Rust f64::asinh

Percentage Accurate: 29.8% → 98.5%

Time: 3.3s

Alternatives: 6

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: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t_0 \leq 10^{-11}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x + x\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

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 (+ (- x x) (/ -0.5 x))) x)
     (if (<= t_0 1e-11)
       (copysign (fma 0.5 (/ (pow x 2.0) (+ x 1.0)) (log1p x)) x)
       (copysign (log (+ (+ x x) (/ 0.5 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(((x - x) + (-0.5 / x))), x);
	} else if (t_0 <= 1e-11) {
		tmp = copysign(fma(0.5, (pow(x, 2.0) / (x + 1.0)), log1p(x)), x);
	} else {
		tmp = copysign(log(((x + x) + (0.5 / 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(log(Float64(Float64(x - x) + Float64(-0.5 / x))), x);
	elseif (t_0 <= 1e-11)
		tmp = copysign(fma(0.5, Float64((x ^ 2.0) / Float64(x + 1.0)), log1p(x)), x);
	else
		tmp = copysign(log(Float64(Float64(x + x) + Float64(0.5 / 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[(x - x), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 1e-11], N[With[{TMP1 = Abs[N[(0.5 * N[(N[Power[x, 2.0], $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(x + x), $MachinePrecision] + N[(0.5 / x), $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) 1)))) x) < -5

   1. Initial program 61.8%

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

      1. +-commutative 61.8%

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

      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. Taylor expanded in x around -inf 98.5%

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

      1. sub-neg 98.5%

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

      2. mul-1-neg 98.5%

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

      3. unsub-neg 98.5%

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

      4. rem-square-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. rem-square-sqrt 99.1%

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

      7. associate-*r/ 99.1%

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

      8. metadata-eval 99.1%

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

      9. distribute-neg-frac 99.1%

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

      10. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

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

   1. Initial program 6.1%

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

      1. +-commutative 6.1%

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

      2. hypot-1-def 6.1%

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

   3. Simplified 6.1%

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

   4. Taylor expanded in x around 0 7.1%

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

      1. +-commutative 7.1%

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

      2. fma-def 7.1%

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

      3. rem-square-sqrt 4.0%

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

      4. fabs-sqr 4.0%

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

      5. rem-square-sqrt 7.1%

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

      6. log1p-def 100.0%

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

      7. rem-square-sqrt 54.5%

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

      8. fabs-sqr 54.5%

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

      9. rem-square-sqrt 100.0%

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

   6. Simplified 100.0%

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

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

   1. Initial program 54.4%

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

      1. +-commutative 54.4%

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

      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. Taylor expanded in x around inf 100.0%

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

      1. rem-square-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. associate-+r+ 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

   6. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(x + x\right) + \frac{0.5}{x}\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 10^{-11}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{{x}^{2}}{x + 1}, \mathsf{log1p}\left(x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x + x\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 2: 98.2% accurate, 0.4× speedup

Math

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

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 (+ (- x x) (/ -0.5 x))) x)
     (if (<= t_0 1e-11)
       (copysign (log1p x) x)
       (copysign (log (+ (+ x x) (/ 0.5 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(((x - x) + (-0.5 / x))), x);
	} else if (t_0 <= 1e-11) {
		tmp = copysign(log1p(x), x);
	} else {
		tmp = copysign(log(((x + x) + (0.5 / 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(((x - x) + (-0.5 / x))), x);
	} else if (t_0 <= 1e-11) {
		tmp = Math.copySign(Math.log1p(x), x);
	} else {
		tmp = Math.copySign(Math.log(((x + x) + (0.5 / 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(((x - x) + (-0.5 / x))), x)
	elif t_0 <= 1e-11:
		tmp = math.copysign(math.log1p(x), x)
	else:
		tmp = math.copysign(math.log(((x + x) + (0.5 / 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(log(Float64(Float64(x - x) + Float64(-0.5 / x))), x);
	elseif (t_0 <= 1e-11)
		tmp = copysign(log1p(x), x);
	else
		tmp = copysign(log(Float64(Float64(x + x) + Float64(0.5 / 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[(x - x), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 1e-11], N[With[{TMP1 = Abs[N[Log[1 + x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(x + x), $MachinePrecision] + N[(0.5 / x), $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) 1)))) x) < -5

   1. Initial program 61.8%

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

      1. +-commutative 61.8%

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

      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. Taylor expanded in x around -inf 98.5%

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

      1. sub-neg 98.5%

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

      2. mul-1-neg 98.5%

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

      3. unsub-neg 98.5%

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

      4. rem-square-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. rem-square-sqrt 99.1%

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

      7. associate-*r/ 99.1%

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

      8. metadata-eval 99.1%

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

      9. distribute-neg-frac 99.1%

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

      10. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

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

   1. Initial program 6.1%

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

      1. +-commutative 6.1%

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

      2. hypot-1-def 6.1%

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

   3. Simplified 6.1%

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

   4. Taylor expanded in x around 0 6.1%

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

      1. log1p-def 99.8%

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

      2. rem-square-sqrt 54.3%

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

      3. fabs-sqr 54.3%

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

      4. rem-square-sqrt 99.8%

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

   6. Simplified 99.8%

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

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

   1. Initial program 54.4%

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

      1. +-commutative 54.4%

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

      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. Taylor expanded in x around inf 100.0%

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

      1. rem-square-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. associate-+r+ 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

   6. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(x + x\right) + \frac{0.5}{x}\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

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

Alternative 3: 81.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x + x\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.5)
   (copysign (log (- x)) x)
   (if (<= x 0.65)
     (copysign (log1p x) x)
     (copysign (log (+ (+ x x) (/ 0.5 x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.5) {
		tmp = copysign(log(-x), x);
	} else if (x <= 0.65) {
		tmp = copysign(log1p(x), x);
	} else {
		tmp = copysign(log(((x + x) + (0.5 / x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.5) {
		tmp = Math.copySign(Math.log(-x), x);
	} else if (x <= 0.65) {
		tmp = Math.copySign(Math.log1p(x), x);
	} else {
		tmp = Math.copySign(Math.log(((x + x) + (0.5 / x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.5:
		tmp = math.copysign(math.log(-x), x)
	elif x <= 0.65:
		tmp = math.copysign(math.log1p(x), x)
	else:
		tmp = math.copysign(math.log(((x + x) + (0.5 / x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.5)
		tmp = copysign(log(Float64(-x)), x);
	elseif (x <= 0.65)
		tmp = copysign(log1p(x), x);
	else
		tmp = copysign(log(Float64(Float64(x + x) + Float64(0.5 / x))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.5], N[With[{TMP1 = Abs[N[Log[(-x)], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 0.65], N[With[{TMP1 = Abs[N[Log[1 + x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(x + x), $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.5

   1. Initial program 61.8%

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

      1. +-commutative 61.8%

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

      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. Taylor expanded in x around -inf 30.9%

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

      1. mul-1-neg 30.9%

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

   6. Simplified 30.9%

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

   if -0.5 < x < 0.650000000000000022

   1. Initial program 6.1%

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

      1. +-commutative 6.1%

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

      2. hypot-1-def 6.1%

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

   3. Simplified 6.1%

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

   4. Taylor expanded in x around 0 6.1%

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

      1. log1p-def 99.8%

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

      2. rem-square-sqrt 54.3%

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

      3. fabs-sqr 54.3%

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

      4. rem-square-sqrt 99.8%

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

   6. Simplified 99.8%

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

   if 0.650000000000000022 < x

   1. Initial program 54.4%

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

      1. +-commutative 54.4%

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

      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. Taylor expanded in x around inf 100.0%

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

      1. rem-square-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. associate-+r+ 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

   6. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(x + x\right) + \frac{0.5}{x}\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x + x\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 4: 81.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -0.5

   1. Initial program 61.8%

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

      1. +-commutative 61.8%

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

      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. Taylor expanded in x around -inf 30.9%

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

      1. mul-1-neg 30.9%

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

   6. Simplified 30.9%

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

   if -0.5 < x < 1

   1. Initial program 6.1%

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

      1. +-commutative 6.1%

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

      2. hypot-1-def 6.1%

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

   3. Simplified 6.1%

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

   4. Taylor expanded in x around 0 6.1%

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

      1. log1p-def 99.8%

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

      2. rem-square-sqrt 54.3%

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

      3. fabs-sqr 54.3%

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

      4. rem-square-sqrt 99.8%

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

   6. Simplified 99.8%

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

   if 1 < x

   1. Initial program 54.4%

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

      1. +-commutative 54.4%

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

      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. Taylor expanded in x around inf 99.5%

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

      1. rem-square-sqrt 99.5%

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

      2. fabs-sqr 99.5%

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

      3. rem-square-sqrt 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 81.9%

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

Alternative 5: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -0.5) {
		tmp = copysign(log(-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(-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(-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(log(Float64(-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[(-x)], $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 61.8%

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

      1. +-commutative 61.8%

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

      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. Taylor expanded in x around -inf 30.9%

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

      1. mul-1-neg 30.9%

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

   6. Simplified 30.9%

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

   if -0.5 < x

   1. Initial program 23.4%

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

      1. +-commutative 23.4%

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

      2. hypot-1-def 39.7%

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

   3. Simplified 39.7%

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

   4. Taylor expanded in x around 0 15.1%

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

      1. log1p-def 75.3%

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

      2. rem-square-sqrt 46.1%

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

      3. fabs-sqr 46.1%

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

      4. rem-square-sqrt 75.3%

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

   6. Simplified 75.3%

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

4. Final simplification 63.8%

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

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 33.3%

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

   1. +-commutative 33.3%

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

   2. hypot-1-def 54.9%

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

3. Simplified 54.9%

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

4. Taylor expanded in x around 0 19.2%

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

   1. log1p-def 63.8%

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

   2. rem-square-sqrt 34.2%

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

   3. fabs-sqr 34.2%

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

   4. rem-square-sqrt 55.9%

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

6. Simplified 55.9%

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

7. Final simplification 55.9%

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

Developer target: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t_0\right) + t_0}\right), x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0)))) x)))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return copysign(log1p((fabs(x) + (fabs(x) / (hypot(1.0, t_0) + t_0)))), x);
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	return Math.copySign(Math.log1p((Math.abs(x) + (Math.abs(x) / (Math.hypot(1.0, t_0) + t_0)))), x);
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	return math.copysign(math.log1p((math.fabs(x) + (math.fabs(x) / (math.hypot(1.0, t_0) + t_0)))), x)

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return copysign(log1p(Float64(abs(x) + Float64(abs(x) / Float64(hypot(1.0, t_0) + t_0)))), x)
end

Wolfram

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

Reproduce

herbie shell --seed 2023334 
(FPCore (x)
  :name "Rust f64::asinh"
  :precision binary64

  :herbie-target
  (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))