Rust f64::asinh

Percentage Accurate: 29.8% → 99.4%

Time: 8.2s

Alternatives: 8

Speedup: 4.0×

Specification

Math

\[\begin{array}{l} \\ \sinh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (asinh x))

C

double code(double x) {
	return asinh(x);
}

Python

def code(x):
	return math.asinh(x)

Julia

function code(x)
	return asinh(x)
end

MATLAB

function tmp = code(x)
	tmp = asinh(x);
end

Wolfram

code[x_] := N[ArcSinh[x], $MachinePrecision]

Initial Program: 29.8% 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: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -6.2e-6)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 1.25) (copysign x x) (copysign (log (* x 2.0)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -6.2e-6) {
		tmp = copysign(-log((hypot(1.0, x) - x)), x);
	} else if (x <= 1.25) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log((x * 2.0)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -6.2e-6) {
		tmp = Math.copySign(-Math.log((Math.hypot(1.0, x) - x)), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log((x * 2.0)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -6.2e-6:
		tmp = math.copysign(-math.log((math.hypot(1.0, x) - x)), x)
	elif x <= 1.25:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log((x * 2.0)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -6.2e-6)
		tmp = copysign(Float64(-log(Float64(hypot(1.0, x) - x))), x);
	elseif (x <= 1.25)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float64(x * 2.0)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -6.2e-6)
		tmp = sign(x) * abs(-log((hypot(1.0, x) - x)));
	elseif (x <= 1.25)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -6.2e-6], 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[x, 1.25], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.1999999999999999e-6

   1. Initial program 46.3%

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

      1. +-commutative 46.3%

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

      2. hypot-1-def 99.7%

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

   3. Simplified 99.7%

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

      1. flip-+ 5.8%

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

      2. clear-num 5.8%

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

      3. log-div 5.8%

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

      4. metadata-eval 5.8%

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

      5. add-sqr-sqrt 0.0%

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

      6. fabs-sqr 0.0%

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

      7. add-sqr-sqrt 5.6%

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

      8. pow2 5.6%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{\left(\left|x\right|\right)}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      9. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      10. fabs-sqr 0.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      11. add-sqr-sqrt 5.6%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{x}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      12. hypot-1-def 5.6%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      13. hypot-1-def 5.6%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      14. add-sqr-sqrt 6.6%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

      15. +-commutative 6.6%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

   6. Applied egg-rr 6.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-sub0 6.6%

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

      2. div-sub 6.6%

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

      3. fma-undefine 6.6%

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

      4. unpow2 6.6%

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

      5. associate--r+ 6.6%

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

      6. +-inverses 6.6%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      7. metadata-eval 6.6%

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

      8. *-rgt-identity 6.6%

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

      9. associate-/l* 6.6%

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

      10. metadata-eval 6.6%

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

      11. *-commutative 6.6%

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

      12. fma-undefine 6.6%

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

      13. unpow2 6.6%

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

      14. associate--r+ 44.5%

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

      15. +-inverses 99.7%

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

      16. metadata-eval 99.7%

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

      17. *-rgt-identity 99.7%

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

      18. associate-/l* 99.7%

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

      19. metadata-eval 99.7%

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

      20. *-commutative 99.7%

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

      21. neg-mul-1 99.7%

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

   8. Simplified 99.7%

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

   if -6.1999999999999999e-6 < x < 1.25

   1. Initial program 6.0%

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

      1. +-commutative 6.0%

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

      2. hypot-1-def 6.0%

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

   3. Simplified 6.0%

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

      1. pow1 6.0%

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

      2. pow-to-exp 6.0%

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

      3. log1p-expm1-u 6.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      4. log1p-undefine 6.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      5. exp-to-pow 6.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}\right)}, x\right) \]

      6. pow1 6.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

      7. expm1-log1p-u 6.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right)}, x\right) \]

      8. pow1 6.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}}\right)\right)\right), x\right) \]

      9. exp-to-pow 6.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right) \cdot 1}}\right)\right)\right), x\right) \]

      10. log1p-undefine 6.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      11. log1p-expm1-u 6.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      12. pow-to-exp 6.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}^{1}}\right)\right)\right), x\right) \]

      13. pow1 6.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right)\right)\right), x\right) \]

      14. add-sqr-sqrt 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      15. fabs-sqr 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      16. add-sqr-sqrt 6.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 6.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0 100.0%

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

   if 1.25 < x

   1. Initial program 59.1%

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

      1. +-commutative 59.1%

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

      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. pow1 100.0%

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

      2. pow-to-exp 100.0%

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

      3. log1p-expm1-u 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      4. log1p-undefine 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      5. exp-to-pow 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}\right)}, x\right) \]

      6. pow1 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

      7. expm1-log1p-u 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right)}, x\right) \]

      8. pow1 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}}\right)\right)\right), x\right) \]

      9. exp-to-pow 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right) \cdot 1}}\right)\right)\right), x\right) \]

      10. log1p-undefine 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      11. log1p-expm1-u 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      12. pow-to-exp 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}^{1}}\right)\right)\right), x\right) \]

      13. pow1 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right)\right)\right), x\right) \]

      14. add-sqr-sqrt 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      15. fabs-sqr 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      16. add-sqr-sqrt 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) -5.0)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (copysign (log1p (+ x (+ (hypot 1.0 x) -1.0))) x)))

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 44.7%

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

      1. +-commutative 44.7%

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

      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-+ 2.7%

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

      2. clear-num 2.7%

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

      3. log-div 2.7%

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

      4. metadata-eval 2.7%

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

      5. add-sqr-sqrt 0.0%

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

      6. fabs-sqr 0.0%

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

      7. add-sqr-sqrt 2.6%

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

      8. pow2 2.6%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{\left(\left|x\right|\right)}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      9. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      10. fabs-sqr 0.0%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      11. add-sqr-sqrt 2.6%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{\color{blue}{x}}^{2} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      12. hypot-1-def 2.6%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

      13. hypot-1-def 2.6%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right), x\right) \]

      14. add-sqr-sqrt 3.6%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

      15. +-commutative 3.6%

          \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

   6. Applied egg-rr 3.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-sub0 3.6%

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

      2. div-sub 3.6%

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

      3. fma-undefine 3.6%

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

      4. unpow2 3.6%

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

      5. associate--r+ 3.6%

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

      6. +-inverses 3.6%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right), x\right) \]

      7. metadata-eval 3.6%

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

      8. *-rgt-identity 3.6%

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

      9. associate-/l* 3.6%

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

      10. metadata-eval 3.6%

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

      11. *-commutative 3.6%

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

      12. fma-undefine 3.6%

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

      13. unpow2 3.6%

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

      14. associate--r+ 42.9%

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

      15. +-inverses 100.0%

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

      16. metadata-eval 100.0%

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

      17. *-rgt-identity 100.0%

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

      18. associate-/l* 100.0%

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

      19. metadata-eval 100.0%

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

      20. *-commutative 100.0%

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

      21. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

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

   1. Initial program 25.7%

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

      1. +-commutative 25.7%

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

      2. hypot-1-def 40.3%

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

   3. Simplified 40.3%

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

      1. pow1 40.3%

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

      2. pow-to-exp 40.3%

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

      3. log1p-expm1-u 40.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      4. log1p-undefine 40.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      5. exp-to-pow 40.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}\right)}, x\right) \]

      6. pow1 40.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

      7. expm1-log1p-u 40.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right)}, x\right) \]

      8. pow1 40.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}}\right)\right)\right), x\right) \]

      9. exp-to-pow 40.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right) \cdot 1}}\right)\right)\right), x\right) \]

      10. log1p-undefine 40.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      11. log1p-expm1-u 40.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      12. pow-to-exp 40.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}^{1}}\right)\right)\right), x\right) \]

      13. pow1 40.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right)\right)\right), x\right) \]

      14. add-sqr-sqrt 37.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      15. fabs-sqr 37.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      16. add-sqr-sqrt 40.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 40.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]
   7. Step-by-step derivation

      1. expm1-log1p-u 40.3%

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

      2. +-commutative 40.3%

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

   8. Applied egg-rr 40.3%

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

      1. log1p-expm1-u 40.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\mathsf{hypot}\left(1, x\right) + x\right)\right)\right)}, x\right) \]

      2. expm1-undefine 40.3%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\mathsf{hypot}\left(1, x\right) + x\right)} - 1}\right), x\right) \]

      3. add-exp-log 40.3%

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

      4. +-commutative 40.3%

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

   10. Applied egg-rr 40.3%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left(x + \mathsf{hypot}\left(1, x\right)\right) - 1\right)}, x\right) \]
   11. Step-by-step derivation

       1. associate--l+ 99.7%

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

   12. Simplified 99.7%

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

4. Final simplification 99.8%

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

Alternative 3: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.25)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (* x 2.0)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = copysign(log((-0.5 / x)), x);
	} else if (x <= 1.25) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log((x * 2.0)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.copySign(Math.log((-0.5 / x)), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(Math.log((x * 2.0)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.copysign(math.log((-0.5 / x)), x)
	elif x <= 1.25:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(math.log((x * 2.0)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = copysign(log(Float64(-0.5 / x)), x);
	elseif (x <= 1.25)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(log(Float64(x * 2.0)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = sign(x) * abs(log((-0.5 / x)));
	elseif (x <= 1.25)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(log((x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[With[{TMP1 = Abs[N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 44.7%

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

      1. +-commutative 44.7%

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

      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. pow1 100.0%

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

      2. pow-to-exp 100.0%

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

      3. log1p-expm1-u 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      4. log1p-undefine 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      5. exp-to-pow 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}\right)}, x\right) \]

      6. pow1 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

      7. expm1-log1p-u 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right)}, x\right) \]

      8. pow1 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}}\right)\right)\right), x\right) \]

      9. exp-to-pow 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right) \cdot 1}}\right)\right)\right), x\right) \]

      10. log1p-undefine 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      11. log1p-expm1-u 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      12. pow-to-exp 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}^{1}}\right)\right)\right), x\right) \]

      13. pow1 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right)\right)\right), x\right) \]

      14. add-sqr-sqrt 0.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      15. fabs-sqr 0.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      16. add-sqr-sqrt 5.6%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 5.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around -inf 98.5%

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

   if -1.25 < x < 1.25

   1. Initial program 7.4%

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

      1. +-commutative 7.4%

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

      2. hypot-1-def 7.4%

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

   3. Simplified 7.4%

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

      1. pow1 7.4%

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

      2. pow-to-exp 7.4%

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

      3. log1p-expm1-u 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      4. log1p-undefine 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      5. exp-to-pow 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}\right)}, x\right) \]

      6. pow1 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

      7. expm1-log1p-u 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right)}, x\right) \]

      8. pow1 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}}\right)\right)\right), x\right) \]

      9. exp-to-pow 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right) \cdot 1}}\right)\right)\right), x\right) \]

      10. log1p-undefine 7.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      11. log1p-expm1-u 7.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      12. pow-to-exp 7.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}^{1}}\right)\right)\right), x\right) \]

      13. pow1 7.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right)\right)\right), x\right) \]

      14. add-sqr-sqrt 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      15. fabs-sqr 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      16. add-sqr-sqrt 7.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 7.4%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0 99.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)}, x\right) \]
   8. Step-by-step derivation

      1. distribute-rgt-in 99.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{1 \cdot x + \left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot x}, x\right) \]

      2. *-lft-identity 99.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x} + \left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot x, x\right) \]

      3. associate-*l* 99.4%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot x\right)}, x\right) \]

      4. unpow2 99.4%

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

      5. unpow3 99.4%

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

   9. Simplified 99.4%

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

   if 1.25 < x

   1. Initial program 59.1%

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

      1. +-commutative 59.1%

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

      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. pow1 100.0%

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

      2. pow-to-exp 100.0%

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

      3. log1p-expm1-u 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      4. log1p-undefine 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      5. exp-to-pow 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}\right)}, x\right) \]

      6. pow1 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

      7. expm1-log1p-u 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right)}, x\right) \]

      8. pow1 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}}\right)\right)\right), x\right) \]

      9. exp-to-pow 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right) \cdot 1}}\right)\right)\right), x\right) \]

      10. log1p-undefine 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      11. log1p-expm1-u 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      12. pow-to-exp 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}^{1}}\right)\right)\right), x\right) \]

      13. pow1 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right)\right)\right), x\right) \]

      14. add-sqr-sqrt 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      15. fabs-sqr 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      16. add-sqr-sqrt 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.2)
   (copysign (log (- x)) x)
   (if (<= x 1.25) (copysign x x) (copysign (log (* x 2.0)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -3.2) {
		tmp = copysign(log(-x), x);
	} else if (x <= 1.25) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log((x * 2.0)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.2) {
		tmp = Math.copySign(Math.log(-x), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log((x * 2.0)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.2:
		tmp = math.copysign(math.log(-x), x)
	elif x <= 1.25:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log((x * 2.0)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.2)
		tmp = copysign(log(Float64(-x)), x);
	elseif (x <= 1.25)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float64(x * 2.0)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.2)
		tmp = sign(x) * abs(log(-x));
	elseif (x <= 1.25)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.2], N[With[{TMP1 = Abs[N[Log[(-x)], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2000000000000002

   1. Initial program 44.7%

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

      1. +-commutative 44.7%

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

      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 31.5%

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

      1. mul-1-neg 31.5%

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

   7. Simplified 31.5%

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

   if -3.2000000000000002 < x < 1.25

   1. Initial program 7.4%

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

      1. +-commutative 7.4%

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

      2. hypot-1-def 7.4%

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

   3. Simplified 7.4%

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

      1. pow1 7.4%

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

      2. pow-to-exp 7.4%

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

      3. log1p-expm1-u 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      4. log1p-undefine 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      5. exp-to-pow 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}\right)}, x\right) \]

      6. pow1 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

      7. expm1-log1p-u 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right)}, x\right) \]

      8. pow1 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}}\right)\right)\right), x\right) \]

      9. exp-to-pow 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right) \cdot 1}}\right)\right)\right), x\right) \]

      10. log1p-undefine 7.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      11. log1p-expm1-u 7.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      12. pow-to-exp 7.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}^{1}}\right)\right)\right), x\right) \]

      13. pow1 7.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right)\right)\right), x\right) \]

      14. add-sqr-sqrt 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      15. fabs-sqr 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      16. add-sqr-sqrt 7.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 7.4%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0 99.1%

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

   if 1.25 < x

   1. Initial program 59.1%

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

      1. +-commutative 59.1%

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

      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. pow1 100.0%

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

      2. pow-to-exp 100.0%

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

      3. log1p-expm1-u 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      4. log1p-undefine 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      5. exp-to-pow 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}\right)}, x\right) \]

      6. pow1 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

      7. expm1-log1p-u 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right)}, x\right) \]

      8. pow1 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}}\right)\right)\right), x\right) \]

      9. exp-to-pow 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right) \cdot 1}}\right)\right)\right), x\right) \]

      10. log1p-undefine 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      11. log1p-expm1-u 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      12. pow-to-exp 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}^{1}}\right)\right)\right), x\right) \]

      13. pow1 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right)\right)\right), x\right) \]

      14. add-sqr-sqrt 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      15. fabs-sqr 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      16. add-sqr-sqrt 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.25) (copysign x x) (copysign (log (* x 2.0)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = copysign(log((-0.5 / x)), x);
	} else if (x <= 1.25) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log((x * 2.0)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.copySign(Math.log((-0.5 / x)), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log((x * 2.0)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.copysign(math.log((-0.5 / x)), x)
	elif x <= 1.25:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log((x * 2.0)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = copysign(log(Float64(-0.5 / x)), x);
	elseif (x <= 1.25)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float64(x * 2.0)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = sign(x) * abs(log((-0.5 / x)));
	elseif (x <= 1.25)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[With[{TMP1 = Abs[N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 44.7%

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

      1. +-commutative 44.7%

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

      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. pow1 100.0%

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

      2. pow-to-exp 100.0%

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

      3. log1p-expm1-u 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      4. log1p-undefine 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      5. exp-to-pow 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}\right)}, x\right) \]

      6. pow1 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

      7. expm1-log1p-u 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right)}, x\right) \]

      8. pow1 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}}\right)\right)\right), x\right) \]

      9. exp-to-pow 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right) \cdot 1}}\right)\right)\right), x\right) \]

      10. log1p-undefine 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      11. log1p-expm1-u 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      12. pow-to-exp 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}^{1}}\right)\right)\right), x\right) \]

      13. pow1 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right)\right)\right), x\right) \]

      14. add-sqr-sqrt 0.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      15. fabs-sqr 0.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      16. add-sqr-sqrt 5.6%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 5.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around -inf 98.5%

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

   if -1.25 < x < 1.25

   1. Initial program 7.4%

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

      1. +-commutative 7.4%

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

      2. hypot-1-def 7.4%

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

   3. Simplified 7.4%

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

      1. pow1 7.4%

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

      2. pow-to-exp 7.4%

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

      3. log1p-expm1-u 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      4. log1p-undefine 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      5. exp-to-pow 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}\right)}, x\right) \]

      6. pow1 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

      7. expm1-log1p-u 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right)}, x\right) \]

      8. pow1 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}}\right)\right)\right), x\right) \]

      9. exp-to-pow 7.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right) \cdot 1}}\right)\right)\right), x\right) \]

      10. log1p-undefine 7.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      11. log1p-expm1-u 7.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      12. pow-to-exp 7.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}^{1}}\right)\right)\right), x\right) \]

      13. pow1 7.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right)\right)\right), x\right) \]

      14. add-sqr-sqrt 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      15. fabs-sqr 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      16. add-sqr-sqrt 7.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 7.4%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0 99.1%

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

   if 1.25 < x

   1. Initial program 59.1%

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

      1. +-commutative 59.1%

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

      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. pow1 100.0%

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

      2. pow-to-exp 100.0%

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

      3. log1p-expm1-u 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      4. log1p-undefine 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

      5. exp-to-pow 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}\right)}, x\right) \]

      6. pow1 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

      7. expm1-log1p-u 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right)}, x\right) \]

      8. pow1 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}}\right)\right)\right), x\right) \]

      9. exp-to-pow 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right) \cdot 1}}\right)\right)\right), x\right) \]

      10. log1p-undefine 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      11. log1p-expm1-u 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

      12. pow-to-exp 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}^{1}}\right)\right)\right), x\right) \]

      13. pow1 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right)\right)\right), x\right) \]

      14. add-sqr-sqrt 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      15. fabs-sqr 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      16. add-sqr-sqrt 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

   7. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 44.7%

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

      1. +-commutative 44.7%

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

      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 31.5%

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

      1. mul-1-neg 31.5%

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

   7. Simplified 31.5%

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

   if -0.5 < x

   1. Initial program 25.7%

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

      1. +-commutative 25.7%

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

      2. hypot-1-def 40.3%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in x around 0 15.3%

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

      1. log1p-define 74.7%

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

      2. rem-square-sqrt 45.9%

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

      3. fabs-sqr 45.9%

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

      4. rem-square-sqrt 74.7%

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

   7. Simplified 74.7%

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

4. Final simplification 65.3%

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

Alternative 7: 56.7% 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 29.9%

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

   1. +-commutative 29.9%

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

   2. hypot-1-def 53.3%

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

3. Simplified 53.3%

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

5. Taylor expanded in x around 0 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 65.3%

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

   2. rem-square-sqrt 35.9%

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

   3. fabs-sqr 35.9%

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

   4. rem-square-sqrt 58.4%

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

7. Simplified 58.4%

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

8. Final simplification 58.4%

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

Alternative 8: 52.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.9%

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

   1. +-commutative 29.9%

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

   2. hypot-1-def 53.3%

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

3. Simplified 53.3%

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

   1. pow1 53.3%

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

   2. pow-to-exp 53.3%

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

   3. log1p-expm1-u 53.3%

      \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

   4. log1p-undefine 53.3%

      \[\leadsto \mathsf{copysign}\left(\log \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right), x\right) \]

   5. exp-to-pow 53.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}\right)}, x\right) \]

   6. pow1 53.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

   7. expm1-log1p-u 53.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right)}, x\right) \]

   8. pow1 53.3%

      \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}^{1}}\right)\right)\right), x\right) \]

   9. exp-to-pow 53.3%

      \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right) \cdot 1}}\right)\right)\right), x\right) \]

   10. log1p-undefine 53.3%

       \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

   11. log1p-expm1-u 53.3%

       \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\color{blue}{\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)} \cdot 1}\right)\right)\right), x\right) \]

   12. pow-to-exp 53.3%

       \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}^{1}}\right)\right)\right), x\right) \]

   13. pow1 53.3%

       \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right| + \mathsf{hypot}\left(1, x\right)}\right)\right)\right), x\right) \]

   14. add-sqr-sqrt 29.2%

       \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

   15. fabs-sqr 29.2%

       \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

   16. add-sqr-sqrt 32.7%

       \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

6. Applied egg-rr 32.7%

   \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]

7. Taylor expanded in x around 0 52.6%

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

8. Final simplification 52.6%

   \[\leadsto \mathsf{copysign}\left(x, x\right) \]
9. 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 2024075 
(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))