Rust f64::asinh

Percentage Accurate: 29.7% → 81.8%

Time: 6.5s

Alternatives: 5

Speedup: 4.0×

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: 29.7% 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: 81.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\frac{{\log x}^{3} + {\log 2}^{3}}{\mathsf{fma}\left(\log x, \log x, \log 2 \cdot \left(\log 2 - \log x\right)\right)}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) 4e-6)
   (copysign (log1p (fabs x)) x)
   (copysign
    (/
     (+ (pow (log x) 3.0) (pow (log 2.0) 3.0))
     (fma (log x) (log x) (* (log 2.0) (- (log 2.0) (log x)))))
    x)))

C

double code(double x) {
	double tmp;
	if (copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x) <= 4e-6) {
		tmp = copysign(log1p(fabs(x)), x);
	} else {
		tmp = copysign(((pow(log(x), 3.0) + pow(log(2.0), 3.0)) / fma(log(x), log(x), (log(2.0) * (log(2.0) - 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) <= 4e-6)
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(Float64(Float64((log(x) ^ 3.0) + (log(2.0) ^ 3.0)) / fma(log(x), log(x), Float64(log(2.0) * Float64(log(2.0) - 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], 4e-6], N[With[{TMP1 = Abs[N[Log[1 + N[Abs[x], $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[(N[(N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision] + N[Power[N[Log[2.0], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Log[x], $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[Log[2.0], $MachinePrecision] * N[(N[Log[2.0], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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) < 3.99999999999999982e-6

   1. Initial program 23.1%

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

      1. +-commutative 23.1%

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

      2. hypot-1-def 35.8%

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

   3. Simplified 35.8%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 14.4%

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

      1. log1p-define 78.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   7. Simplified 78.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   if 3.99999999999999982e-6 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 53.1%

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

      1. +-commutative 53.1%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 97.6%

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

      1. rem-square-sqrt 97.6%

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

      2. fabs-sqr 97.6%

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

      3. rem-square-sqrt 97.6%

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

   7. Simplified 97.6%

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

      1. log-prod 98.8%

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

      2. flip3-+ 98.2%

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

      3. *-inverses 98.2%

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

      4. metadata-eval 98.2%

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

      5. fma-define 98.2%

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

      6. *-inverses 98.2%

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

      7. metadata-eval 98.2%

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

      8. *-inverses 98.2%

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

      9. metadata-eval 98.2%

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

      10. *-inverses 98.2%

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

      11. metadata-eval 98.2%

          \[\leadsto \mathsf{copysign}\left(\frac{{\log x}^{3} + {\log 2}^{3}}{\mathsf{fma}\left(\log x, \log x, \log 2 \cdot \log 2 - \log x \cdot \log \color{blue}{2}\right)}, x\right) \]

   9. Applied egg-rr 98.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{{\log x}^{3} + {\log 2}^{3}}{\mathsf{fma}\left(\log x, \log x, \log 2 \cdot \log 2 - \log x \cdot \log 2\right)}}, x\right) \]
   10. Step-by-step derivation

       1. distribute-rgt-out-- 98.2%

          \[\leadsto \mathsf{copysign}\left(\frac{{\log x}^{3} + {\log 2}^{3}}{\mathsf{fma}\left(\log x, \log x, \color{blue}{\log 2 \cdot \left(\log 2 - \log x\right)}\right)}, x\right) \]

   11. Simplified 98.2%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{{\log x}^{3} + {\log 2}^{3}}{\mathsf{fma}\left(\log x, \log x, \log 2 \cdot \left(\log 2 - \log x\right)\right)}}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\frac{{\log x}^{3} + {\log 2}^{3}}{\mathsf{fma}\left(\log x, \log x, \log 2 \cdot \left(\log 2 - \log x\right)\right)}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{-0.125}{x + 1} + \frac{-0.125}{{\left(x + 1\right)}^{2}}\right) + \frac{0.5}{x + 1}, \mathsf{log1p}\left(x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) -10.0)
   (copysign (log (/ -0.5 x)) x)
   (copysign
    (fma
     (pow x 2.0)
     (+
      (* (pow x 2.0) (+ (/ -0.125 (+ x 1.0)) (/ -0.125 (pow (+ x 1.0) 2.0))))
      (/ 0.5 (+ x 1.0)))
     (log1p x))
    x)))

C

double code(double x) {
	double tmp;
	if (copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x) <= -10.0) {
		tmp = copysign(log((-0.5 / x)), x);
	} else {
		tmp = copysign(fma(pow(x, 2.0), ((pow(x, 2.0) * ((-0.125 / (x + 1.0)) + (-0.125 / pow((x + 1.0), 2.0)))) + (0.5 / (x + 1.0))), log1p(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) <= -10.0)
		tmp = copysign(log(Float64(-0.5 / x)), x);
	else
		tmp = copysign(fma((x ^ 2.0), Float64(Float64((x ^ 2.0) * Float64(Float64(-0.125 / Float64(x + 1.0)) + Float64(-0.125 / (Float64(x + 1.0) ^ 2.0)))) + Float64(0.5 / Float64(x + 1.0))), log1p(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], -10.0], N[With[{TMP1 = Abs[N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(-0.125 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-0.125 / N[Power[N[(x + 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Log[1 + 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) < -10

   1. Initial program 58.3%

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

      1. +-commutative 58.3%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. log-prod 100.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. add-sqr-sqrt 3.7%

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

      7. metadata-eval 3.7%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   6. Applied egg-rr 3.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]
   7. Step-by-step derivation

      1. +-rgt-identity 3.7%

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

   8. Simplified 3.7%

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

   9. Taylor expanded in x around -inf 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)}, x\right) \]

   if -10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 22.5%

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

      1. +-commutative 22.5%

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

      2. hypot-1-def 37.2%

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

   3. Simplified 37.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 7.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + {x}^{2} \cdot \left(-0.041666666666666664 \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + 0.5 \cdot \frac{1}{1 + \left|x\right|}\right)}, x\right) \]
   6. Step-by-step derivation

      1. +-commutative 7.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{2} \cdot \left(-0.041666666666666664 \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + 0.5 \cdot \frac{1}{1 + \left|x\right|}\right) + \log \left(1 + \left|x\right|\right)}, x\right) \]

      2. fma-define 7.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{2}, -0.041666666666666664 \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + 0.5 \cdot \frac{1}{1 + \left|x\right|}, \log \left(1 + \left|x\right|\right)\right)}, x\right) \]

   7. Simplified 68.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{-0.125}{1 + x} + \frac{-0.125}{{\left(1 + x\right)}^{2}}\right) + \frac{0.5}{1 + x}, \mathsf{log1p}\left(x\right)\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{-0.125}{x + 1} + \frac{-0.125}{{\left(x + 1\right)}^{2}}\right) + \frac{0.5}{x + 1}, \mathsf{log1p}\left(x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (copysign (log1p (fabs x)) x))

C

double code(double x) {
	return copysign(log1p(fabs(x)), x);
}

Java

public static double code(double x) {
	return Math.copySign(Math.log1p(Math.abs(x)), x);
}

Python

def code(x):
	return math.copysign(math.log1p(math.fabs(x)), x)

Julia

function code(x)
	return copysign(log1p(abs(x)), x)
end

Wolfram

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

Derivation

1. Initial program 30.6%

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

   1. +-commutative 30.6%

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

   2. hypot-1-def 51.4%

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

3. Simplified 51.4%

   \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 18.6%

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

   1. log1p-define 66.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

7. Simplified 66.4%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

8. Final simplification 66.4%

   \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right) \]
9. Add Preprocessing

Alternative 4: 57.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (copysign (log1p x) x))

C

double code(double x) {
	return copysign(log1p(x), x);
}

Java

public static double code(double x) {
	return Math.copySign(Math.log1p(x), x);
}

Python

def code(x):
	return math.copysign(math.log1p(x), x)

Julia

function code(x)
	return copysign(log1p(x), x)
end

Wolfram

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

Derivation

1. Initial program 30.6%

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

   1. +-commutative 30.6%

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

   2. hypot-1-def 51.4%

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

3. Simplified 51.4%

   \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 18.6%

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

   1. log1p-define 66.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   2. rem-square-sqrt 33.9%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right), x\right) \]

   3. fabs-sqr 33.9%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right), x\right) \]

   4. rem-square-sqrt 59.2%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x}\right), x\right) \]

7. Simplified 59.2%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right)}, x\right) \]

8. Final simplification 59.2%

   \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right) \]
9. Add Preprocessing

Alternative 5: 53.1% accurate, 4.0× 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 30.6%

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

   1. +-commutative 30.6%

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

   2. hypot-1-def 51.4%

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

3. Simplified 51.4%

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

   1. *-un-lft-identity 51.4%

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

   2. *-commutative 51.4%

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

   3. log-prod 51.4%

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

   4. add-sqr-sqrt 26.5%

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

   5. fabs-sqr 26.5%

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

   6. add-sqr-sqrt 29.6%

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

   7. metadata-eval 29.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

6. Applied egg-rr 29.6%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]
7. Step-by-step derivation

   1. +-rgt-identity 29.6%

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

8. Simplified 29.6%

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

9. Taylor expanded in x around 0 54.4%

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

10. Final simplification 54.4%

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

Developer target: 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 2024066 
(FPCore (x)
  :name "Rust f64::asinh"
  :precision binary64

  :alt
  (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)

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