Rust f64::asinh

Percentage Accurate: 30.5% → 99.8%

Time: 2.1s

Alternatives: 4

Speedup: 1.5×

Specification

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]

Initial Program: 30.5% 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, 1.5× 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 30.5%

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

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

3. Applied rewrites 99.8%

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

Alternative 2: 28.4% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (copysign (log (+ (fabs x) x)) x))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.5%

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

2. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-fabs.f64 28.4

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

4. Applied rewrites 28.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. sum-to-mult-rev N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 28.4

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

6. Applied rewrites 28.4%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + x\right)}, x\right) \]
7. Add Preprocessing

Alternative 3: 18.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.5%

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

2. Taylor expanded in x around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-fabs.f64 27.0

      \[\leadsto \mathsf{copysign}\left(\log \left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right), x\right) \]

4. Applied rewrites 27.0%

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

5. Taylor expanded in x around 0

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

   1. lower-fabs.f64 18.5

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

7. Applied rewrites 18.5%

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

Alternative 4: 9.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x)
	return copysign(log(Float64(-x)), x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.5%

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

2. Taylor expanded in x around -inf

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

   1. lower-*.f64 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 9.0

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

6. Applied rewrites 9.0%

   \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(-x\right), x\right)} \]
7. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.4× 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 2025154 
(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))