x}^{2} + 1}}\right), x\right) \]

   10. pow2 N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{x \cdot x} + 1}\right), x\right) \]

   11. sqr-abs-rev N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{\left|x\right| \cdot \left|x\right|} + 1}\right), x\right) \]

   12. asinh-def-rev N/A

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\sinh^{-1} \left(\left|x\right|\right)}, x\right) \]

   13. rem-sqrt-square-rev N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left(\sqrt{x \cdot x}\right)}, x\right) \]

   14. pow2 N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left(\sqrt{\color{blue}{{x}^{2}}}\right), x\right) \]

   15. sqrt-pow1 N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left({x}^{\left(\frac{2}{2}\right)}\right)}, x\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left({x}^{\color{blue}{1}}\right), x\right) \]

   17. unpow1 N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{x}, x\right) \]

   18. lower-asinh.f64 99.8

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\sinh^{-1} x}, x\right) \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{copysign}\left(\sinh^{-1} x, x\right)} \]
5. Add Preprocessing

Alternative 2: 51.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + x\right)}^{2}\\ t_1 := {t\_0}^{-1}\\ t_2 := {\left(1 + x\right)}^{-1}\\ \mathsf{copysign}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), \mathsf{fma}\left(45, t\_2, \mathsf{fma}\left(45, t\_1, \frac{30}{{t\_0}^{1.5}}\right)\right), \mathsf{fma}\left(t\_1, 3, t\_2 \cdot 3\right) \cdot -0.041666666666666664\right), x \cdot x, t\_2 \cdot 0.5\right), x \cdot x, \mathsf{log1p}\left(x\right)\right), x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 x) 2.0))
        (t_1 (pow t_0 -1.0))
        (t_2 (pow (+ 1.0 x) -1.0)))
   (copysign
    (fma
     (fma
      (fma
       (* 0.001388888888888889 (* x x))
       (fma 45.0 t_2 (fma 45.0 t_1 (/ 30.0 (pow t_0 1.5))))
       (* (fma t_1 3.0 (* t_2 3.0)) -0.041666666666666664))
      (* x x)
      (* t_2 0.5))
     (* x x)
     (log1p x))
    x)))

C

double code(double x) {
	double t_0 = pow((1.0 + x), 2.0);
	double t_1 = pow(t_0, -1.0);
	double t_2 = pow((1.0 + x), -1.0);
	return copysign(fma(fma(fma((0.001388888888888889 * (x * x)), fma(45.0, t_2, fma(45.0, t_1, (30.0 / pow(t_0, 1.5)))), (fma(t_1, 3.0, (t_2 * 3.0)) * -0.041666666666666664)), (x * x), (t_2 * 0.5)), (x * x), log1p(x)), x);
}

Julia

function code(x)
	t_0 = Float64(1.0 + x) ^ 2.0
	t_1 = t_0 ^ -1.0
	t_2 = Float64(1.0 + x) ^ -1.0
	return copysign(fma(fma(fma(Float64(0.001388888888888889 * Float64(x * x)), fma(45.0, t_2, fma(45.0, t_1, Float64(30.0 / (t_0 ^ 1.5)))), Float64(fma(t_1, 3.0, Float64(t_2 * 3.0)) * -0.041666666666666664)), Float64(x * x), Float64(t_2 * 0.5)), Float64(x * x), log1p(x)), x)
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, -1.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(1.0 + x), $MachinePrecision], -1.0], $MachinePrecision]}, N[With[{TMP1 = Abs[N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(45.0 * t$95$2 + N[(45.0 * t$95$1 + N[(30.0 / N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 * 3.0 + N[(t$95$2 * 3.0), $MachinePrecision]), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Initial program 31.8%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{24} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right) + \frac{1}{720} \cdot \left({x}^{2} \cdot \left(45 \cdot \frac{1}{1 + \left|x\right|} + \left(45 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}} + 30 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{3}}\right)\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)}, x\right) \]

5. Applied rewrites 50.5%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), \mathsf{fma}\left(45, {\left(1 + x\right)}^{-1}, \mathsf{fma}\left(45, {\left({\left(1 + x\right)}^{2}\right)}^{-1}, \frac{30}{{\left({\left(1 + x\right)}^{2}\right)}^{1.5}}\right)\right), \mathsf{fma}\left({\left({\left(1 + x\right)}^{2}\right)}^{-1}, 3, {\left(1 + x\right)}^{-1} \cdot 3\right) \cdot -0.041666666666666664\right), x \cdot x, {\left(1 + x\right)}^{-1} \cdot 0.5\right), x \cdot x, \mathsf{log1p}\left(x\right)\right)}, x\right) \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(FPCore (x)
  :name "Rust f64::asinh"
  :precision binary64

  :alt
  (! :herbie-platform c (let* ((ax (fabs x)) (ix (/ 1 ax))) (copysign (log1p (+ ax (/ ax (+ (hypot 1 ix) ix)))) x)))

  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))