Rust f64::asinh

Percentage Accurate: 29.8% → 99.2%

Time: 5.1s

Alternatives: 7

Speedup: 2.0×

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: 29.8% 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: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-11}:\\ \;\;\;\;\mathsf{copysign}\left(\left(x \cdot x\right) \cdot 0.5 + \mathsf{log1p}\left(x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -5.0)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 1e-11)
       (copysign (+ (* (* x x) 0.5) (log1p x)) x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = copysign(-log((hypot(1.0, x) - x)), x);
	} else if (t_0 <= 1e-11) {
		tmp = copysign((((x * x) * 0.5) + log1p(x)), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = Math.copySign(-Math.log((Math.hypot(1.0, x) - x)), x);
	} else if (t_0 <= 1e-11) {
		tmp = Math.copySign((((x * x) * 0.5) + Math.log1p(x)), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -5.0:
		tmp = math.copysign(-math.log((math.hypot(1.0, x) - x)), x)
	elif t_0 <= 1e-11:
		tmp = math.copysign((((x * x) * 0.5) + math.log1p(x)), x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = copysign(Float64(-log(Float64(hypot(1.0, x) - x))), x);
	elseif (t_0 <= 1e-11)
		tmp = copysign(Float64(Float64(Float64(x * x) * 0.5) + log1p(x)), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -5.0], N[With[{TMP1 = Abs[(-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 1e-11], N[With[{TMP1 = Abs[N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] + N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -5

   1. Initial program 48.3%

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

      1. +-commutative 48.3%

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

      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. flip-+ 2.5%

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

      2. frac-2neg 2.5%

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

      3. log-div 2.5%

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

      4. pow2 2.5%

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

      5. add-sqr-sqrt 0.0%

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

      6. fabs-sqr 0.0%

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

      7. add-sqr-sqrt 2.5%

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

      8. pow2 2.5%

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

      9. hypot-1-def 2.5%

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

      10. hypot-1-def 2.5%

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

      11. add-sqr-sqrt 2.5%

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

   6. Applied egg-rr 4.7%

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

      1. neg-mul-1 4.7%

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

      2. unpow2 4.7%

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

      3. unpow2 4.7%

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

      4. +-commutative 4.7%

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

      5. associate--r+ 46.7%

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

      6. +-inverses 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-sub0 100.0%

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

      11. associate--r- 100.0%

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

      12. neg-sub0 100.0%

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

      13. +-commutative 100.0%

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

      14. sub-neg 100.0%

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

   8. Simplified 100.0%

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

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 9.99999999999999939e-12

   1. Initial program 5.8%

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

      1. +-commutative 5.8%

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

      2. hypot-1-def 5.8%

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

   3. Simplified 5.8%

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

   5. Taylor expanded in x around 0 6.7%

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

      1. log1p-define 100.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)} + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}, x\right) \]

      2. *-commutative 100.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{{x}^{2}}{1 + \left|x\right|} \cdot 0.5}, x\right) \]

      3. associate-*l/ 100.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{{x}^{2} \cdot 0.5}{1 + \left|x\right|}}, x\right) \]

      4. associate-*r/ 100.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{{x}^{2} \cdot \frac{0.5}{1 + \left|x\right|}}, x\right) \]

      5. unpow2 100.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\left(x \cdot x\right)} \cdot \frac{0.5}{1 + \left|x\right|}, x\right) \]

   7. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right) + \left(x \cdot x\right) \cdot \frac{0.5}{1 + \left|x\right|}}, x\right) \]
   8. Step-by-step derivation

      1. frac-2neg 100.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \left(x \cdot x\right) \cdot \color{blue}{\frac{-0.5}{-\left(1 + \left|x\right|\right)}}, x\right) \]

      2. associate-*r/ 100.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(-0.5\right)}{-\left(1 + \left|x\right|\right)}}, x\right) \]

      3. metadata-eval 100.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot \color{blue}{-0.5}}{-\left(1 + \left|x\right|\right)}, x\right) \]

      4. +-commutative 100.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\color{blue}{\left(\left|x\right| + 1\right)}}, x\right) \]

      5. add-sqr-sqrt 42.4%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1\right)}, x\right) \]

      6. fabs-sqr 42.4%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1\right)}, x\right) \]

      7. add-sqr-sqrt 100.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\left(\color{blue}{x} + 1\right)}, x\right) \]

      8. distribute-neg-in 100.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\color{blue}{\left(-x\right) + \left(-1\right)}}, x\right) \]

      9. *-un-lft-identity 100.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\color{blue}{1 \cdot \left(-x\right)} + \left(-1\right)}, x\right) \]

      10. metadata-eval 100.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{1 \cdot \left(-x\right) + \color{blue}{-1}}, x\right) \]

      11. fma-define 100.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\color{blue}{\mathsf{fma}\left(1, -x, -1\right)}}, x\right) \]

      12. add-sqr-sqrt 57.6%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}, -1\right)}, x\right) \]

      13. pow1/2 57.6%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{{\left(-x\right)}^{0.5}} \cdot \sqrt{-x}, -1\right)}, x\right) \]

      14. pow1/2 57.6%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\left(-x\right)}^{0.5} \cdot \color{blue}{{\left(-x\right)}^{0.5}}, -1\right)}, x\right) \]

      15. unpow-prod-down 100.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{{\left(\left(-x\right) \cdot \left(-x\right)\right)}^{0.5}}, -1\right)}, x\right) \]

      16. sqr-abs 100.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\color{blue}{\left(\left|-x\right| \cdot \left|-x\right|\right)}}^{0.5}, -1\right)}, x\right) \]

      17. neg-fabs 100.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\left(\color{blue}{\left|x\right|} \cdot \left|-x\right|\right)}^{0.5}, -1\right)}, x\right) \]

      18. neg-fabs 100.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\left(\left|x\right| \cdot \color{blue}{\left|x\right|}\right)}^{0.5}, -1\right)}, x\right) \]

      19. sqr-abs 100.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\color{blue}{\left(x \cdot x\right)}}^{0.5}, -1\right)}, x\right) \]

      20. pow1/2 100.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{\sqrt{x \cdot x}}, -1\right)}, x\right) \]

      21. sqrt-prod 42.4%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, -1\right)}, x\right) \]

      22. add-sqr-sqrt 100.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{x}, -1\right)}, x\right) \]

      23. metadata-eval 100.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, x, \color{blue}{-1}\right)}, x\right) \]

   9. Applied egg-rr 100.0%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{\left(x \cdot x\right) \cdot -0.5}{x - 1}}, x\right) \]

   10. Taylor expanded in x around 0 6.7%

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

       1. +-commutative 6.7%

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

       2. unpow2 6.7%

          \[\leadsto \mathsf{copysign}\left(0.5 \cdot \color{blue}{\left(x \cdot x\right)} + \log \left(1 + \left|x\right|\right), x\right) \]

       3. log1p-define 100.0%

          \[\leadsto \mathsf{copysign}\left(0.5 \cdot \left(x \cdot x\right) + \color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

       4. rem-square-sqrt 42.1%

          \[\leadsto \mathsf{copysign}\left(0.5 \cdot \left(x \cdot x\right) + \mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right), x\right) \]

       5. unpow1/2 42.1%

          \[\leadsto \mathsf{copysign}\left(0.5 \cdot \left(x \cdot x\right) + \mathsf{log1p}\left(\left|\color{blue}{{x}^{0.5}} \cdot \sqrt{x}\right|\right), x\right) \]

       6. unpow1/2 42.1%

          \[\leadsto \mathsf{copysign}\left(0.5 \cdot \left(x \cdot x\right) + \mathsf{log1p}\left(\left|{x}^{0.5} \cdot \color{blue}{{x}^{0.5}}\right|\right), x\right) \]

       7. fabs-sqr 42.1%

          \[\leadsto \mathsf{copysign}\left(0.5 \cdot \left(x \cdot x\right) + \mathsf{log1p}\left(\color{blue}{{x}^{0.5} \cdot {x}^{0.5}}\right), x\right) \]

       8. unpow1/2 42.1%

          \[\leadsto \mathsf{copysign}\left(0.5 \cdot \left(x \cdot x\right) + \mathsf{log1p}\left(\color{blue}{\sqrt{x}} \cdot {x}^{0.5}\right), x\right) \]

       9. unpow1/2 42.1%

          \[\leadsto \mathsf{copysign}\left(0.5 \cdot \left(x \cdot x\right) + \mathsf{log1p}\left(\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right), x\right) \]

       10. rem-square-sqrt 100.0%

           \[\leadsto \mathsf{copysign}\left(0.5 \cdot \left(x \cdot x\right) + \mathsf{log1p}\left(\color{blue}{x}\right), x\right) \]

   12. Simplified 100.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{0.5 \cdot \left(x \cdot x\right) + \mathsf{log1p}\left(x\right)}, x\right) \]

   if 9.99999999999999939e-12 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 53.6%

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

      1. +-commutative 53.6%

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

      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. add-sqr-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 10^{-11}:\\ \;\;\;\;\mathsf{copysign}\left(\left(x \cdot x\right) \cdot 0.5 + \mathsf{log1p}\left(x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.7e-8)
   (copysign (log1p (fabs x)) x)
   (copysign (log (+ x (hypot 1.0 x))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.7e-8) {
		tmp = copysign(log1p(fabs(x)), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.7e-8) {
		tmp = Math.copySign(Math.log1p(Math.abs(x)), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.7e-8:
		tmp = math.copysign(math.log1p(math.fabs(x)), x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.7e-8)
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.7e-8], 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[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7e-8

   1. Initial program 19.6%

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

      1. +-commutative 19.6%

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

      2. hypot-1-def 36.3%

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

   3. Simplified 36.3%

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

   5. Taylor expanded in x around 0 14.1%

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

      1. log1p-define 77.6%

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

   7. Simplified 77.6%

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

   if 1.7e-8 < x

   1. Initial program 53.6%

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

      1. +-commutative 53.6%

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

      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. add-sqr-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 64.7% 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 29.0%

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

   1. +-commutative 29.0%

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

   2. hypot-1-def 54.0%

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

3. Simplified 54.0%

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

5. Taylor expanded in x around 0 18.8%

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

   1. log1p-define 64.7%

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

7. Simplified 64.7%

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

Alternative 4: 64.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.5) (copysign (- (log (/ -1.0 x))) x) (copysign (log1p x) x)))

C

double code(double x) {
	double tmp;
	if (x <= -0.5) {
		tmp = copysign(-log((-1.0 / x)), x);
	} else {
		tmp = copysign(log1p(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.5) {
		tmp = Math.copySign(-Math.log((-1.0 / x)), x);
	} else {
		tmp = Math.copySign(Math.log1p(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.5:
		tmp = math.copysign(-math.log((-1.0 / x)), x)
	else:
		tmp = math.copysign(math.log1p(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.5)
		tmp = copysign(Float64(-log(Float64(-1.0 / x))), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.5], N[With[{TMP1 = Abs[(-N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.5

   1. Initial program 48.3%

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

      1. +-commutative 48.3%

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

      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. Taylor expanded in x around -inf 31.3%

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

      1. mul-1-neg 31.3%

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

      2. neg-sub0 31.3%

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

   7. Simplified 31.3%

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

   if -0.5 < x

   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 39.9%

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

   3. Simplified 39.9%

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

   5. Taylor expanded in x around 0 6.3%

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

      1. log1p-define 65.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)} + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}, x\right) \]

      2. *-commutative 65.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{{x}^{2}}{1 + \left|x\right|} \cdot 0.5}, x\right) \]

      3. associate-*l/ 65.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{{x}^{2} \cdot 0.5}{1 + \left|x\right|}}, x\right) \]

      4. associate-*r/ 65.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{{x}^{2} \cdot \frac{0.5}{1 + \left|x\right|}}, x\right) \]

      5. unpow2 65.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\left(x \cdot x\right)} \cdot \frac{0.5}{1 + \left|x\right|}, x\right) \]

   7. Simplified 65.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right) + \left(x \cdot x\right) \cdot \frac{0.5}{1 + \left|x\right|}}, x\right) \]
   8. Step-by-step derivation

      1. frac-2neg 65.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \left(x \cdot x\right) \cdot \color{blue}{\frac{-0.5}{-\left(1 + \left|x\right|\right)}}, x\right) \]

      2. associate-*r/ 65.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(-0.5\right)}{-\left(1 + \left|x\right|\right)}}, x\right) \]

      3. metadata-eval 65.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot \color{blue}{-0.5}}{-\left(1 + \left|x\right|\right)}, x\right) \]

      4. +-commutative 65.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\color{blue}{\left(\left|x\right| + 1\right)}}, x\right) \]

      5. add-sqr-sqrt 29.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1\right)}, x\right) \]

      6. fabs-sqr 29.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1\right)}, x\right) \]

      7. add-sqr-sqrt 65.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\left(\color{blue}{x} + 1\right)}, x\right) \]

      8. distribute-neg-in 65.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\color{blue}{\left(-x\right) + \left(-1\right)}}, x\right) \]

      9. *-un-lft-identity 65.8%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\color{blue}{1 \cdot \left(-x\right)} + \left(-1\right)}, x\right) \]

      10. metadata-eval 65.8%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{1 \cdot \left(-x\right) + \color{blue}{-1}}, x\right) \]

      11. fma-define 65.8%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\color{blue}{\mathsf{fma}\left(1, -x, -1\right)}}, x\right) \]

      12. add-sqr-sqrt 36.7%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}, -1\right)}, x\right) \]

      13. pow1/2 36.7%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{{\left(-x\right)}^{0.5}} \cdot \sqrt{-x}, -1\right)}, x\right) \]

      14. pow1/2 36.7%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\left(-x\right)}^{0.5} \cdot \color{blue}{{\left(-x\right)}^{0.5}}, -1\right)}, x\right) \]

      15. unpow-prod-down 65.2%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{{\left(\left(-x\right) \cdot \left(-x\right)\right)}^{0.5}}, -1\right)}, x\right) \]

      16. sqr-abs 65.2%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\color{blue}{\left(\left|-x\right| \cdot \left|-x\right|\right)}}^{0.5}, -1\right)}, x\right) \]

      17. neg-fabs 65.2%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\left(\color{blue}{\left|x\right|} \cdot \left|-x\right|\right)}^{0.5}, -1\right)}, x\right) \]

      18. neg-fabs 65.2%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\left(\left|x\right| \cdot \color{blue}{\left|x\right|}\right)}^{0.5}, -1\right)}, x\right) \]

      19. sqr-abs 65.2%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\color{blue}{\left(x \cdot x\right)}}^{0.5}, -1\right)}, x\right) \]

      20. pow1/2 65.2%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{\sqrt{x \cdot x}}, -1\right)}, x\right) \]

      21. sqrt-prod 29.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, -1\right)}, x\right) \]

      22. add-sqr-sqrt 65.8%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{x}, -1\right)}, x\right) \]

      23. metadata-eval 65.8%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, x, \color{blue}{-1}\right)}, x\right) \]

   9. Applied egg-rr 65.8%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{\left(x \cdot x\right) \cdot -0.5}{x - 1}}, x\right) \]

   10. Taylor expanded in x around 0 14.9%

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

       1. log1p-define 74.9%

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

       2. rem-square-sqrt 38.0%

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

       3. unpow1/2 38.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{{x}^{0.5}} \cdot \sqrt{x}\right|\right), x\right) \]

       4. unpow1/2 38.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|{x}^{0.5} \cdot \color{blue}{{x}^{0.5}}\right|\right), x\right) \]

       5. fabs-sqr 38.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{{x}^{0.5} \cdot {x}^{0.5}}\right), x\right) \]

       6. unpow1/2 38.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x}} \cdot {x}^{0.5}\right), x\right) \]

       7. unpow1/2 38.0%

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

       8. rem-square-sqrt 74.9%

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

   12. Simplified 74.9%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 18.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.8:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.8) (copysign (* x -0.5) x) (copysign (log x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 6.8) {
		tmp = copysign((x * -0.5), x);
	} else {
		tmp = copysign(log(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 6.8) {
		tmp = Math.copySign((x * -0.5), x);
	} else {
		tmp = Math.copySign(Math.log(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.8:
		tmp = math.copysign((x * -0.5), x)
	else:
		tmp = math.copysign(math.log(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.8)
		tmp = copysign(Float64(x * -0.5), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 6.8)
		tmp = sign(x) * abs((x * -0.5));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 6.8], N[With[{TMP1 = Abs[N[(x * -0.5), $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 x < 6.79999999999999982

   1. Initial program 20.0%

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

      1. +-commutative 20.0%

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

      2. hypot-1-def 36.7%

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

   3. Simplified 36.7%

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

   5. Taylor expanded in x around 0 6.3%

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

      1. log1p-define 69.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)} + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}, x\right) \]

      2. *-commutative 69.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{{x}^{2}}{1 + \left|x\right|} \cdot 0.5}, x\right) \]

      3. associate-*l/ 69.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{{x}^{2} \cdot 0.5}{1 + \left|x\right|}}, x\right) \]

      4. associate-*r/ 69.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{{x}^{2} \cdot \frac{0.5}{1 + \left|x\right|}}, x\right) \]

      5. unpow2 69.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\left(x \cdot x\right)} \cdot \frac{0.5}{1 + \left|x\right|}, x\right) \]

   7. Simplified 69.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right) + \left(x \cdot x\right) \cdot \frac{0.5}{1 + \left|x\right|}}, x\right) \]
   8. Step-by-step derivation

      1. frac-2neg 69.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \left(x \cdot x\right) \cdot \color{blue}{\frac{-0.5}{-\left(1 + \left|x\right|\right)}}, x\right) \]

      2. associate-*r/ 69.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(-0.5\right)}{-\left(1 + \left|x\right|\right)}}, x\right) \]

      3. metadata-eval 69.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot \color{blue}{-0.5}}{-\left(1 + \left|x\right|\right)}, x\right) \]

      4. +-commutative 69.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\color{blue}{\left(\left|x\right| + 1\right)}}, x\right) \]

      5. add-sqr-sqrt 28.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1\right)}, x\right) \]

      6. fabs-sqr 28.6%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1\right)}, x\right) \]

      7. add-sqr-sqrt 69.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\left(\color{blue}{x} + 1\right)}, x\right) \]

      8. distribute-neg-in 69.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\color{blue}{\left(-x\right) + \left(-1\right)}}, x\right) \]

      9. *-un-lft-identity 69.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\color{blue}{1 \cdot \left(-x\right)} + \left(-1\right)}, x\right) \]

      10. metadata-eval 69.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{1 \cdot \left(-x\right) + \color{blue}{-1}}, x\right) \]

      11. fma-define 69.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\color{blue}{\mathsf{fma}\left(1, -x, -1\right)}}, x\right) \]

      12. add-sqr-sqrt 40.4%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}, -1\right)}, x\right) \]

      13. pow1/2 40.4%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{{\left(-x\right)}^{0.5}} \cdot \sqrt{-x}, -1\right)}, x\right) \]

      14. pow1/2 40.4%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\left(-x\right)}^{0.5} \cdot \color{blue}{{\left(-x\right)}^{0.5}}, -1\right)}, x\right) \]

      15. unpow-prod-down 68.4%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{{\left(\left(-x\right) \cdot \left(-x\right)\right)}^{0.5}}, -1\right)}, x\right) \]

      16. sqr-abs 68.4%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\color{blue}{\left(\left|-x\right| \cdot \left|-x\right|\right)}}^{0.5}, -1\right)}, x\right) \]

      17. neg-fabs 68.4%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\left(\color{blue}{\left|x\right|} \cdot \left|-x\right|\right)}^{0.5}, -1\right)}, x\right) \]

      18. neg-fabs 68.4%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\left(\left|x\right| \cdot \color{blue}{\left|x\right|}\right)}^{0.5}, -1\right)}, x\right) \]

      19. sqr-abs 68.4%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\color{blue}{\left(x \cdot x\right)}}^{0.5}, -1\right)}, x\right) \]

      20. pow1/2 68.4%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{\sqrt{x \cdot x}}, -1\right)}, x\right) \]

      21. sqrt-prod 28.6%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, -1\right)}, x\right) \]

      22. add-sqr-sqrt 69.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{x}, -1\right)}, x\right) \]

      23. metadata-eval 69.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, x, \color{blue}{-1}\right)}, x\right) \]

   9. Applied egg-rr 69.0%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{\left(x \cdot x\right) \cdot -0.5}{x - 1}}, x\right) \]

   10. Taylor expanded in x around inf 14.6%

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

   if 6.79999999999999982 < x

   1. Initial program 53.0%

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

      1. +-commutative 53.0%

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

      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. Taylor expanded in x around inf 31.2%

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

      1. mul-1-neg 31.2%

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

      2. log-rec 31.2%

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

      3. remove-double-neg 31.2%

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

   7. Simplified 31.2%

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

4. Final simplification 19.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.8% 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 29.0%

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

   1. +-commutative 29.0%

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

   2. hypot-1-def 54.0%

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

3. Simplified 54.0%

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

5. Taylor expanded in x around 0 6.0%

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

   1. log1p-define 51.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)} + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}, x\right) \]

   2. *-commutative 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{{x}^{2}}{1 + \left|x\right|} \cdot 0.5}, x\right) \]

   3. associate-*l/ 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{{x}^{2} \cdot 0.5}{1 + \left|x\right|}}, x\right) \]

   4. associate-*r/ 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{{x}^{2} \cdot \frac{0.5}{1 + \left|x\right|}}, x\right) \]

   5. unpow2 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\left(x \cdot x\right)} \cdot \frac{0.5}{1 + \left|x\right|}, x\right) \]

7. Simplified 51.6%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right) + \left(x \cdot x\right) \cdot \frac{0.5}{1 + \left|x\right|}}, x\right) \]
8. Step-by-step derivation

   1. frac-2neg 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \left(x \cdot x\right) \cdot \color{blue}{\frac{-0.5}{-\left(1 + \left|x\right|\right)}}, x\right) \]

   2. associate-*r/ 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(-0.5\right)}{-\left(1 + \left|x\right|\right)}}, x\right) \]

   3. metadata-eval 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot \color{blue}{-0.5}}{-\left(1 + \left|x\right|\right)}, x\right) \]

   4. +-commutative 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\color{blue}{\left(\left|x\right| + 1\right)}}, x\right) \]

   5. add-sqr-sqrt 22.2%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1\right)}, x\right) \]

   6. fabs-sqr 22.2%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1\right)}, x\right) \]

   7. add-sqr-sqrt 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\left(\color{blue}{x} + 1\right)}, x\right) \]

   8. distribute-neg-in 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\color{blue}{\left(-x\right) + \left(-1\right)}}, x\right) \]

   9. *-un-lft-identity 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\color{blue}{1 \cdot \left(-x\right)} + \left(-1\right)}, x\right) \]

   10. metadata-eval 51.6%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{1 \cdot \left(-x\right) + \color{blue}{-1}}, x\right) \]

   11. fma-define 51.6%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\color{blue}{\mathsf{fma}\left(1, -x, -1\right)}}, x\right) \]

   12. add-sqr-sqrt 29.3%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}, -1\right)}, x\right) \]

   13. pow1/2 29.3%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{{\left(-x\right)}^{0.5}} \cdot \sqrt{-x}, -1\right)}, x\right) \]

   14. pow1/2 29.3%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\left(-x\right)}^{0.5} \cdot \color{blue}{{\left(-x\right)}^{0.5}}, -1\right)}, x\right) \]

   15. unpow-prod-down 50.8%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{{\left(\left(-x\right) \cdot \left(-x\right)\right)}^{0.5}}, -1\right)}, x\right) \]

   16. sqr-abs 50.8%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\color{blue}{\left(\left|-x\right| \cdot \left|-x\right|\right)}}^{0.5}, -1\right)}, x\right) \]

   17. neg-fabs 50.8%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\left(\color{blue}{\left|x\right|} \cdot \left|-x\right|\right)}^{0.5}, -1\right)}, x\right) \]

   18. neg-fabs 50.8%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\left(\left|x\right| \cdot \color{blue}{\left|x\right|}\right)}^{0.5}, -1\right)}, x\right) \]

   19. sqr-abs 50.8%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\color{blue}{\left(x \cdot x\right)}}^{0.5}, -1\right)}, x\right) \]

   20. pow1/2 50.8%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{\sqrt{x \cdot x}}, -1\right)}, x\right) \]

   21. sqrt-prod 22.2%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, -1\right)}, x\right) \]

   22. add-sqr-sqrt 51.6%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{x}, -1\right)}, x\right) \]

   23. metadata-eval 51.6%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, x, \color{blue}{-1}\right)}, x\right) \]

9. Applied egg-rr 51.6%

   \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{\left(x \cdot x\right) \cdot -0.5}{x - 1}}, x\right) \]

10. Taylor expanded in x around 0 18.8%

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

    1. log1p-define 64.7%

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

    2. rem-square-sqrt 29.1%

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

    3. unpow1/2 29.1%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{{x}^{0.5}} \cdot \sqrt{x}\right|\right), x\right) \]

    4. unpow1/2 29.1%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|{x}^{0.5} \cdot \color{blue}{{x}^{0.5}}\right|\right), x\right) \]

    5. fabs-sqr 29.1%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{{x}^{0.5} \cdot {x}^{0.5}}\right), x\right) \]

    6. unpow1/2 29.1%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x}} \cdot {x}^{0.5}\right), x\right) \]

    7. unpow1/2 29.1%

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

    8. rem-square-sqrt 57.4%

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

12. Simplified 57.4%

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

Alternative 7: 12.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(x \cdot -0.5, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (copysign (* x -0.5) x))

C

double code(double x) {
	return copysign((x * -0.5), x);
}

Java

public static double code(double x) {
	return Math.copySign((x * -0.5), x);
}

Python

def code(x):
	return math.copysign((x * -0.5), x)

Julia

function code(x)
	return copysign(Float64(x * -0.5), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs((x * -0.5));
end

Wolfram

code[x_] := N[With[{TMP1 = Abs[N[(x * -0.5), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

Derivation

1. Initial program 29.0%

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

   1. +-commutative 29.0%

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

   2. hypot-1-def 54.0%

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

3. Simplified 54.0%

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

5. Taylor expanded in x around 0 6.0%

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

   1. log1p-define 51.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)} + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}, x\right) \]

   2. *-commutative 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{{x}^{2}}{1 + \left|x\right|} \cdot 0.5}, x\right) \]

   3. associate-*l/ 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{{x}^{2} \cdot 0.5}{1 + \left|x\right|}}, x\right) \]

   4. associate-*r/ 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{{x}^{2} \cdot \frac{0.5}{1 + \left|x\right|}}, x\right) \]

   5. unpow2 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\left(x \cdot x\right)} \cdot \frac{0.5}{1 + \left|x\right|}, x\right) \]

7. Simplified 51.6%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right) + \left(x \cdot x\right) \cdot \frac{0.5}{1 + \left|x\right|}}, x\right) \]
8. Step-by-step derivation

   1. frac-2neg 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \left(x \cdot x\right) \cdot \color{blue}{\frac{-0.5}{-\left(1 + \left|x\right|\right)}}, x\right) \]

   2. associate-*r/ 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(-0.5\right)}{-\left(1 + \left|x\right|\right)}}, x\right) \]

   3. metadata-eval 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot \color{blue}{-0.5}}{-\left(1 + \left|x\right|\right)}, x\right) \]

   4. +-commutative 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\color{blue}{\left(\left|x\right| + 1\right)}}, x\right) \]

   5. add-sqr-sqrt 22.2%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1\right)}, x\right) \]

   6. fabs-sqr 22.2%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1\right)}, x\right) \]

   7. add-sqr-sqrt 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{-\left(\color{blue}{x} + 1\right)}, x\right) \]

   8. distribute-neg-in 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\color{blue}{\left(-x\right) + \left(-1\right)}}, x\right) \]

   9. *-un-lft-identity 51.6%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\color{blue}{1 \cdot \left(-x\right)} + \left(-1\right)}, x\right) \]

   10. metadata-eval 51.6%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{1 \cdot \left(-x\right) + \color{blue}{-1}}, x\right) \]

   11. fma-define 51.6%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\color{blue}{\mathsf{fma}\left(1, -x, -1\right)}}, x\right) \]

   12. add-sqr-sqrt 29.3%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}, -1\right)}, x\right) \]

   13. pow1/2 29.3%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{{\left(-x\right)}^{0.5}} \cdot \sqrt{-x}, -1\right)}, x\right) \]

   14. pow1/2 29.3%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\left(-x\right)}^{0.5} \cdot \color{blue}{{\left(-x\right)}^{0.5}}, -1\right)}, x\right) \]

   15. unpow-prod-down 50.8%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{{\left(\left(-x\right) \cdot \left(-x\right)\right)}^{0.5}}, -1\right)}, x\right) \]

   16. sqr-abs 50.8%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\color{blue}{\left(\left|-x\right| \cdot \left|-x\right|\right)}}^{0.5}, -1\right)}, x\right) \]

   17. neg-fabs 50.8%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\left(\color{blue}{\left|x\right|} \cdot \left|-x\right|\right)}^{0.5}, -1\right)}, x\right) \]

   18. neg-fabs 50.8%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\left(\left|x\right| \cdot \color{blue}{\left|x\right|}\right)}^{0.5}, -1\right)}, x\right) \]

   19. sqr-abs 50.8%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, {\color{blue}{\left(x \cdot x\right)}}^{0.5}, -1\right)}, x\right) \]

   20. pow1/2 50.8%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{\sqrt{x \cdot x}}, -1\right)}, x\right) \]

   21. sqrt-prod 22.2%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{\sqrt{x} \cdot \sqrt{x}}, -1\right)}, x\right) \]

   22. add-sqr-sqrt 51.6%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, \color{blue}{x}, -1\right)}, x\right) \]

   23. metadata-eval 51.6%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{\mathsf{fma}\left(1, x, \color{blue}{-1}\right)}, x\right) \]

9. Applied egg-rr 51.6%

   \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \color{blue}{\frac{\left(x \cdot x\right) \cdot -0.5}{x - 1}}, x\right) \]

10. Taylor expanded in x around inf 12.1%

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

11. Final simplification 12.1%

    \[\leadsto \mathsf{copysign}\left(x \cdot -0.5, x\right) \]
12. 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 2024107 
(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))