Rust f64::asinh

Percentage Accurate: 30.6% → 99.8%

Time: 3.7s

Alternatives: 3

Speedup: 2.2×

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: 30.6% 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: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.9%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
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-sqrt.f64 N/A

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

   4. lift-+.f64 N/A

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

   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. sqr-abs-rev N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. asinh-def-rev N/A

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

   10. lower-asinh.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 57.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.02:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) 0.02)
   (copysign x x)
   (copysign (log x) x)))

C

double code(double x) {
	double tmp;
	if (copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x) <= 0.02) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x) <= 0.02) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x) <= 0.02:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x) <= 0.02)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))))) <= 0.02)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[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], 0.02], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.0200000000000000004

   1. Initial program 23.1%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      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. sqr-abs-rev N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. asinh-def-rev N/A

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

      10. lower-asinh.f64 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\sinh^{-1} \left(\left|x\right|\right), x\right)} \]
   5. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

      2. neg-fabs N/A

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

      3. lift-neg.f64 N/A

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

      4. rem-sqrt-square N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 67.0

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

   6. Applied rewrites 67.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot {\left(\sqrt{-1}\right)}^{2}}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 70.9%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign}\left(-1 \cdot \color{blue}{x}, x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 70.9%

       \[\leadsto \mathsf{copysign}\left(x, x\right) \]

   if 0.0200000000000000004 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 52.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{x}, x\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 31.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 51.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.copySign(x, x);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.9%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
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-sqrt.f64 N/A

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

   4. lift-+.f64 N/A

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

   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. sqr-abs-rev N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. asinh-def-rev N/A

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

   10. lower-asinh.f64 99.8

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

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{copysign}\left(\sinh^{-1} \left(\left|x\right|\right), x\right)} \]
5. Step-by-step derivation

   1. lift-fabs.f64 N/A

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

   2. neg-fabs N/A

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

   3. lift-neg.f64 N/A

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

   4. rem-sqrt-square N/A

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

   5. sqrt-prod N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-sqrt.f64 48.9

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

6. Applied rewrites 48.9%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot {\left(\sqrt{-1}\right)}^{2}}, x\right) \]
8. Step-by-step derivation

9. Applied rewrites 53.2%

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

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{copysign}\left(-1 \cdot \color{blue}{x}, x\right) \]
11. Step-by-step derivation

12. Applied rewrites 53.2%

    \[\leadsto \mathsf{copysign}\left(x, x\right) \]
13. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0)))) x)))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return copysign(log1p((fabs(x) + (fabs(x) / (hypot(1.0, t_0) + t_0)))), x);
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	return Math.copySign(Math.log1p((Math.abs(x) + (Math.abs(x) / (Math.hypot(1.0, t_0) + t_0)))), x);
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	return math.copysign(math.log1p((math.fabs(x) + (math.fabs(x) / (math.hypot(1.0, t_0) + t_0)))), x)

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return copysign(log1p(Float64(abs(x) + Float64(abs(x) / Float64(hypot(1.0, t_0) + t_0)))), x)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[With[{TMP1 = Abs[N[Log[1 + N[(N[Abs[x], $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + t$95$0 ^ 2], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Reproduce

herbie shell --seed 2025026 
(FPCore (x)
  :name "Rust f64::asinh"
  :precision binary64

  :alt
  (! :herbie-platform default (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))