Rust f64::asinh

Percentage Accurate: 29.8% → 99.8%

Time: 5.5s

Alternatives: 3

Speedup: 1.9×

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: 29.8% 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

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

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 29.5%

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

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

   3. sqrt-prod N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 47.0

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

6. Applied rewrites 47.0%

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

7. Taylor expanded in x around -inf

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

   1. associate-*r* N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. metadata-eval N/A

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

   7. *-rgt-identity 99.8

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

9. Applied rewrites 99.8%

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

Alternative 2: 58.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.05:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot \left(x \cdot x\right), x, x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) 0.05)
   (copysign (fma (* -0.16666666666666666 (* x x)) x x) x)
   (copysign (log x) x)))

C

double code(double x) {
	double tmp;
	if (copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x) <= 0.05) {
		tmp = copysign(fma((-0.16666666666666666 * (x * x)), x, x), x);
	} else {
		tmp = copysign(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) <= 0.05)
		tmp = copysign(fma(Float64(-0.16666666666666666 * Float64(x * x)), x, x), x);
	else
		tmp = copysign(log(x), x);
	end
	return 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[N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]], 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) < 0.050000000000000003

   1. Initial program 22.5%

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 31.6

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

   6. Applied rewrites 31.6%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign}\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}, x\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x + 1 \cdot x}, x\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{copysign}\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \color{blue}{x}, x\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)} + x, x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{copysign}\left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} + x, x\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(x \cdot {x}^{2}\right) \cdot \frac{-1}{6}} + x, x\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{copysign}\left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \frac{-1}{6} + x, x\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{3}} \cdot \frac{-1}{6} + x, x\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{3}, \frac{-1}{6}, x\right)}, x\right) \]

      10. lower-pow.f64 65.2

          \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(\color{blue}{{x}^{3}}, -0.16666666666666666, x\right), x\right) \]

   9. Applied rewrites 65.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)}, x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 65.2%

       \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot \left(x \cdot x\right), \color{blue}{x}, x\right), x\right) \]

   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 53.2%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

      4. lower-log.f64 31.5

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

   5. Applied rewrites 31.5%

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

Alternative 3: 51.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot \left(x \cdot x\right), x, x\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (copysign (fma (* -0.16666666666666666 (* x x)) x x) x))

C

double code(double x) {
	return copysign(fma((-0.16666666666666666 * (x * x)), x, x), x);
}

Julia

function code(x)
	return copysign(fma(Float64(-0.16666666666666666 * Float64(x * x)), x, x), x)
end

Wolfram

code[x_] := N[With[{TMP1 = Abs[N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + 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. 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. rem-sqrt-square-rev N/A

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

   3. sqrt-prod N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 47.0

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

6. Applied rewrites 47.0%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \mathsf{copysign}\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}, x\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x + 1 \cdot x}, x\right) \]

   3. *-lft-identity N/A

      \[\leadsto \mathsf{copysign}\left(\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot x + \color{blue}{x}, x\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right)} + x, x\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{copysign}\left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} + x, x\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(x \cdot {x}^{2}\right) \cdot \frac{-1}{6}} + x, x\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{copysign}\left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \frac{-1}{6} + x, x\right) \]

   8. cube-mult N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{3}} \cdot \frac{-1}{6} + x, x\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{3}, \frac{-1}{6}, x\right)}, x\right) \]

   10. lower-pow.f64 51.3

       \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(\color{blue}{{x}^{3}}, -0.16666666666666666, x\right), x\right) \]

9. Applied rewrites 51.3%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)}, x\right) \]
10. Step-by-step derivation

11. Applied rewrites 51.3%

    \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot \left(x \cdot x\right), \color{blue}{x}, x\right), x\right) \]
12. Add Preprocessing

Developer Target 1: 100.0% 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 2024360 
(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))