Rust f64::asinh

Percentage Accurate: 30.9% → 99.5%

Time: 8.3s

Alternatives: 12

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: 30.9% 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: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log 0.5 + \log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x}{\frac{1 + \mathsf{hypot}\left(1, x\right)}{x}}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) -20.0)
   (copysign (+ (log 0.5) (log (/ -1.0 x))) x)
   (copysign (log1p (+ x (/ x (/ (+ 1.0 (hypot 1.0 x)) x)))) x)))

C

double code(double x) {
	double tmp;
	if (copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x) <= -20.0) {
		tmp = copysign((log(0.5) + log((-1.0 / x))), x);
	} else {
		tmp = copysign(log1p((x + (x / ((1.0 + hypot(1.0, x)) / x)))), 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) <= -20.0) {
		tmp = Math.copySign((Math.log(0.5) + Math.log((-1.0 / x))), x);
	} else {
		tmp = Math.copySign(Math.log1p((x + (x / ((1.0 + Math.hypot(1.0, x)) / x)))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x) <= -20.0:
		tmp = math.copysign((math.log(0.5) + math.log((-1.0 / x))), x)
	else:
		tmp = math.copysign(math.log1p((x + (x / ((1.0 + math.hypot(1.0, x)) / x)))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x) <= -20.0)
		tmp = copysign(Float64(log(0.5) + log(Float64(-1.0 / x))), x);
	else
		tmp = copysign(log1p(Float64(x + Float64(x / Float64(Float64(1.0 + hypot(1.0, x)) / x)))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], -20.0], N[With[{TMP1 = Abs[N[(N[Log[0.5], $MachinePrecision] + N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + N[(x + N[(x / N[(N[(1.0 + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < -20

   1. Initial program 47.7%

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

      1. +-commutative 47.7%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

      1. log1p-expm1-u 98.5%

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

      2. expm1-udef 98.5%

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

      3. add-exp-log 98.5%

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

      4. add-sqr-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. add-sqr-sqrt 3.1%

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

   6. Applied egg-rr 3.1%

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

      1. associate--l+ 3.1%

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

   8. Simplified 3.1%

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

   9. Taylor expanded in x around -inf 99.7%

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

   if -20 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x)

   1. Initial program 23.7%

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

      1. +-commutative 23.7%

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

      2. hypot-1-def 37.7%

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

   3. Simplified 37.7%

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

      1. log1p-expm1-u 37.7%

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

      2. expm1-udef 37.7%

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

      3. add-exp-log 37.7%

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

      4. add-sqr-sqrt 34.0%

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

      5. fabs-sqr 34.0%

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

      6. add-sqr-sqrt 37.7%

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

   6. Applied egg-rr 37.7%

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

      1. associate--l+ 99.0%

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

   8. Simplified 99.0%

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

      1. sub-neg 99.0%

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

      2. flip-+ 84.9%

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

      3. hypot-1-def 84.9%

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

      4. hypot-1-def 84.9%

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

      5. add-sqr-sqrt 84.9%

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

      6. add-exp-log 84.9%

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

      7. log1p-udef 84.9%

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

      8. metadata-eval 84.9%

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

      9. metadata-eval 84.9%

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

      10. metadata-eval 84.9%

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

      11. expm1-udef 85.9%

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

      12. expm1-log1p-u 85.9%

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

      13. pow2 85.9%

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

      14. metadata-eval 85.9%

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

   10. Applied egg-rr 85.9%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\frac{{x}^{2}}{\mathsf{hypot}\left(1, x\right) - -1}}\right), x\right) \]
   11. Step-by-step derivation

       1. unpow2 85.9%

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

       2. *-un-lft-identity 85.9%

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

       3. times-frac 100.0%

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

       4. sub-neg 100.0%

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

       5. metadata-eval 100.0%

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

       6. +-commutative 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. /-rgt-identity 100.0%

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

       2. clear-num 100.0%

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

       3. un-div-inv 100.0%

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

   14. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log 0.5 + \log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x}{\frac{1 + \mathsf{hypot}\left(1, x\right)}{x}}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10500:\\ \;\;\;\;\mathsf{copysign}\left(\log 0.5 + \log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(-1 + \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -10500.0)
   (copysign (+ (log 0.5) (log (/ -1.0 x))) x)
   (copysign (log1p (+ x (+ -1.0 (hypot 1.0 x)))) x)))

C

double code(double x) {
	double tmp;
	if (x <= -10500.0) {
		tmp = copysign((log(0.5) + log((-1.0 / x))), x);
	} else {
		tmp = copysign(log1p((x + (-1.0 + hypot(1.0, x)))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -10500.0) {
		tmp = Math.copySign((Math.log(0.5) + Math.log((-1.0 / x))), x);
	} else {
		tmp = Math.copySign(Math.log1p((x + (-1.0 + Math.hypot(1.0, x)))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -10500.0:
		tmp = math.copysign((math.log(0.5) + math.log((-1.0 / x))), x)
	else:
		tmp = math.copysign(math.log1p((x + (-1.0 + math.hypot(1.0, x)))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -10500.0)
		tmp = copysign(Float64(log(0.5) + log(Float64(-1.0 / x))), x);
	else
		tmp = copysign(log1p(Float64(x + Float64(-1.0 + hypot(1.0, x)))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -10500.0], N[With[{TMP1 = Abs[N[(N[Log[0.5], $MachinePrecision] + N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + N[(x + N[(-1.0 + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -10500

   1. Initial program 47.7%

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

      1. +-commutative 47.7%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

      1. log1p-expm1-u 98.5%

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

      2. expm1-udef 98.5%

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

      3. add-exp-log 98.5%

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

      4. add-sqr-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. add-sqr-sqrt 3.1%

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

   6. Applied egg-rr 3.1%

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

      1. associate--l+ 3.1%

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

   8. Simplified 3.1%

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

   9. Taylor expanded in x around -inf 99.7%

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

   if -10500 < x

   1. Initial program 23.7%

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

      1. +-commutative 23.7%

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

      2. hypot-1-def 37.7%

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

   3. Simplified 37.7%

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

      1. log1p-expm1-u 37.7%

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

      2. expm1-udef 37.7%

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

      3. add-exp-log 37.7%

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

      4. add-sqr-sqrt 34.0%

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

      5. fabs-sqr 34.0%

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

      6. add-sqr-sqrt 37.7%

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

   6. Applied egg-rr 37.7%

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

      1. associate--l+ 99.0%

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

   8. Simplified 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10500:\\ \;\;\;\;\mathsf{copysign}\left(\log 0.5 + \log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(-1 + \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0011:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0011)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 1.3)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (- (log (/ 0.5 x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0011) {
		tmp = copysign(-log((hypot(1.0, x) - x)), x);
	} else if (x <= 1.3) {
		tmp = copysign((x + (pow(x, 3.0) * -0.16666666666666666)), x);
	} else {
		tmp = copysign(-log((0.5 / x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.0011) {
		tmp = Math.copySign(-Math.log((Math.hypot(1.0, x) - x)), x);
	} else if (x <= 1.3) {
		tmp = Math.copySign((x + (Math.pow(x, 3.0) * -0.16666666666666666)), x);
	} else {
		tmp = Math.copySign(-Math.log((0.5 / x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.0011:
		tmp = math.copysign(-math.log((math.hypot(1.0, x) - x)), x)
	elif x <= 1.3:
		tmp = math.copysign((x + (math.pow(x, 3.0) * -0.16666666666666666)), x)
	else:
		tmp = math.copysign(-math.log((0.5 / x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0011)
		tmp = copysign(Float64(-log(Float64(hypot(1.0, x) - x))), x);
	elseif (x <= 1.3)
		tmp = copysign(Float64(x + Float64((x ^ 3.0) * -0.16666666666666666)), x);
	else
		tmp = copysign(Float64(-log(Float64(0.5 / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0011)
		tmp = sign(x) * abs(-log((hypot(1.0, x) - x)));
	elseif (x <= 1.3)
		tmp = sign(x) * abs((x + ((x ^ 3.0) * -0.16666666666666666)));
	else
		tmp = sign(x) * abs(-log((0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.0011], 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, 1.3], N[With[{TMP1 = Abs[N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[(-N[Log[N[(0.5 / x), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00110000000000000007

   1. Initial program 48.5%

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

      1. +-commutative 48.5%

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

      2. hypot-1-def 98.4%

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

   3. Simplified 98.4%

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

      1. flip-+ 1.5%

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

      2. frac-2neg 1.5%

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

      3. log-div 1.5%

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

   6. Applied egg-rr 2.9%

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

      1. neg-sub0 2.9%

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

      2. associate--r- 2.9%

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

      3. neg-sub0 2.9%

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

      4. +-commutative 2.9%

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

      5. sub-neg 2.9%

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

      6. neg-sub0 2.9%

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

      7. associate--r- 2.9%

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

      8. neg-sub0 2.9%

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

      9. +-commutative 2.9%

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

      10. sub-neg 2.9%

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

      11. fma-udef 2.9%

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

      12. unpow2 2.9%

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

      13. +-commutative 2.9%

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

      14. associate--l+ 46.8%

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

      15. +-inverses 98.4%

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

      16. metadata-eval 98.4%

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

      17. metadata-eval 98.4%

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

      18. neg-sub0 98.4%

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

   8. Simplified 98.4%

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

   if -0.00110000000000000007 < x < 1.30000000000000004

   1. Initial program 8.3%

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

      1. +-commutative 8.3%

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

      2. hypot-1-def 8.3%

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

   3. Simplified 8.3%

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

      1. log1p-expm1-u 8.3%

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

      2. expm1-udef 8.3%

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

      3. add-exp-log 8.3%

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

      4. add-sqr-sqrt 3.5%

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

      5. fabs-sqr 3.5%

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

      6. add-sqr-sqrt 8.3%

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

   6. Applied egg-rr 8.3%

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

      1. associate--l+ 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in x around 0 99.9%

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

       1. *-commutative 99.9%

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

   11. Simplified 99.9%

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

   if 1.30000000000000004 < x

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      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-+ 0.0%

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

      2. frac-2neg 0.0%

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

      3. log-div 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. neg-sub0 0.0%

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

      2. associate--r- 0.0%

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

      3. neg-sub0 0.0%

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

      4. +-commutative 0.0%

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

      5. sub-neg 0.0%

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

      6. neg-sub0 0.0%

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

      7. associate--r- 0.0%

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

      8. neg-sub0 0.0%

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

      9. +-commutative 0.0%

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

      10. sub-neg 0.0%

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

      11. fma-udef 0.0%

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

      12. unpow2 0.0%

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

      13. +-commutative 0.0%

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

      14. associate--l+ 1.7%

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

      15. +-inverses 3.1%

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

      16. metadata-eval 3.1%

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

      17. metadata-eval 3.1%

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

      18. neg-sub0 3.1%

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

   8. Simplified 3.1%

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

   9. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0011:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log 0.5 + \log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (+ (log 0.5) (log (/ -1.0 x))) x)
   (if (<= x 1.3)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (- (log (/ 0.5 x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = copysign((log(0.5) + log((-1.0 / x))), x);
	} else if (x <= 1.3) {
		tmp = copysign((x + (pow(x, 3.0) * -0.16666666666666666)), x);
	} else {
		tmp = copysign(-log((0.5 / x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.copySign((Math.log(0.5) + Math.log((-1.0 / x))), x);
	} else if (x <= 1.3) {
		tmp = Math.copySign((x + (Math.pow(x, 3.0) * -0.16666666666666666)), x);
	} else {
		tmp = Math.copySign(-Math.log((0.5 / x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.copysign((math.log(0.5) + math.log((-1.0 / x))), x)
	elif x <= 1.3:
		tmp = math.copysign((x + (math.pow(x, 3.0) * -0.16666666666666666)), x)
	else:
		tmp = math.copysign(-math.log((0.5 / x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = copysign(Float64(log(0.5) + log(Float64(-1.0 / x))), x);
	elseif (x <= 1.3)
		tmp = copysign(Float64(x + Float64((x ^ 3.0) * -0.16666666666666666)), x);
	else
		tmp = copysign(Float64(-log(Float64(0.5 / x))), 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) + log((-1.0 / x))));
	elseif (x <= 1.3)
		tmp = sign(x) * abs((x + ((x ^ 3.0) * -0.16666666666666666)));
	else
		tmp = sign(x) * abs(-log((0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[With[{TMP1 = Abs[N[(N[Log[0.5], $MachinePrecision] + N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.3], N[With[{TMP1 = Abs[N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[(-N[Log[N[(0.5 / x), $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 47.7%

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

      1. +-commutative 47.7%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

      1. log1p-expm1-u 98.5%

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

      2. expm1-udef 98.5%

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

      3. add-exp-log 98.5%

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

      4. add-sqr-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. add-sqr-sqrt 3.1%

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

   6. Applied egg-rr 3.1%

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

      1. associate--l+ 3.1%

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

   8. Simplified 3.1%

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

   9. Taylor expanded in x around -inf 99.7%

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

   if -1.25 < x < 1.30000000000000004

   1. Initial program 8.9%

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

      1. +-commutative 8.9%

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

      2. hypot-1-def 8.9%

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

   3. Simplified 8.9%

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

      1. log1p-expm1-u 8.9%

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

      2. expm1-udef 8.9%

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

      3. add-exp-log 8.9%

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

      4. add-sqr-sqrt 3.5%

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

      5. fabs-sqr 3.5%

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

      6. add-sqr-sqrt 9.0%

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

   6. Applied egg-rr 9.0%

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

      1. associate--l+ 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in x around 0 99.5%

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

       1. *-commutative 99.5%

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

   11. Simplified 99.5%

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

   if 1.30000000000000004 < x

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      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-+ 0.0%

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

      2. frac-2neg 0.0%

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

      3. log-div 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. neg-sub0 0.0%

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

      2. associate--r- 0.0%

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

      3. neg-sub0 0.0%

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

      4. +-commutative 0.0%

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

      5. sub-neg 0.0%

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

      6. neg-sub0 0.0%

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

      7. associate--r- 0.0%

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

      8. neg-sub0 0.0%

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

      9. +-commutative 0.0%

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

      10. sub-neg 0.0%

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

      11. fma-udef 0.0%

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

      12. unpow2 0.0%

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

      13. +-commutative 0.0%

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

      14. associate--l+ 1.7%

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

      15. +-inverses 3.1%

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

      16. metadata-eval 3.1%

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

      17. metadata-eval 3.1%

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

      18. neg-sub0 3.1%

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

   8. Simplified 3.1%

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

   9. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log 0.5 + \log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (- (log (* x -2.0))) x)
   (if (<= x 1.3)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (- (log (/ 0.5 x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = copysign(-log((x * -2.0)), x);
	} else if (x <= 1.3) {
		tmp = copysign((x + (pow(x, 3.0) * -0.16666666666666666)), x);
	} else {
		tmp = copysign(-log((0.5 / x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.copySign(-Math.log((x * -2.0)), x);
	} else if (x <= 1.3) {
		tmp = Math.copySign((x + (Math.pow(x, 3.0) * -0.16666666666666666)), x);
	} else {
		tmp = Math.copySign(-Math.log((0.5 / x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.copysign(-math.log((x * -2.0)), x)
	elif x <= 1.3:
		tmp = math.copysign((x + (math.pow(x, 3.0) * -0.16666666666666666)), x)
	else:
		tmp = math.copysign(-math.log((0.5 / x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = copysign(Float64(-log(Float64(x * -2.0))), x);
	elseif (x <= 1.3)
		tmp = copysign(Float64(x + Float64((x ^ 3.0) * -0.16666666666666666)), x);
	else
		tmp = copysign(Float64(-log(Float64(0.5 / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = sign(x) * abs(-log((x * -2.0)));
	elseif (x <= 1.3)
		tmp = sign(x) * abs((x + ((x ^ 3.0) * -0.16666666666666666)));
	else
		tmp = sign(x) * abs(-log((0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[With[{TMP1 = Abs[(-N[Log[N[(x * -2.0), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.3], N[With[{TMP1 = Abs[N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[(-N[Log[N[(0.5 / x), $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 47.7%

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

      1. +-commutative 47.7%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

      1. flip-+ 0.0%

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

      2. frac-2neg 0.0%

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

      3. log-div 0.0%

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

   6. Applied egg-rr 1.4%

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

      1. neg-sub0 1.4%

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

      2. associate--r- 1.4%

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

      3. neg-sub0 1.4%

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

      4. +-commutative 1.4%

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

      5. sub-neg 1.4%

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

      6. neg-sub0 1.4%

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

      7. associate--r- 1.4%

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

      8. neg-sub0 1.4%

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

      9. +-commutative 1.4%

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

      10. sub-neg 1.4%

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

      11. fma-udef 1.4%

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

      12. unpow2 1.4%

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

      13. +-commutative 1.4%

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

      14. associate--l+ 46.0%

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

      15. +-inverses 98.5%

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

      16. metadata-eval 98.5%

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

      17. metadata-eval 98.5%

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

      18. neg-sub0 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in x around -inf 98.5%

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

       1. *-commutative 98.5%

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

   11. Simplified 98.5%

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

   if -1.25 < x < 1.30000000000000004

   1. Initial program 8.9%

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

      1. +-commutative 8.9%

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

      2. hypot-1-def 8.9%

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

   3. Simplified 8.9%

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

      1. log1p-expm1-u 8.9%

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

      2. expm1-udef 8.9%

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

      3. add-exp-log 8.9%

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

      4. add-sqr-sqrt 3.5%

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

      5. fabs-sqr 3.5%

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

      6. add-sqr-sqrt 9.0%

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

   6. Applied egg-rr 9.0%

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

      1. associate--l+ 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in x around 0 99.5%

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

       1. *-commutative 99.5%

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

   11. Simplified 99.5%

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

   if 1.30000000000000004 < x

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      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-+ 0.0%

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

      2. frac-2neg 0.0%

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

      3. log-div 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. neg-sub0 0.0%

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

      2. associate--r- 0.0%

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

      3. neg-sub0 0.0%

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

      4. +-commutative 0.0%

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

      5. sub-neg 0.0%

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

      6. neg-sub0 0.0%

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

      7. associate--r- 0.0%

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

      8. neg-sub0 0.0%

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

      9. +-commutative 0.0%

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

      10. sub-neg 0.0%

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

      11. fma-udef 0.0%

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

      12. unpow2 0.0%

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

      13. +-commutative 0.0%

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

      14. associate--l+ 1.7%

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

      15. +-inverses 3.1%

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

      16. metadata-eval 3.1%

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

      17. metadata-eval 3.1%

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

      18. neg-sub0 3.1%

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

   8. Simplified 3.1%

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

   9. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (- (log (* x -2.0))) x)
   (if (<= x 1.3) (copysign x x) (copysign (- (log (/ 0.5 x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = copysign(-log((x * -2.0)), x);
	} else if (x <= 1.3) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(-log((0.5 / x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.copySign(-Math.log((x * -2.0)), x);
	} else if (x <= 1.3) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(-Math.log((0.5 / x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.copysign(-math.log((x * -2.0)), x)
	elif x <= 1.3:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(-math.log((0.5 / x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = copysign(Float64(-log(Float64(x * -2.0))), x);
	elseif (x <= 1.3)
		tmp = copysign(x, x);
	else
		tmp = copysign(Float64(-log(Float64(0.5 / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = sign(x) * abs(-log((x * -2.0)));
	elseif (x <= 1.3)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(-log((0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[With[{TMP1 = Abs[(-N[Log[N[(x * -2.0), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.3], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[(-N[Log[N[(0.5 / x), $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 47.7%

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

      1. +-commutative 47.7%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

      1. flip-+ 0.0%

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

      2. frac-2neg 0.0%

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

      3. log-div 0.0%

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

   6. Applied egg-rr 1.4%

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

      1. neg-sub0 1.4%

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

      2. associate--r- 1.4%

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

      3. neg-sub0 1.4%

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

      4. +-commutative 1.4%

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

      5. sub-neg 1.4%

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

      6. neg-sub0 1.4%

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

      7. associate--r- 1.4%

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

      8. neg-sub0 1.4%

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

      9. +-commutative 1.4%

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

      10. sub-neg 1.4%

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

      11. fma-udef 1.4%

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

      12. unpow2 1.4%

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

      13. +-commutative 1.4%

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

      14. associate--l+ 46.0%

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

      15. +-inverses 98.5%

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

      16. metadata-eval 98.5%

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

      17. metadata-eval 98.5%

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

      18. neg-sub0 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in x around -inf 98.5%

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

       1. *-commutative 98.5%

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

   11. Simplified 98.5%

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

   if -1.25 < x < 1.30000000000000004

   1. Initial program 8.9%

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

      1. +-commutative 8.9%

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

      2. hypot-1-def 8.9%

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

   3. Simplified 8.9%

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

      1. log1p-expm1-u 8.9%

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

      2. expm1-udef 8.9%

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

      3. add-exp-log 8.9%

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

      4. add-sqr-sqrt 3.5%

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

      5. fabs-sqr 3.5%

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

      6. add-sqr-sqrt 9.0%

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

   6. Applied egg-rr 9.0%

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

      1. associate--l+ 98.5%

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

   8. Simplified 98.5%

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

      1. sub-neg 98.5%

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

      2. flip-+ 98.5%

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

      3. hypot-1-def 98.5%

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

      4. hypot-1-def 98.5%

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

      5. add-sqr-sqrt 98.5%

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

      6. add-exp-log 98.5%

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

      7. log1p-udef 98.5%

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

      8. metadata-eval 98.5%

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

      9. metadata-eval 98.5%

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

      10. metadata-eval 98.5%

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

      11. expm1-udef 100.0%

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

      12. expm1-log1p-u 100.0%

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

      13. pow2 100.0%

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

      14. metadata-eval 100.0%

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

   10. Applied egg-rr 100.0%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\frac{{x}^{2}}{\mathsf{hypot}\left(1, x\right) - -1}}\right), x\right) \]
   11. Step-by-step derivation

       1. unpow2 100.0%

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

       2. *-un-lft-identity 100.0%

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

       3. times-frac 100.0%

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

       4. sub-neg 100.0%

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

       5. metadata-eval 100.0%

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

       6. +-commutative 100.0%

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

   12. Applied egg-rr 100.0%

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

   13. Taylor expanded in x around 0 99.1%

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

   if 1.30000000000000004 < x

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      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-+ 0.0%

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

      2. frac-2neg 0.0%

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

      3. log-div 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. neg-sub0 0.0%

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

      2. associate--r- 0.0%

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

      3. neg-sub0 0.0%

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

      4. +-commutative 0.0%

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

      5. sub-neg 0.0%

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

      6. neg-sub0 0.0%

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

      7. associate--r- 0.0%

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

      8. neg-sub0 0.0%

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

      9. +-commutative 0.0%

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

      10. sub-neg 0.0%

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

      11. fma-udef 0.0%

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

      12. unpow2 0.0%

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

      13. +-commutative 0.0%

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

      14. associate--l+ 1.7%

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

      15. +-inverses 3.1%

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

      16. metadata-eval 3.1%

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

      17. metadata-eval 3.1%

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

      18. neg-sub0 3.1%

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

   8. Simplified 3.1%

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

   9. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (- (log (* x -2.0))) x)
   (if (<= x 1.3) (copysign x x) (copysign (log1p (+ x x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = copysign(-log((x * -2.0)), x);
	} else if (x <= 1.3) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log1p((x + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.copySign(-Math.log((x * -2.0)), x);
	} else if (x <= 1.3) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log1p((x + x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.copysign(-math.log((x * -2.0)), x)
	elif x <= 1.3:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log1p((x + x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = copysign(Float64(-log(Float64(x * -2.0))), x);
	elseif (x <= 1.3)
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(Float64(x + x)), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[With[{TMP1 = Abs[(-N[Log[N[(x * -2.0), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.3], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + N[(x + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 47.7%

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

      1. +-commutative 47.7%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

      1. flip-+ 0.0%

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

      2. frac-2neg 0.0%

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

      3. log-div 0.0%

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

   6. Applied egg-rr 1.4%

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

      1. neg-sub0 1.4%

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

      2. associate--r- 1.4%

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

      3. neg-sub0 1.4%

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

      4. +-commutative 1.4%

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

      5. sub-neg 1.4%

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

      6. neg-sub0 1.4%

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

      7. associate--r- 1.4%

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

      8. neg-sub0 1.4%

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

      9. +-commutative 1.4%

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

      10. sub-neg 1.4%

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

      11. fma-udef 1.4%

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

      12. unpow2 1.4%

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

      13. +-commutative 1.4%

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

      14. associate--l+ 46.0%

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

      15. +-inverses 98.5%

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

      16. metadata-eval 98.5%

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

      17. metadata-eval 98.5%

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

      18. neg-sub0 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in x around -inf 98.5%

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

       1. *-commutative 98.5%

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

   11. Simplified 98.5%

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

   if -1.25 < x < 1.30000000000000004

   1. Initial program 8.9%

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

      1. +-commutative 8.9%

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

      2. hypot-1-def 8.9%

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

   3. Simplified 8.9%

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

      1. log1p-expm1-u 8.9%

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

      2. expm1-udef 8.9%

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

      3. add-exp-log 8.9%

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

      4. add-sqr-sqrt 3.5%

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

      5. fabs-sqr 3.5%

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

      6. add-sqr-sqrt 9.0%

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

   6. Applied egg-rr 9.0%

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

      1. associate--l+ 98.5%

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

   8. Simplified 98.5%

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

      1. sub-neg 98.5%

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

      2. flip-+ 98.5%

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

      3. hypot-1-def 98.5%

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

      4. hypot-1-def 98.5%

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

      5. add-sqr-sqrt 98.5%

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

      6. add-exp-log 98.5%

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

      7. log1p-udef 98.5%

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

      8. metadata-eval 98.5%

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

      9. metadata-eval 98.5%

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

      10. metadata-eval 98.5%

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

      11. expm1-udef 100.0%

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

      12. expm1-log1p-u 100.0%

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

      13. pow2 100.0%

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

      14. metadata-eval 100.0%

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

   10. Applied egg-rr 100.0%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\frac{{x}^{2}}{\mathsf{hypot}\left(1, x\right) - -1}}\right), x\right) \]
   11. Step-by-step derivation

       1. unpow2 100.0%

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

       2. *-un-lft-identity 100.0%

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

       3. times-frac 100.0%

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

       4. sub-neg 100.0%

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

       5. metadata-eval 100.0%

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

       6. +-commutative 100.0%

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

   12. Applied egg-rr 100.0%

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

   13. Taylor expanded in x around 0 99.1%

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

   if 1.30000000000000004 < x

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      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. log1p-expm1-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. add-exp-log 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. fabs-sqr 100.0%

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

      6. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 99.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + 1\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.6) (copysign x x) (copysign (log (+ x 1.0)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log((x + 1.0)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log((x + 1.0)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.6:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log((x + 1.0)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.6)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float64(x + 1.0)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.6)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.6], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   1. Initial program 21.5%

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

      1. +-commutative 21.5%

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

      2. hypot-1-def 37.9%

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

   3. Simplified 37.9%

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

      1. log1p-expm1-u 37.9%

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

      2. expm1-udef 37.9%

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

      3. add-exp-log 37.9%

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

      4. add-sqr-sqrt 2.4%

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

      5. fabs-sqr 2.4%

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

      6. add-sqr-sqrt 7.1%

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

   6. Applied egg-rr 7.1%

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

      1. associate--l+ 67.7%

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

   8. Simplified 67.7%

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

      1. sub-neg 67.7%

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

      2. flip-+ 67.8%

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

      3. hypot-1-def 67.8%

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

      4. hypot-1-def 67.8%

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

      5. add-sqr-sqrt 67.8%

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

      6. add-exp-log 68.4%

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

      7. log1p-udef 68.4%

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

      8. metadata-eval 68.4%

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

      9. metadata-eval 68.4%

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

      10. metadata-eval 68.4%

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

      11. expm1-udef 69.4%

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

      12. expm1-log1p-u 68.7%

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

      13. pow2 68.7%

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

      14. metadata-eval 68.7%

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

   10. Applied egg-rr 68.7%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\frac{{x}^{2}}{\mathsf{hypot}\left(1, x\right) - -1}}\right), x\right) \]
   11. Step-by-step derivation

       1. unpow2 68.7%

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

       2. *-un-lft-identity 68.7%

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

       3. times-frac 68.7%

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

       4. sub-neg 68.7%

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

       5. metadata-eval 68.7%

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

       6. +-commutative 68.7%

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

   12. Applied egg-rr 68.7%

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

   13. Taylor expanded in x around 0 68.7%

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

   if 1.6000000000000001 < x

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      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 0 31.3%

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

      1. log1p-def 31.3%

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

      2. unpow1 31.3%

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

      3. sqr-pow 31.3%

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

      4. fabs-sqr 31.3%

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

      5. sqr-pow 31.3%

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

      6. unpow1 31.3%

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

   7. Simplified 31.3%

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

      1. log1p-udef 31.3%

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

   9. Applied egg-rr 31.3%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + 1\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.3) (copysign x x) (copysign (log1p (+ x x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.3) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log1p((x + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.3) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log1p((x + x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.3:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log1p((x + x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.3)
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(Float64(x + x)), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.3], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + N[(x + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.30000000000000004

   1. Initial program 21.5%

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

      1. +-commutative 21.5%

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

      2. hypot-1-def 37.9%

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

   3. Simplified 37.9%

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

      1. log1p-expm1-u 37.9%

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

      2. expm1-udef 37.9%

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

      3. add-exp-log 37.9%

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

      4. add-sqr-sqrt 2.4%

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

      5. fabs-sqr 2.4%

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

      6. add-sqr-sqrt 7.1%

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

   6. Applied egg-rr 7.1%

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

      1. associate--l+ 67.7%

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

   8. Simplified 67.7%

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

      1. sub-neg 67.7%

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

      2. flip-+ 67.8%

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

      3. hypot-1-def 67.8%

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

      4. hypot-1-def 67.8%

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

      5. add-sqr-sqrt 67.8%

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

      6. add-exp-log 68.4%

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

      7. log1p-udef 68.4%

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

      8. metadata-eval 68.4%

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

      9. metadata-eval 68.4%

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

      10. metadata-eval 68.4%

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

      11. expm1-udef 69.4%

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

      12. expm1-log1p-u 68.7%

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

      13. pow2 68.7%

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

      14. metadata-eval 68.7%

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

   10. Applied egg-rr 68.7%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\frac{{x}^{2}}{\mathsf{hypot}\left(1, x\right) - -1}}\right), x\right) \]
   11. Step-by-step derivation

       1. unpow2 68.7%

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

       2. *-un-lft-identity 68.7%

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

       3. times-frac 68.7%

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

       4. sub-neg 68.7%

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

       5. metadata-eval 68.7%

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

       6. +-commutative 68.7%

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

   12. Applied egg-rr 68.7%

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

   13. Taylor expanded in x around 0 68.7%

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

   if 1.30000000000000004 < x

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      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. log1p-expm1-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. add-exp-log 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. fabs-sqr 100.0%

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

      6. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 99.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 0.5], N[With[{TMP1 = Abs[(-N[Log[1 + (-x)], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + N[(x + x), $MachinePrecision]], $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 21.5%

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

      1. +-commutative 21.5%

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

      2. hypot-1-def 37.9%

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

   3. Simplified 37.9%

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

      1. flip-+ 6.0%

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

      2. frac-2neg 6.0%

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

      3. log-div 6.1%

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

   6. Applied egg-rr 6.5%

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

      1. neg-sub0 6.5%

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

      2. associate--r- 6.5%

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

      3. neg-sub0 6.5%

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

      4. +-commutative 6.5%

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

      5. sub-neg 6.5%

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

      6. neg-sub0 6.5%

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

      7. associate--r- 6.5%

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

      8. neg-sub0 6.5%

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

      9. +-commutative 6.5%

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

      10. sub-neg 6.5%

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

      11. fma-udef 6.5%

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

      12. unpow2 6.5%

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

      13. +-commutative 6.5%

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

      14. associate--l+ 20.9%

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

      15. +-inverses 37.9%

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

      16. metadata-eval 37.9%

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

      17. metadata-eval 37.9%

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

      18. neg-sub0 37.9%

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

   8. Simplified 37.9%

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

   9. Taylor expanded in x around 0 15.6%

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

       1. neg-mul-1 15.6%

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

       2. unsub-neg 15.6%

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

   11. Simplified 15.6%

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

       1. *-un-lft-identity 15.6%

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

       2. log-prod 15.6%

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

       3. metadata-eval 15.6%

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

       4. sub-neg 15.6%

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

       5. log1p-def 76.2%

          \[\leadsto \mathsf{copysign}\left(-\left(0 + \color{blue}{\mathsf{log1p}\left(-x\right)}\right), x\right) \]

   13. Applied egg-rr 76.2%

       \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0 + \mathsf{log1p}\left(-x\right)\right)}, x\right) \]
   14. Step-by-step derivation

       1. +-lft-identity 76.2%

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

   15. Simplified 76.2%

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

   if 0.5 < x

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      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. log1p-expm1-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. add-exp-log 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. fabs-sqr 100.0%

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

      6. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 99.1%

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

4. Final simplification 81.7%

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

Alternative 11: 58.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.6) (copysign x x) (copysign (log1p x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log1p(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log1p(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.6:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log1p(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.6)
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.6], N[With[{TMP1 = Abs[x], 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 < 1.6000000000000001

   1. Initial program 21.5%

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

      1. +-commutative 21.5%

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

      2. hypot-1-def 37.9%

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

   3. Simplified 37.9%

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

      1. log1p-expm1-u 37.9%

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

      2. expm1-udef 37.9%

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

      3. add-exp-log 37.9%

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

      4. add-sqr-sqrt 2.4%

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

      5. fabs-sqr 2.4%

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

      6. add-sqr-sqrt 7.1%

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

   6. Applied egg-rr 7.1%

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

      1. associate--l+ 67.7%

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

   8. Simplified 67.7%

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

      1. sub-neg 67.7%

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

      2. flip-+ 67.8%

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

      3. hypot-1-def 67.8%

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

      4. hypot-1-def 67.8%

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

      5. add-sqr-sqrt 67.8%

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

      6. add-exp-log 68.4%

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

      7. log1p-udef 68.4%

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

      8. metadata-eval 68.4%

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

      9. metadata-eval 68.4%

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

      10. metadata-eval 68.4%

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

      11. expm1-udef 69.4%

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

      12. expm1-log1p-u 68.7%

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

      13. pow2 68.7%

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

      14. metadata-eval 68.7%

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

   10. Applied egg-rr 68.7%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\frac{{x}^{2}}{\mathsf{hypot}\left(1, x\right) - -1}}\right), x\right) \]
   11. Step-by-step derivation

       1. unpow2 68.7%

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

       2. *-un-lft-identity 68.7%

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

       3. times-frac 68.7%

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

       4. sub-neg 68.7%

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

       5. metadata-eval 68.7%

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

       6. +-commutative 68.7%

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

   12. Applied egg-rr 68.7%

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

   13. Taylor expanded in x around 0 68.7%

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

   if 1.6000000000000001 < x

   1. Initial program 55.5%

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

      1. +-commutative 55.5%

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

      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 0 31.3%

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

      1. log1p-def 31.3%

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

      2. unpow1 31.3%

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

      3. sqr-pow 31.3%

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

      4. fabs-sqr 31.3%

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

      5. sqr-pow 31.3%

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

      6. unpow1 31.3%

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

   7. Simplified 31.3%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.0% 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 29.6%

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

   1. +-commutative 29.6%

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

   2. hypot-1-def 52.7%

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

3. Simplified 52.7%

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

   1. log1p-expm1-u 52.7%

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

   2. expm1-udef 52.7%

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

   3. add-exp-log 52.7%

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

   4. add-sqr-sqrt 25.6%

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

   5. fabs-sqr 25.6%

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

   6. add-sqr-sqrt 29.2%

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

6. Applied egg-rr 29.2%

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

   1. associate--l+ 75.4%

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

8. Simplified 75.4%

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

   1. sub-neg 75.4%

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

   2. flip-+ 64.8%

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

   3. hypot-1-def 64.8%

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

   4. hypot-1-def 64.8%

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

   5. add-sqr-sqrt 64.9%

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

   6. add-exp-log 65.3%

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

   7. log1p-udef 65.3%

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

   8. metadata-eval 65.3%

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

   9. metadata-eval 65.3%

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

   10. metadata-eval 65.3%

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

   11. expm1-udef 66.0%

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

   12. expm1-log1p-u 65.6%

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

   13. pow2 65.6%

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

   14. metadata-eval 65.6%

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

10. Applied egg-rr 65.6%

    \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\frac{{x}^{2}}{\mathsf{hypot}\left(1, x\right) - -1}}\right), x\right) \]
11. Step-by-step derivation

    1. unpow2 65.6%

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

    2. *-un-lft-identity 65.6%

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

    3. times-frac 76.1%

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

    4. sub-neg 76.1%

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

    5. metadata-eval 76.1%

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

    6. +-commutative 76.1%

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

12. Applied egg-rr 76.1%

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

13. Taylor expanded in x around 0 53.6%

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

14. Final simplification 53.6%

    \[\leadsto \mathsf{copysign}\left(x, x\right) \]
15. 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 2024010 
(FPCore (x)
  :name "Rust f64::asinh"
  :precision binary64

  :herbie-target
  (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)

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