Rust f64::asinh

Percentage Accurate: 30.6% → 99.4%

Time: 5.7s

Alternatives: 8

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 30.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))

C

double code(double x) {
	return copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
}

Java

public static double code(double x) {
	return Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
}

Python

def code(x):
	return math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)

Julia

function code(x)
	return copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
end

Wolfram

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

Alternative 1: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;\mathsf{copysign}\left(\left(-\log \left(x \cdot -2\right)\right) - \frac{0.25}{x \cdot x}, x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.06)
   (copysign (- (- (log (* x -2.0))) (/ 0.25 (* x x))) x)
   (if (<= x 2.0)
     (copysign (log1p (+ x (/ (* x x) (+ 2.0 (* (* x x) 0.5))))) x)
     (copysign (log (/ 0.5 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.06) {
		tmp = copysign((-log((x * -2.0)) - (0.25 / (x * x))), x);
	} else if (x <= 2.0) {
		tmp = copysign(log1p((x + ((x * x) / (2.0 + ((x * x) * 0.5))))), x);
	} else {
		tmp = copysign(log((0.5 / x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.06) {
		tmp = Math.copySign((-Math.log((x * -2.0)) - (0.25 / (x * x))), x);
	} else if (x <= 2.0) {
		tmp = Math.copySign(Math.log1p((x + ((x * x) / (2.0 + ((x * x) * 0.5))))), x);
	} else {
		tmp = Math.copySign(Math.log((0.5 / x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.06:
		tmp = math.copysign((-math.log((x * -2.0)) - (0.25 / (x * x))), x)
	elif x <= 2.0:
		tmp = math.copysign(math.log1p((x + ((x * x) / (2.0 + ((x * x) * 0.5))))), x)
	else:
		tmp = math.copysign(math.log((0.5 / x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.06)
		tmp = copysign(Float64(Float64(-log(Float64(x * -2.0))) - Float64(0.25 / Float64(x * x))), x);
	elseif (x <= 2.0)
		tmp = copysign(log1p(Float64(x + Float64(Float64(x * x) / Float64(2.0 + Float64(Float64(x * x) * 0.5))))), x);
	else
		tmp = copysign(log(Float64(0.5 / x)), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.06], N[With[{TMP1 = Abs[N[((-N[Log[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]) - N[(0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 2.0], N[With[{TMP1 = Abs[N[Log[1 + N[(x + N[(N[(x * x), $MachinePrecision] / N[(2.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < -1.0600000000000001

   1. Initial program 50.0%

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

      1. +-commutative 50.0%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. log-prod 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 5.4%

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

   5. Applied egg-rr 5.4%

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

      1. +-lft-identity 5.4%

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

   7. Simplified 5.4%

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

   8. Taylor expanded in x around -inf 99.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - 0.25 \cdot \frac{1}{{x}^{2}}}, x\right) \]
   9. Step-by-step derivation

      1. unpow2 99.8%

         \[\leadsto \mathsf{copysign}\left(\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - 0.25 \cdot \frac{1}{\color{blue}{x \cdot x}}, x\right) \]

      2. associate-*r/ 99.8%

         \[\leadsto \mathsf{copysign}\left(\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - \color{blue}{\frac{0.25 \cdot 1}{x \cdot x}}, x\right) \]

      3. metadata-eval 99.8%

         \[\leadsto \mathsf{copysign}\left(\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - \frac{\color{blue}{0.25}}{x \cdot x}, x\right) \]

   10. Simplified 99.8%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - \frac{0.25}{x \cdot x}}, x\right) \]
   11. Step-by-step derivation

       1. +-commutative 99.8%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log 0.5 + \log \left(\frac{-1}{x}\right)\right)} - \frac{0.25}{x \cdot x}, x\right) \]

       2. sum-log 100.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(0.5 \cdot \frac{-1}{x}\right)} - \frac{0.25}{x \cdot x}, x\right) \]

       3. frac-2neg 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(0.5 \cdot \color{blue}{\frac{--1}{-x}}\right) - \frac{0.25}{x \cdot x}, x\right) \]

       4. metadata-eval 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(0.5 \cdot \frac{\color{blue}{1}}{-x}\right) - \frac{0.25}{x \cdot x}, x\right) \]

       5. div-inv 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.5}{-x}\right)} - \frac{0.25}{x \cdot x}, x\right) \]

       6. metadata-eval 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{--0.5}}{-x}\right) - \frac{0.25}{x \cdot x}, x\right) \]

       7. frac-2neg 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)} - \frac{0.25}{x \cdot x}, x\right) \]

       8. clear-num 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{x}{-0.5}}\right)} - \frac{0.25}{x \cdot x}, x\right) \]

       9. log-div 100.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log 1 - \log \left(\frac{x}{-0.5}\right)\right)} - \frac{0.25}{x \cdot x}, x\right) \]

       10. metadata-eval 100.0%

           \[\leadsto \mathsf{copysign}\left(\left(\color{blue}{0} - \log \left(\frac{x}{-0.5}\right)\right) - \frac{0.25}{x \cdot x}, x\right) \]

       11. div-inv 100.0%

           \[\leadsto \mathsf{copysign}\left(\left(0 - \log \color{blue}{\left(x \cdot \frac{1}{-0.5}\right)}\right) - \frac{0.25}{x \cdot x}, x\right) \]

       12. metadata-eval 100.0%

           \[\leadsto \mathsf{copysign}\left(\left(0 - \log \left(x \cdot \color{blue}{-2}\right)\right) - \frac{0.25}{x \cdot x}, x\right) \]

   12. Applied egg-rr 100.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(0 - \log \left(x \cdot -2\right)\right)} - \frac{0.25}{x \cdot x}, x\right) \]
   13. Step-by-step derivation

       1. neg-sub0 100.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(-\log \left(x \cdot -2\right)\right)} - \frac{0.25}{x \cdot x}, x\right) \]

   14. Simplified 100.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(-\log \left(x \cdot -2\right)\right)} - \frac{0.25}{x \cdot x}, x\right) \]

   if -1.0600000000000001 < x < 2

   1. Initial program 8.1%

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

      1. +-commutative 8.1%

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

      2. hypot-1-def 8.1%

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

   3. Simplified 8.1%

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

      1. log1p-expm1-u 8.1%

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

      2. expm1-udef 8.1%

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

      3. add-exp-log 8.1%

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

      4. add-sqr-sqrt 3.6%

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

      5. fabs-sqr 3.6%

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

      6. add-sqr-sqrt 8.0%

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

   5. Applied egg-rr 8.0%

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

      1. associate--l+ 98.9%

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

   7. Simplified 98.9%

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

      1. flip-- 98.9%

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

      2. div-inv 98.9%

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

      3. metadata-eval 98.9%

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

      4. sub-neg 98.9%

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

      5. hypot-udef 98.9%

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

      6. hypot-udef 98.9%

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

      7. add-sqr-sqrt 98.9%

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

      8. metadata-eval 98.9%

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

      9. metadata-eval 98.9%

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

      10. +-commutative 98.9%

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

   9. Applied egg-rr 98.9%

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

       1. associate-*r/ 98.9%

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

       2. *-rgt-identity 98.9%

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

       3. +-commutative 98.9%

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

       4. unpow2 98.9%

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

       5. associate-+r+ 100.0%

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

       6. metadata-eval 100.0%

          \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{\color{blue}{0} + {x}^{2}}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

       7. +-lft-identity 100.0%

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

       8. unpow2 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around 0 100.0%

       \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{\color{blue}{2 + 0.5 \cdot {x}^{2}}}\right), x\right) \]
   13. Step-by-step derivation

       1. unpow2 100.0%

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

   14. Simplified 100.0%

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

   if 2 < x

   1. Initial program 59.0%

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

      1. +-commutative 59.0%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

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

   5. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around -inf 3.1%

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

      1. +-commutative 3.1%

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

      2. associate--r+ 3.1%

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

      3. sub-neg 3.1%

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

      4. sub-neg 3.1%

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

      5. metadata-eval 3.1%

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

      6. +-commutative 3.1%

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

      7. mul-1-neg 3.1%

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

      8. unsub-neg 3.1%

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

      9. associate-*r/ 3.1%

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

      10. metadata-eval 3.1%

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

      11. distribute-neg-frac 3.1%

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

      12. metadata-eval 3.1%

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

   10. Simplified 3.1%

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

       1. expm1-log1p-u 3.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x + \left(\left(-1 - x\right) + \frac{-0.5}{x}\right)\right)\right)\right)}, x\right) \]

       2. expm1-udef 3.0%

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

       3. +-commutative 3.0%

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

       4. associate-+r- 3.0%

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

   12. Applied egg-rr 3.0%

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

       1. expm1-def 3.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x + \left(\left(\frac{-0.5}{x} + -1\right) - x\right)\right)\right)\right)}, x\right) \]

       2. expm1-log1p 3.1%

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

       3. +-commutative 3.1%

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

       4. associate-+l- 3.0%

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

       5. metadata-eval 3.0%

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

       6. distribute-neg-frac 3.0%

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

       7. metadata-eval 3.0%

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

       8. associate-*r/ 3.0%

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

       9. metadata-eval 3.0%

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

       10. distribute-neg-in 3.0%

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

       11. sub-neg 3.0%

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

       12. +-inverses 3.0%

           \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(-\left(0.5 \cdot \frac{1}{x} + 1\right)\right) + \left(-\color{blue}{0}\right)\right), x\right) \]

       13. metadata-eval 3.0%

           \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(-\left(0.5 \cdot \frac{1}{x} + 1\right)\right) + \color{blue}{0}\right), x\right) \]

       14. +-rgt-identity 3.0%

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

       15. +-commutative 3.0%

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

       16. distribute-neg-in 3.0%

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

       17. metadata-eval 3.0%

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

       18. associate-*r/ 3.0%

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

       19. metadata-eval 3.0%

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

       20. distribute-neg-frac 3.0%

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

       21. metadata-eval 3.0%

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

   14. Simplified 3.0%

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

       1. expm1-log1p-u 0.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(-1 + \frac{-0.5}{x}\right)\right)\right)}, x\right) \]

       2. expm1-udef 0.0%

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

       3. log1p-udef 0.0%

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

       4. associate-+r+ 0.0%

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

       5. metadata-eval 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(\color{blue}{0} + \frac{-0.5}{x}\right)\right)} - 1, x\right) \]

       6. add-sqr-sqrt 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \color{blue}{\sqrt{\frac{-0.5}{x}} \cdot \sqrt{\frac{-0.5}{x}}}\right)\right)} - 1, x\right) \]

       7. sqrt-unprod 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \color{blue}{\sqrt{\frac{-0.5}{x} \cdot \frac{-0.5}{x}}}\right)\right)} - 1, x\right) \]

       8. frac-times 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \sqrt{\color{blue}{\frac{-0.5 \cdot -0.5}{x \cdot x}}}\right)\right)} - 1, x\right) \]

       9. metadata-eval 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \sqrt{\frac{\color{blue}{0.25}}{x \cdot x}}\right)\right)} - 1, x\right) \]

       10. sqrt-div 0.0%

           \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \color{blue}{\frac{\sqrt{0.25}}{\sqrt{x \cdot x}}}\right)\right)} - 1, x\right) \]

       11. metadata-eval 0.0%

           \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \frac{\color{blue}{0.5}}{\sqrt{x \cdot x}}\right)\right)} - 1, x\right) \]

       12. sqrt-prod 0.0%

           \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \frac{0.5}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)\right)} - 1, x\right) \]

       13. add-sqr-sqrt 0.0%

           \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \frac{0.5}{\color{blue}{x}}\right)\right)} - 1, x\right) \]

   16. Applied egg-rr 0.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{e^{\mathsf{log1p}\left(\log \left(0 + \frac{0.5}{x}\right)\right)} - 1}, x\right) \]
   17. Step-by-step derivation

       1. expm1-def 0.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(0 + \frac{0.5}{x}\right)\right)\right)}, x\right) \]

       2. expm1-log1p 100.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(0 + \frac{0.5}{x}\right)}, x\right) \]

       3. +-lft-identity 100.0%

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

   18. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;\mathsf{copysign}\left(\left(-\log \left(x \cdot -2\right)\right) - \frac{0.25}{x \cdot x}, x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.0005:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) -0.0005)
   (copysign (log (+ (fabs x) (hypot 1.0 x))) x)
   (copysign (log1p (+ x (+ (hypot 1.0 x) -1.0))) x)))

C

double code(double x) {
	double tmp;
	if (copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x) <= -0.0005) {
		tmp = copysign(log((fabs(x) + hypot(1.0, x))), x);
	} else {
		tmp = copysign(log1p((x + (hypot(1.0, x) + -1.0))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x) <= -0.0005) {
		tmp = Math.copySign(Math.log((Math.abs(x) + Math.hypot(1.0, x))), x);
	} else {
		tmp = Math.copySign(Math.log1p((x + (Math.hypot(1.0, x) + -1.0))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x) <= -0.0005:
		tmp = math.copysign(math.log((math.fabs(x) + math.hypot(1.0, x))), x)
	else:
		tmp = math.copysign(math.log1p((x + (math.hypot(1.0, x) + -1.0))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x) <= -0.0005)
		tmp = copysign(log(Float64(abs(x) + hypot(1.0, x))), x);
	else
		tmp = copysign(log1p(Float64(x + Float64(hypot(1.0, x) + -1.0))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], -0.0005], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + N[(x + N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < -5.0000000000000001e-4

   1. Initial program 50.7%

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

      1. +-commutative 50.7%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   if -5.0000000000000001e-4 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x)

   1. Initial program 28.0%

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

      1. +-commutative 28.0%

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

      2. hypot-1-def 44.4%

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

   3. Simplified 44.4%

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

      1. log1p-expm1-u 44.4%

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

      2. expm1-udef 44.4%

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

      3. add-exp-log 44.4%

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

      4. add-sqr-sqrt 42.2%

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

      5. fabs-sqr 42.2%

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

      6. add-sqr-sqrt 44.4%

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

   5. Applied egg-rr 44.4%

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

      1. associate--l+ 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.0005:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]

Alternative 3: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -310:\\ \;\;\;\;\mathsf{copysign}\left(\left(-\log \left(x \cdot -2\right)\right) - \frac{0.25}{x \cdot x}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -310.0)
   (copysign (- (- (log (* x -2.0))) (/ 0.25 (* x x))) x)
   (copysign (log1p (+ x (+ (hypot 1.0 x) -1.0))) x)))

C

double code(double x) {
	double tmp;
	if (x <= -310.0) {
		tmp = copysign((-log((x * -2.0)) - (0.25 / (x * x))), x);
	} else {
		tmp = copysign(log1p((x + (hypot(1.0, x) + -1.0))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -310.0) {
		tmp = Math.copySign((-Math.log((x * -2.0)) - (0.25 / (x * x))), x);
	} else {
		tmp = Math.copySign(Math.log1p((x + (Math.hypot(1.0, x) + -1.0))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -310.0:
		tmp = math.copysign((-math.log((x * -2.0)) - (0.25 / (x * x))), x)
	else:
		tmp = math.copysign(math.log1p((x + (math.hypot(1.0, x) + -1.0))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -310.0)
		tmp = copysign(Float64(Float64(-log(Float64(x * -2.0))) - Float64(0.25 / Float64(x * x))), x);
	else
		tmp = copysign(log1p(Float64(x + Float64(hypot(1.0, x) + -1.0))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -310.0], N[With[{TMP1 = Abs[N[((-N[Log[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]) - N[(0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + N[(x + N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -310

   1. Initial program 50.0%

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

      1. +-commutative 50.0%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. log-prod 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 5.4%

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

   5. Applied egg-rr 5.4%

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

      1. +-lft-identity 5.4%

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

   7. Simplified 5.4%

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

   8. Taylor expanded in x around -inf 99.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - 0.25 \cdot \frac{1}{{x}^{2}}}, x\right) \]
   9. Step-by-step derivation

      1. unpow2 99.8%

         \[\leadsto \mathsf{copysign}\left(\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - 0.25 \cdot \frac{1}{\color{blue}{x \cdot x}}, x\right) \]

      2. associate-*r/ 99.8%

         \[\leadsto \mathsf{copysign}\left(\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - \color{blue}{\frac{0.25 \cdot 1}{x \cdot x}}, x\right) \]

      3. metadata-eval 99.8%

         \[\leadsto \mathsf{copysign}\left(\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - \frac{\color{blue}{0.25}}{x \cdot x}, x\right) \]

   10. Simplified 99.8%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - \frac{0.25}{x \cdot x}}, x\right) \]
   11. Step-by-step derivation

       1. +-commutative 99.8%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log 0.5 + \log \left(\frac{-1}{x}\right)\right)} - \frac{0.25}{x \cdot x}, x\right) \]

       2. sum-log 100.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(0.5 \cdot \frac{-1}{x}\right)} - \frac{0.25}{x \cdot x}, x\right) \]

       3. frac-2neg 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(0.5 \cdot \color{blue}{\frac{--1}{-x}}\right) - \frac{0.25}{x \cdot x}, x\right) \]

       4. metadata-eval 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(0.5 \cdot \frac{\color{blue}{1}}{-x}\right) - \frac{0.25}{x \cdot x}, x\right) \]

       5. div-inv 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.5}{-x}\right)} - \frac{0.25}{x \cdot x}, x\right) \]

       6. metadata-eval 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{--0.5}}{-x}\right) - \frac{0.25}{x \cdot x}, x\right) \]

       7. frac-2neg 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)} - \frac{0.25}{x \cdot x}, x\right) \]

       8. clear-num 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{x}{-0.5}}\right)} - \frac{0.25}{x \cdot x}, x\right) \]

       9. log-div 100.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log 1 - \log \left(\frac{x}{-0.5}\right)\right)} - \frac{0.25}{x \cdot x}, x\right) \]

       10. metadata-eval 100.0%

           \[\leadsto \mathsf{copysign}\left(\left(\color{blue}{0} - \log \left(\frac{x}{-0.5}\right)\right) - \frac{0.25}{x \cdot x}, x\right) \]

       11. div-inv 100.0%

           \[\leadsto \mathsf{copysign}\left(\left(0 - \log \color{blue}{\left(x \cdot \frac{1}{-0.5}\right)}\right) - \frac{0.25}{x \cdot x}, x\right) \]

       12. metadata-eval 100.0%

           \[\leadsto \mathsf{copysign}\left(\left(0 - \log \left(x \cdot \color{blue}{-2}\right)\right) - \frac{0.25}{x \cdot x}, x\right) \]

   12. Applied egg-rr 100.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(0 - \log \left(x \cdot -2\right)\right)} - \frac{0.25}{x \cdot x}, x\right) \]
   13. Step-by-step derivation

       1. neg-sub0 100.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(-\log \left(x \cdot -2\right)\right)} - \frac{0.25}{x \cdot x}, x\right) \]

   14. Simplified 100.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(-\log \left(x \cdot -2\right)\right)} - \frac{0.25}{x \cdot x}, x\right) \]

   if -310 < x

   1. Initial program 28.3%

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

      1. +-commutative 28.3%

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

      2. hypot-1-def 44.6%

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

   3. Simplified 44.6%

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

      1. log1p-expm1-u 44.6%

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

      2. expm1-udef 44.6%

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

      3. add-exp-log 44.6%

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

      4. add-sqr-sqrt 41.9%

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

      5. fabs-sqr 41.9%

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

      6. add-sqr-sqrt 44.6%

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

   5. Applied egg-rr 44.6%

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

      1. associate--l+ 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -310:\\ \;\;\;\;\mathsf{copysign}\left(\left(-\log \left(x \cdot -2\right)\right) - \frac{0.25}{x \cdot x}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]

Alternative 4: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\mathsf{copysign}\left(\left(-\log \left(x \cdot -2\right)\right) - \frac{0.25}{x \cdot x}, x\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (copysign (- (- (log (* x -2.0))) (/ 0.25 (* x x))) x)
   (if (<= x 1.3)
     (copysign (+ x (* -0.16666666666666666 (* x (* x x)))) x)
     (copysign (log (/ 0.5 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = copysign((-log((x * -2.0)) - (0.25 / (x * x))), x);
	} else if (x <= 1.3) {
		tmp = copysign((x + (-0.16666666666666666 * (x * (x * x)))), x);
	} else {
		tmp = copysign(log((0.5 / x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = Math.copySign((-Math.log((x * -2.0)) - (0.25 / (x * x))), x);
	} else if (x <= 1.3) {
		tmp = Math.copySign((x + (-0.16666666666666666 * (x * (x * x)))), x);
	} else {
		tmp = Math.copySign(Math.log((0.5 / x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = math.copysign((-math.log((x * -2.0)) - (0.25 / (x * x))), x)
	elif x <= 1.3:
		tmp = math.copysign((x + (-0.16666666666666666 * (x * (x * x)))), x)
	else:
		tmp = math.copysign(math.log((0.5 / x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = copysign(Float64(Float64(-log(Float64(x * -2.0))) - Float64(0.25 / Float64(x * x))), x);
	elseif (x <= 1.3)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * Float64(x * Float64(x * x)))), x);
	else
		tmp = copysign(log(Float64(0.5 / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = sign(x) * abs((-log((x * -2.0)) - (0.25 / (x * x))));
	elseif (x <= 1.3)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x * (x * x)))));
	else
		tmp = sign(x) * abs(log((0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[With[{TMP1 = Abs[N[((-N[Log[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]) - N[(0.25 / N[(x * 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[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $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 < -1.05000000000000004

   1. Initial program 50.0%

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

      1. +-commutative 50.0%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. log-prod 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 5.4%

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

   5. Applied egg-rr 5.4%

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

      1. +-lft-identity 5.4%

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

   7. Simplified 5.4%

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

   8. Taylor expanded in x around -inf 99.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - 0.25 \cdot \frac{1}{{x}^{2}}}, x\right) \]
   9. Step-by-step derivation

      1. unpow2 99.8%

         \[\leadsto \mathsf{copysign}\left(\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - 0.25 \cdot \frac{1}{\color{blue}{x \cdot x}}, x\right) \]

      2. associate-*r/ 99.8%

         \[\leadsto \mathsf{copysign}\left(\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - \color{blue}{\frac{0.25 \cdot 1}{x \cdot x}}, x\right) \]

      3. metadata-eval 99.8%

         \[\leadsto \mathsf{copysign}\left(\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - \frac{\color{blue}{0.25}}{x \cdot x}, x\right) \]

   10. Simplified 99.8%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - \frac{0.25}{x \cdot x}}, x\right) \]
   11. Step-by-step derivation

       1. +-commutative 99.8%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log 0.5 + \log \left(\frac{-1}{x}\right)\right)} - \frac{0.25}{x \cdot x}, x\right) \]

       2. sum-log 100.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(0.5 \cdot \frac{-1}{x}\right)} - \frac{0.25}{x \cdot x}, x\right) \]

       3. frac-2neg 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(0.5 \cdot \color{blue}{\frac{--1}{-x}}\right) - \frac{0.25}{x \cdot x}, x\right) \]

       4. metadata-eval 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(0.5 \cdot \frac{\color{blue}{1}}{-x}\right) - \frac{0.25}{x \cdot x}, x\right) \]

       5. div-inv 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.5}{-x}\right)} - \frac{0.25}{x \cdot x}, x\right) \]

       6. metadata-eval 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\color{blue}{--0.5}}{-x}\right) - \frac{0.25}{x \cdot x}, x\right) \]

       7. frac-2neg 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)} - \frac{0.25}{x \cdot x}, x\right) \]

       8. clear-num 100.0%

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{x}{-0.5}}\right)} - \frac{0.25}{x \cdot x}, x\right) \]

       9. log-div 100.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log 1 - \log \left(\frac{x}{-0.5}\right)\right)} - \frac{0.25}{x \cdot x}, x\right) \]

       10. metadata-eval 100.0%

           \[\leadsto \mathsf{copysign}\left(\left(\color{blue}{0} - \log \left(\frac{x}{-0.5}\right)\right) - \frac{0.25}{x \cdot x}, x\right) \]

       11. div-inv 100.0%

           \[\leadsto \mathsf{copysign}\left(\left(0 - \log \color{blue}{\left(x \cdot \frac{1}{-0.5}\right)}\right) - \frac{0.25}{x \cdot x}, x\right) \]

       12. metadata-eval 100.0%

           \[\leadsto \mathsf{copysign}\left(\left(0 - \log \left(x \cdot \color{blue}{-2}\right)\right) - \frac{0.25}{x \cdot x}, x\right) \]

   12. Applied egg-rr 100.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(0 - \log \left(x \cdot -2\right)\right)} - \frac{0.25}{x \cdot x}, x\right) \]
   13. Step-by-step derivation

       1. neg-sub0 100.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(-\log \left(x \cdot -2\right)\right)} - \frac{0.25}{x \cdot x}, x\right) \]

   14. Simplified 100.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(-\log \left(x \cdot -2\right)\right)} - \frac{0.25}{x \cdot x}, x\right) \]

   if -1.05000000000000004 < x < 1.30000000000000004

   1. Initial program 8.1%

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

      1. +-commutative 8.1%

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

      2. hypot-1-def 8.1%

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

   3. Simplified 8.1%

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

      1. *-un-lft-identity 8.1%

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

      2. log-prod 8.1%

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

      3. metadata-eval 8.1%

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

      4. *-un-lft-identity 8.1%

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

      5. *-un-lft-identity 8.1%

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

      6. add-sqr-sqrt 3.6%

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

      7. fabs-sqr 3.6%

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

      8. add-sqr-sqrt 8.0%

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

   5. Applied egg-rr 8.0%

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

      1. +-lft-identity 8.0%

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

   7. Simplified 8.0%

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

   8. Taylor expanded in x around 0 99.9%

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

      1. unpow3 99.9%

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

   10. Applied egg-rr 99.9%

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

   if 1.30000000000000004 < x

   1. Initial program 59.0%

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

      1. +-commutative 59.0%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

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

   5. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around -inf 3.1%

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

      1. +-commutative 3.1%

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

      2. associate--r+ 3.1%

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

      3. sub-neg 3.1%

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

      4. sub-neg 3.1%

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

      5. metadata-eval 3.1%

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

      6. +-commutative 3.1%

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

      7. mul-1-neg 3.1%

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

      8. unsub-neg 3.1%

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

      9. associate-*r/ 3.1%

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

      10. metadata-eval 3.1%

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

      11. distribute-neg-frac 3.1%

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

      12. metadata-eval 3.1%

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

   10. Simplified 3.1%

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

       1. expm1-log1p-u 3.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x + \left(\left(-1 - x\right) + \frac{-0.5}{x}\right)\right)\right)\right)}, x\right) \]

       2. expm1-udef 3.0%

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

       3. +-commutative 3.0%

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

       4. associate-+r- 3.0%

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

   12. Applied egg-rr 3.0%

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

       1. expm1-def 3.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x + \left(\left(\frac{-0.5}{x} + -1\right) - x\right)\right)\right)\right)}, x\right) \]

       2. expm1-log1p 3.1%

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

       3. +-commutative 3.1%

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

       4. associate-+l- 3.0%

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

       5. metadata-eval 3.0%

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

       6. distribute-neg-frac 3.0%

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

       7. metadata-eval 3.0%

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

       8. associate-*r/ 3.0%

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

       9. metadata-eval 3.0%

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

       10. distribute-neg-in 3.0%

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

       11. sub-neg 3.0%

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

       12. +-inverses 3.0%

           \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(-\left(0.5 \cdot \frac{1}{x} + 1\right)\right) + \left(-\color{blue}{0}\right)\right), x\right) \]

       13. metadata-eval 3.0%

           \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(-\left(0.5 \cdot \frac{1}{x} + 1\right)\right) + \color{blue}{0}\right), x\right) \]

       14. +-rgt-identity 3.0%

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

       15. +-commutative 3.0%

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

       16. distribute-neg-in 3.0%

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

       17. metadata-eval 3.0%

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

       18. associate-*r/ 3.0%

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

       19. metadata-eval 3.0%

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

       20. distribute-neg-frac 3.0%

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

       21. metadata-eval 3.0%

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

   14. Simplified 3.0%

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

       1. expm1-log1p-u 0.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(-1 + \frac{-0.5}{x}\right)\right)\right)}, x\right) \]

       2. expm1-udef 0.0%

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

       3. log1p-udef 0.0%

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

       4. associate-+r+ 0.0%

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

       5. metadata-eval 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(\color{blue}{0} + \frac{-0.5}{x}\right)\right)} - 1, x\right) \]

       6. add-sqr-sqrt 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \color{blue}{\sqrt{\frac{-0.5}{x}} \cdot \sqrt{\frac{-0.5}{x}}}\right)\right)} - 1, x\right) \]

       7. sqrt-unprod 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \color{blue}{\sqrt{\frac{-0.5}{x} \cdot \frac{-0.5}{x}}}\right)\right)} - 1, x\right) \]

       8. frac-times 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \sqrt{\color{blue}{\frac{-0.5 \cdot -0.5}{x \cdot x}}}\right)\right)} - 1, x\right) \]

       9. metadata-eval 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \sqrt{\frac{\color{blue}{0.25}}{x \cdot x}}\right)\right)} - 1, x\right) \]

       10. sqrt-div 0.0%

           \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \color{blue}{\frac{\sqrt{0.25}}{\sqrt{x \cdot x}}}\right)\right)} - 1, x\right) \]

       11. metadata-eval 0.0%

           \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \frac{\color{blue}{0.5}}{\sqrt{x \cdot x}}\right)\right)} - 1, x\right) \]

       12. sqrt-prod 0.0%

           \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \frac{0.5}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)\right)} - 1, x\right) \]

       13. add-sqr-sqrt 0.0%

           \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \frac{0.5}{\color{blue}{x}}\right)\right)} - 1, x\right) \]

   16. Applied egg-rr 0.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{e^{\mathsf{log1p}\left(\log \left(0 + \frac{0.5}{x}\right)\right)} - 1}, x\right) \]
   17. Step-by-step derivation

       1. expm1-def 0.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(0 + \frac{0.5}{x}\right)\right)\right)}, x\right) \]

       2. expm1-log1p 100.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(0 + \frac{0.5}{x}\right)}, x\right) \]

       3. +-lft-identity 100.0%

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

   18. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\mathsf{copysign}\left(\left(-\log \left(x \cdot -2\right)\right) - \frac{0.25}{x \cdot x}, x\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 5: 82.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.0)
   (copysign (- (log (/ -1.0 x))) x)
   (if (<= x 1.3)
     (copysign (+ x (* -0.16666666666666666 (* x (* x x)))) x)
     (copysign (log (/ 0.5 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = copysign(-log((-1.0 / x)), x);
	} else if (x <= 1.3) {
		tmp = copysign((x + (-0.16666666666666666 * (x * (x * x)))), x);
	} else {
		tmp = copysign(log((0.5 / x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = Math.copySign(-Math.log((-1.0 / x)), x);
	} else if (x <= 1.3) {
		tmp = Math.copySign((x + (-0.16666666666666666 * (x * (x * x)))), x);
	} else {
		tmp = Math.copySign(Math.log((0.5 / x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.0:
		tmp = math.copysign(-math.log((-1.0 / x)), x)
	elif x <= 1.3:
		tmp = math.copysign((x + (-0.16666666666666666 * (x * (x * x)))), x)
	else:
		tmp = math.copysign(math.log((0.5 / x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.0)
		tmp = copysign(Float64(-log(Float64(-1.0 / x))), x);
	elseif (x <= 1.3)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * Float64(x * Float64(x * x)))), x);
	else
		tmp = copysign(log(Float64(0.5 / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.0)
		tmp = sign(x) * abs(-log((-1.0 / x)));
	elseif (x <= 1.3)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x * (x * x)))));
	else
		tmp = sign(x) * abs(log((0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.0], N[With[{TMP1 = Abs[(-N[Log[N[(-1.0 / 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[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $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 < -2

   1. Initial program 50.0%

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

      1. +-commutative 50.0%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 31.5%

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

      1. mul-1-neg 31.5%

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

   6. Simplified 31.5%

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

   if -2 < x < 1.30000000000000004

   1. Initial program 8.1%

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

      1. +-commutative 8.1%

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

      2. hypot-1-def 8.1%

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

   3. Simplified 8.1%

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

      1. *-un-lft-identity 8.1%

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

      2. log-prod 8.1%

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

      3. metadata-eval 8.1%

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

      4. *-un-lft-identity 8.1%

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

      5. *-un-lft-identity 8.1%

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

      6. add-sqr-sqrt 3.6%

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

      7. fabs-sqr 3.6%

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

      8. add-sqr-sqrt 8.0%

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

   5. Applied egg-rr 8.0%

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

      1. +-lft-identity 8.0%

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

   7. Simplified 8.0%

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

   8. Taylor expanded in x around 0 99.9%

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

      1. unpow3 99.9%

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

   10. Applied egg-rr 99.9%

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

   if 1.30000000000000004 < x

   1. Initial program 59.0%

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

      1. +-commutative 59.0%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

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

   5. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around -inf 3.1%

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

      1. +-commutative 3.1%

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

      2. associate--r+ 3.1%

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

      3. sub-neg 3.1%

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

      4. sub-neg 3.1%

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

      5. metadata-eval 3.1%

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

      6. +-commutative 3.1%

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

      7. mul-1-neg 3.1%

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

      8. unsub-neg 3.1%

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

      9. associate-*r/ 3.1%

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

      10. metadata-eval 3.1%

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

      11. distribute-neg-frac 3.1%

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

      12. metadata-eval 3.1%

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

   10. Simplified 3.1%

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

       1. expm1-log1p-u 3.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x + \left(\left(-1 - x\right) + \frac{-0.5}{x}\right)\right)\right)\right)}, x\right) \]

       2. expm1-udef 3.0%

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

       3. +-commutative 3.0%

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

       4. associate-+r- 3.0%

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

   12. Applied egg-rr 3.0%

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

       1. expm1-def 3.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x + \left(\left(\frac{-0.5}{x} + -1\right) - x\right)\right)\right)\right)}, x\right) \]

       2. expm1-log1p 3.1%

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

       3. +-commutative 3.1%

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

       4. associate-+l- 3.0%

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

       5. metadata-eval 3.0%

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

       6. distribute-neg-frac 3.0%

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

       7. metadata-eval 3.0%

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

       8. associate-*r/ 3.0%

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

       9. metadata-eval 3.0%

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

       10. distribute-neg-in 3.0%

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

       11. sub-neg 3.0%

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

       12. +-inverses 3.0%

           \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(-\left(0.5 \cdot \frac{1}{x} + 1\right)\right) + \left(-\color{blue}{0}\right)\right), x\right) \]

       13. metadata-eval 3.0%

           \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(-\left(0.5 \cdot \frac{1}{x} + 1\right)\right) + \color{blue}{0}\right), x\right) \]

       14. +-rgt-identity 3.0%

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

       15. +-commutative 3.0%

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

       16. distribute-neg-in 3.0%

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

       17. metadata-eval 3.0%

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

       18. associate-*r/ 3.0%

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

       19. metadata-eval 3.0%

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

       20. distribute-neg-frac 3.0%

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

       21. metadata-eval 3.0%

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

   14. Simplified 3.0%

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

       1. expm1-log1p-u 0.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(-1 + \frac{-0.5}{x}\right)\right)\right)}, x\right) \]

       2. expm1-udef 0.0%

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

       3. log1p-udef 0.0%

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

       4. associate-+r+ 0.0%

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

       5. metadata-eval 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(\color{blue}{0} + \frac{-0.5}{x}\right)\right)} - 1, x\right) \]

       6. add-sqr-sqrt 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \color{blue}{\sqrt{\frac{-0.5}{x}} \cdot \sqrt{\frac{-0.5}{x}}}\right)\right)} - 1, x\right) \]

       7. sqrt-unprod 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \color{blue}{\sqrt{\frac{-0.5}{x} \cdot \frac{-0.5}{x}}}\right)\right)} - 1, x\right) \]

       8. frac-times 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \sqrt{\color{blue}{\frac{-0.5 \cdot -0.5}{x \cdot x}}}\right)\right)} - 1, x\right) \]

       9. metadata-eval 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \sqrt{\frac{\color{blue}{0.25}}{x \cdot x}}\right)\right)} - 1, x\right) \]

       10. sqrt-div 0.0%

           \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \color{blue}{\frac{\sqrt{0.25}}{\sqrt{x \cdot x}}}\right)\right)} - 1, x\right) \]

       11. metadata-eval 0.0%

           \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \frac{\color{blue}{0.5}}{\sqrt{x \cdot x}}\right)\right)} - 1, x\right) \]

       12. sqrt-prod 0.0%

           \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \frac{0.5}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)\right)} - 1, x\right) \]

       13. add-sqr-sqrt 0.0%

           \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \frac{0.5}{\color{blue}{x}}\right)\right)} - 1, x\right) \]

   16. Applied egg-rr 0.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{e^{\mathsf{log1p}\left(\log \left(0 + \frac{0.5}{x}\right)\right)} - 1}, x\right) \]
   17. Step-by-step derivation

       1. expm1-def 0.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(0 + \frac{0.5}{x}\right)\right)\right)}, x\right) \]

       2. expm1-log1p 100.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(0 + \frac{0.5}{x}\right)}, x\right) \]

       3. +-lft-identity 100.0%

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

   18. Simplified 100.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 6: 75.3% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.3) (copysign x x) (copysign (log (/ 0.5 x)) x)))

C

double code(double x) {
	double tmp;
	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 <= 1.3) {
		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 <= 1.3:
		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 <= 1.3)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float64(0.5 / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (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, 1.3], 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 2 regimes

2. if x < 1.30000000000000004

   1. Initial program 22.2%

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

      1. +-commutative 22.2%

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

      2. hypot-1-def 39.1%

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

   3. Simplified 39.1%

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

   4. Taylor expanded in x around 0 15.6%

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

      1. expm1-log1p-u 15.6%

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

      2. expm1-udef 15.6%

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

      3. log1p-udef 15.6%

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

      4. add-exp-log 15.6%

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

      5. log1p-def 15.6%

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

      6. add-sqr-sqrt 2.4%

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

      7. fabs-sqr 2.4%

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

      8. add-sqr-sqrt 5.0%

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

   6. Applied egg-rr 5.0%

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

      1. associate--l+ 5.0%

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

      2. +-commutative 5.0%

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

      3. sub-neg 5.0%

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

      4. metadata-eval 5.0%

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

   8. Applied egg-rr 5.0%

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

   9. Taylor expanded in x around 0 67.8%

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

   if 1.30000000000000004 < x

   1. Initial program 59.0%

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

      1. +-commutative 59.0%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

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

   5. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around -inf 3.1%

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

      1. +-commutative 3.1%

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

      2. associate--r+ 3.1%

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

      3. sub-neg 3.1%

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

      4. sub-neg 3.1%

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

      5. metadata-eval 3.1%

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

      6. +-commutative 3.1%

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

      7. mul-1-neg 3.1%

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

      8. unsub-neg 3.1%

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

      9. associate-*r/ 3.1%

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

      10. metadata-eval 3.1%

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

      11. distribute-neg-frac 3.1%

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

      12. metadata-eval 3.1%

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

   10. Simplified 3.1%

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

       1. expm1-log1p-u 3.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x + \left(\left(-1 - x\right) + \frac{-0.5}{x}\right)\right)\right)\right)}, x\right) \]

       2. expm1-udef 3.0%

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

       3. +-commutative 3.0%

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

       4. associate-+r- 3.0%

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

   12. Applied egg-rr 3.0%

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

       1. expm1-def 3.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(x + \left(\left(\frac{-0.5}{x} + -1\right) - x\right)\right)\right)\right)}, x\right) \]

       2. expm1-log1p 3.1%

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

       3. +-commutative 3.1%

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

       4. associate-+l- 3.0%

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

       5. metadata-eval 3.0%

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

       6. distribute-neg-frac 3.0%

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

       7. metadata-eval 3.0%

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

       8. associate-*r/ 3.0%

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

       9. metadata-eval 3.0%

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

       10. distribute-neg-in 3.0%

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

       11. sub-neg 3.0%

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

       12. +-inverses 3.0%

           \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(-\left(0.5 \cdot \frac{1}{x} + 1\right)\right) + \left(-\color{blue}{0}\right)\right), x\right) \]

       13. metadata-eval 3.0%

           \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(-\left(0.5 \cdot \frac{1}{x} + 1\right)\right) + \color{blue}{0}\right), x\right) \]

       14. +-rgt-identity 3.0%

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

       15. +-commutative 3.0%

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

       16. distribute-neg-in 3.0%

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

       17. metadata-eval 3.0%

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

       18. associate-*r/ 3.0%

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

       19. metadata-eval 3.0%

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

       20. distribute-neg-frac 3.0%

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

       21. metadata-eval 3.0%

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

   14. Simplified 3.0%

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

       1. expm1-log1p-u 0.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(-1 + \frac{-0.5}{x}\right)\right)\right)}, x\right) \]

       2. expm1-udef 0.0%

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

       3. log1p-udef 0.0%

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

       4. associate-+r+ 0.0%

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

       5. metadata-eval 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(\color{blue}{0} + \frac{-0.5}{x}\right)\right)} - 1, x\right) \]

       6. add-sqr-sqrt 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \color{blue}{\sqrt{\frac{-0.5}{x}} \cdot \sqrt{\frac{-0.5}{x}}}\right)\right)} - 1, x\right) \]

       7. sqrt-unprod 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \color{blue}{\sqrt{\frac{-0.5}{x} \cdot \frac{-0.5}{x}}}\right)\right)} - 1, x\right) \]

       8. frac-times 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \sqrt{\color{blue}{\frac{-0.5 \cdot -0.5}{x \cdot x}}}\right)\right)} - 1, x\right) \]

       9. metadata-eval 0.0%

          \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \sqrt{\frac{\color{blue}{0.25}}{x \cdot x}}\right)\right)} - 1, x\right) \]

       10. sqrt-div 0.0%

           \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \color{blue}{\frac{\sqrt{0.25}}{\sqrt{x \cdot x}}}\right)\right)} - 1, x\right) \]

       11. metadata-eval 0.0%

           \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \frac{\color{blue}{0.5}}{\sqrt{x \cdot x}}\right)\right)} - 1, x\right) \]

       12. sqrt-prod 0.0%

           \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \frac{0.5}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)\right)} - 1, x\right) \]

       13. add-sqr-sqrt 0.0%

           \[\leadsto \mathsf{copysign}\left(e^{\mathsf{log1p}\left(\log \left(0 + \frac{0.5}{\color{blue}{x}}\right)\right)} - 1, x\right) \]

   16. Applied egg-rr 0.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{e^{\mathsf{log1p}\left(\log \left(0 + \frac{0.5}{x}\right)\right)} - 1}, x\right) \]
   17. Step-by-step derivation

       1. expm1-def 0.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(0 + \frac{0.5}{x}\right)\right)\right)}, x\right) \]

       2. expm1-log1p 100.0%

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(0 + \frac{0.5}{x}\right)}, x\right) \]

       3. +-lft-identity 100.0%

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

   18. Simplified 100.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\frac{0.5}{x}\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.6%

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

Alternative 7: 58.6% 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 22.2%

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

      1. +-commutative 22.2%

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

      2. hypot-1-def 39.1%

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

   3. Simplified 39.1%

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

   4. Taylor expanded in x around 0 15.6%

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

      1. expm1-log1p-u 15.6%

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

      2. expm1-udef 15.6%

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

      3. log1p-udef 15.6%

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

      4. add-exp-log 15.6%

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

      5. log1p-def 15.6%

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

      6. add-sqr-sqrt 2.4%

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

      7. fabs-sqr 2.4%

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

      8. add-sqr-sqrt 5.0%

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

   6. Applied egg-rr 5.0%

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

      1. associate--l+ 5.0%

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

      2. +-commutative 5.0%

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

      3. sub-neg 5.0%

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

      4. metadata-eval 5.0%

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

   8. Applied egg-rr 5.0%

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

   9. Taylor expanded in x around 0 67.8%

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

   if 1.6000000000000001 < x

   1. Initial program 59.0%

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

      1. +-commutative 59.0%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 31.6%

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

      1. log1p-def 31.6%

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

      2. unpow1 31.6%

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

      3. sqr-pow 31.6%

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

      4. fabs-sqr 31.6%

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

      5. sqr-pow 31.6%

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

      6. unpow1 31.6%

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

   6. Simplified 31.6%

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

4. Final simplification 56.8%

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

Alternative 8: 52.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

   1. +-commutative 33.4%

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

   2. hypot-1-def 57.6%

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

3. Simplified 57.6%

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

4. Taylor expanded in x around 0 20.4%

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

   1. expm1-log1p-u 20.4%

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

   2. expm1-udef 20.4%

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

   3. log1p-udef 20.4%

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

   4. add-exp-log 20.4%

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

   5. log1p-def 20.4%

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

   6. add-sqr-sqrt 11.3%

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

   7. fabs-sqr 11.3%

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

   8. add-sqr-sqrt 13.1%

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

6. Applied egg-rr 13.1%

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

   1. associate--l+ 13.1%

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

   2. +-commutative 13.1%

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

   3. sub-neg 13.1%

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

   4. metadata-eval 13.1%

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

8. Applied egg-rr 13.1%

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

9. Taylor expanded in x around 0 48.8%

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

10. Final simplification 48.8%

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

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 2023199 
(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))