Rust f64::asinh

Percentage Accurate: 29.5% → 99.9%

Time: 2.0s

Alternatives: 4

Speedup: 5.4×

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: 29.5% 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: 99.9% accurate, 1.5× speedup

Math

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

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(Float64(-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 29.5%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
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 99.9

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

3. Applied rewrites 99.9%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\sinh^{-1} \left(\left|x\right|\right)}, x\right) \]
4. 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. rem-sqrt-square-rev N/A

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

   3. *-rgt-identity N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. sqrt-div N/A

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

   9. lower-unsound-/.f64 N/A

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

   10. lower-unsound-sqrt.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-unsound-sqrt.f64 53.2

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

5. Applied rewrites 53.2%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-undiv N/A

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

   5. pow1/2 N/A

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

   6. sqr-pow N/A

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

7. Applied rewrites 99.9%

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

Alternative 2: 75.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.25) (copysign x x) (copysign (log (+ (fabs x) x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log((fabs(x) + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log((Math.abs(x) + x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.25:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log((math.fabs(x) + x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.25)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float64(abs(x) + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.25)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((abs(x) + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], 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. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 29.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 99.9

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

   3. Applied rewrites 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\sinh^{-1} \left(\left|x\right|\right)}, x\right) \]
   4. 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. rem-sqrt-square-rev N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. sqrt-div N/A

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

      9. lower-unsound-/.f64 N/A

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

      10. lower-unsound-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-unsound-sqrt.f64 53.2

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

   5. Applied rewrites 53.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 52.9%

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

   if 1.25 < x

   1. Initial program 29.5%

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

   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 26.9

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

   4. Applied rewrites 26.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right)}, x\right) \]
   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 26.9

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

   6. Applied rewrites 26.9%

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

Alternative 3: 59.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.05:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) 0.05)
   (copysign x x)
   (copysign (log (fabs x)) x)))

C

double code(double x) {
	double tmp;
	if (copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x) <= 0.05) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log(fabs(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) <= 0.05) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log(Math.abs(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) <= 0.05:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log(math.fabs(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) <= 0.05)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(abs(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)))))) <= 0.05)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(abs(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], 0.05], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[Abs[x], $MachinePrecision]], $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) < 0.050000000000000003

   1. Initial program 29.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 99.9

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

   3. Applied rewrites 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\sinh^{-1} \left(\left|x\right|\right)}, x\right) \]
   4. 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. rem-sqrt-square-rev N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. sqrt-div N/A

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

      9. lower-unsound-/.f64 N/A

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

      10. lower-unsound-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-unsound-sqrt.f64 53.2

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

   5. Applied rewrites 53.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 52.9%

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

   if 0.050000000000000003 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 29.5%

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

   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 26.9

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

   4. Applied rewrites 26.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right)}, x\right) \]
   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 26.9

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

   6. Applied rewrites 26.9%

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

   7. Taylor expanded in x around 0

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

      1. lower-fabs.f64 18.2

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

   9. Applied rewrites 18.2%

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

Alternative 4: 52.9% accurate, 5.4× 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 29.5%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
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 99.9

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

3. Applied rewrites 99.9%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\sinh^{-1} \left(\left|x\right|\right)}, x\right) \]
4. 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. rem-sqrt-square-rev N/A

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

   3. *-rgt-identity N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. sqrt-div N/A

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

   9. lower-unsound-/.f64 N/A

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

   10. lower-unsound-sqrt.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-unsound-sqrt.f64 53.2

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

5. Applied rewrites 53.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 52.9%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]
9. 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 2025159 
(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))