Rust f64::asinh

Percentage Accurate: 29.2% → 99.4%

Time: 8.3s

Alternatives: 10

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 29.2% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.005:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -20.0)
     (copysign (log (/ -0.5 x)) x)
     (if (<= t_0 0.005)
       (copysign
        (+ (* -0.16666666666666666 (pow x 3.0)) (+ x (* 0.075 (pow x 5.0))))
        x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = copysign(log((-0.5 / x)), x);
	} else if (t_0 <= 0.005) {
		tmp = copysign(((-0.16666666666666666 * pow(x, 3.0)) + (x + (0.075 * pow(x, 5.0)))), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = Math.copySign(Math.log((-0.5 / x)), x);
	} else if (t_0 <= 0.005) {
		tmp = Math.copySign(((-0.16666666666666666 * Math.pow(x, 3.0)) + (x + (0.075 * Math.pow(x, 5.0)))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -20.0:
		tmp = math.copysign(math.log((-0.5 / x)), x)
	elif t_0 <= 0.005:
		tmp = math.copysign(((-0.16666666666666666 * math.pow(x, 3.0)) + (x + (0.075 * math.pow(x, 5.0)))), x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -20.0)
		tmp = copysign(log(Float64(-0.5 / x)), x);
	elseif (t_0 <= 0.005)
		tmp = copysign(Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + Float64(x + Float64(0.075 * (x ^ 5.0)))), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
	tmp = 0.0;
	if (t_0 <= -20.0)
		tmp = sign(x) * abs(log((-0.5 / x)));
	elseif (t_0 <= 0.005)
		tmp = sign(x) * abs(((-0.16666666666666666 * (x ^ 3.0)) + (x + (0.075 * (x ^ 5.0)))));
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -20.0], N[With[{TMP1 = Abs[N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.005], N[With[{TMP1 = Abs[N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(0.075 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 48.5%

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

      1. +-commutative 48.5%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. unpow1 100.0%

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

      3. sqr-pow 0.0%

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

      4. fabs-sqr 0.0%

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

      5. sqr-pow 3.1%

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

      6. unpow1 3.1%

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

      7. associate-+r- 100.0%

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

      8. mul-1-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. +-inverses 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate-*r/ 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-neg-frac 100.0%

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

      15. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if -20 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < 0.0050000000000000001

   1. Initial program 9.0%

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

      1. +-commutative 9.0%

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

      2. hypot-1-def 9.0%

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

   3. Simplified 9.0%

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

      1. flip-+ 9.0%

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

      2. div-sub 9.0%

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

      3. pow2 9.0%

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

      4. add-sqr-sqrt 4.1%

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

      5. fabs-sqr 4.1%

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

      6. add-sqr-sqrt 9.0%

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

      7. pow2 9.0%

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

      8. add-sqr-sqrt 4.1%

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

      9. fabs-sqr 4.1%

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

      10. add-sqr-sqrt 8.9%

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

      11. hypot-udef 8.9%

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

      12. hypot-udef 8.9%

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

      13. add-sqr-sqrt 8.9%

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

      14. metadata-eval 8.9%

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

      15. add-sqr-sqrt 4.1%

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

      16. fabs-sqr 4.1%

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

      17. add-sqr-sqrt 9.0%

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

   5. Applied egg-rr 9.0%

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

      1. unpow2 9.0%

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

      2. div-sub 9.0%

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

      3. unpow2 9.0%

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

      4. unpow2 9.0%

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

      5. unpow2 9.0%

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

      6. +-commutative 9.0%

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

      7. associate--r+ 9.0%

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

      8. +-inverses 9.0%

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

      9. metadata-eval 9.0%

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

      10. metadata-eval 9.0%

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

      11. associate-/r* 9.0%

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

      12. neg-mul-1 9.0%

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

      13. sub-neg 9.0%

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

      14. +-commutative 9.0%

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

      15. distribute-neg-in 9.0%

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

      16. remove-double-neg 9.0%

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

      17. sub-neg 9.0%

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

   7. Simplified 9.0%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + x\right)}, x\right) \]

   if 0.0050000000000000001 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x)

   1. Initial program 52.3%

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

      1. +-commutative 52.3%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. 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 100.0%

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

      7. fabs-sqr 100.0%

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

      8. add-sqr-sqrt 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.005:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.28:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.00112:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\left(1 + \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right) + -1, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.28)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.00112)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (+ (+ 1.0 (log (+ x (hypot 1.0 x)))) -1.0) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.28) {
		tmp = copysign(log((-0.5 / x)), x);
	} else if (x <= 0.00112) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(((1.0 + log((x + hypot(1.0, x)))) + -1.0), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.28) {
		tmp = Math.copySign(Math.log((-0.5 / x)), x);
	} else if (x <= 0.00112) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(((1.0 + Math.log((x + Math.hypot(1.0, x)))) + -1.0), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.28:
		tmp = math.copysign(math.log((-0.5 / x)), x)
	elif x <= 0.00112:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(((1.0 + math.log((x + math.hypot(1.0, x)))) + -1.0), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.28)
		tmp = copysign(log(Float64(-0.5 / x)), x);
	elseif (x <= 0.00112)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(Float64(Float64(1.0 + log(Float64(x + hypot(1.0, x)))) + -1.0), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.28)
		tmp = sign(x) * abs(log((-0.5 / x)));
	elseif (x <= 0.00112)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(((1.0 + log((x + hypot(1.0, x)))) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.28], N[With[{TMP1 = Abs[N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 0.00112], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[(N[(1.0 + N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.28000000000000003

   1. Initial program 48.5%

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

      1. +-commutative 48.5%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. unpow1 100.0%

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

      3. sqr-pow 0.0%

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

      4. fabs-sqr 0.0%

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

      5. sqr-pow 3.1%

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

      6. unpow1 3.1%

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

      7. associate-+r- 100.0%

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

      8. mul-1-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. +-inverses 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate-*r/ 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-neg-frac 100.0%

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

      15. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if -1.28000000000000003 < x < 0.0011199999999999999

   1. Initial program 8.4%

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

      1. +-commutative 8.4%

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

      2. hypot-1-def 8.4%

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

   3. Simplified 8.4%

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

      1. flip-+ 8.4%

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

      2. div-sub 8.4%

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

      3. pow2 8.4%

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

      4. add-sqr-sqrt 3.5%

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

      5. fabs-sqr 3.5%

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

      6. add-sqr-sqrt 8.4%

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

      7. pow2 8.4%

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

      8. add-sqr-sqrt 3.5%

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

      9. fabs-sqr 3.5%

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

      10. add-sqr-sqrt 8.3%

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

      11. hypot-udef 8.3%

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

      12. hypot-udef 8.3%

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

      13. add-sqr-sqrt 8.3%

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

      14. metadata-eval 8.3%

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

      15. add-sqr-sqrt 3.5%

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

      16. fabs-sqr 3.5%

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

      17. add-sqr-sqrt 8.4%

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

   5. Applied egg-rr 8.4%

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

      1. unpow2 8.4%

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

      2. div-sub 8.4%

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

      3. unpow2 8.4%

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

      4. unpow2 8.4%

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

      5. unpow2 8.4%

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

      6. +-commutative 8.4%

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

      7. associate--r+ 8.4%

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

      8. +-inverses 8.4%

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

      9. metadata-eval 8.4%

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

      10. metadata-eval 8.4%

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

      11. associate-/r* 8.4%

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

      12. neg-mul-1 8.4%

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

      13. sub-neg 8.4%

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

      14. +-commutative 8.4%

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

      15. distribute-neg-in 8.4%

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

      16. remove-double-neg 8.4%

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

      17. sub-neg 8.4%

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

   7. Simplified 8.4%

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

   8. Taylor expanded in x around 0 100.0%

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

   if 0.0011199999999999999 < x

   1. Initial program 52.9%

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

      1. +-commutative 52.9%

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

      2. hypot-1-def 99.8%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 98.1%

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

      2. expm1-udef 98.3%

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

      3. log1p-udef 98.3%

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

      4. add-exp-log 99.8%

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

      5. *-un-lft-identity 99.8%

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

      6. *-un-lft-identity 99.8%

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

      7. add-sqr-sqrt 99.8%

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

      8. fabs-sqr 99.8%

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

      9. add-sqr-sqrt 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.28:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.00112:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\left(1 + \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right) + -1, x\right)\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.28:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.00105:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.28)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.00105)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.28) {
		tmp = copysign(log((-0.5 / x)), x);
	} else if (x <= 0.00105) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.28) {
		tmp = Math.copySign(Math.log((-0.5 / x)), x);
	} else if (x <= 0.00105) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.28:
		tmp = math.copysign(math.log((-0.5 / x)), x)
	elif x <= 0.00105:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.28)
		tmp = copysign(log(Float64(-0.5 / x)), x);
	elseif (x <= 0.00105)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.28)
		tmp = sign(x) * abs(log((-0.5 / x)));
	elseif (x <= 0.00105)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.28], N[With[{TMP1 = Abs[N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 0.00105], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.28000000000000003

   1. Initial program 48.5%

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

      1. +-commutative 48.5%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. unpow1 100.0%

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

      3. sqr-pow 0.0%

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

      4. fabs-sqr 0.0%

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

      5. sqr-pow 3.1%

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

      6. unpow1 3.1%

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

      7. associate-+r- 100.0%

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

      8. mul-1-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. +-inverses 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate-*r/ 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-neg-frac 100.0%

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

      15. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if -1.28000000000000003 < x < 0.00104999999999999994

   1. Initial program 8.4%

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

      1. +-commutative 8.4%

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

      2. hypot-1-def 8.4%

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

   3. Simplified 8.4%

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

      1. flip-+ 8.4%

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

      2. div-sub 8.4%

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

      3. pow2 8.4%

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

      4. add-sqr-sqrt 3.5%

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

      5. fabs-sqr 3.5%

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

      6. add-sqr-sqrt 8.4%

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

      7. pow2 8.4%

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

      8. add-sqr-sqrt 3.5%

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

      9. fabs-sqr 3.5%

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

      10. add-sqr-sqrt 8.3%

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

      11. hypot-udef 8.3%

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

      12. hypot-udef 8.3%

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

      13. add-sqr-sqrt 8.3%

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

      14. metadata-eval 8.3%

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

      15. add-sqr-sqrt 3.5%

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

      16. fabs-sqr 3.5%

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

      17. add-sqr-sqrt 8.4%

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

   5. Applied egg-rr 8.4%

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

      1. unpow2 8.4%

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

      2. div-sub 8.4%

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

      3. unpow2 8.4%

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

      4. unpow2 8.4%

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

      5. unpow2 8.4%

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

      6. +-commutative 8.4%

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

      7. associate--r+ 8.4%

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

      8. +-inverses 8.4%

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

      9. metadata-eval 8.4%

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

      10. metadata-eval 8.4%

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

      11. associate-/r* 8.4%

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

      12. neg-mul-1 8.4%

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

      13. sub-neg 8.4%

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

      14. +-commutative 8.4%

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

      15. distribute-neg-in 8.4%

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

      16. remove-double-neg 8.4%

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

      17. sub-neg 8.4%

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

   7. Simplified 8.4%

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

   8. Taylor expanded in x around 0 100.0%

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

   if 0.00104999999999999994 < x

   1. Initial program 52.9%

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

      1. +-commutative 52.9%

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

      2. hypot-1-def 99.8%

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

   3. Simplified 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. log-prod 99.8%

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

      3. metadata-eval 99.8%

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

      4. *-un-lft-identity 99.8%

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

      5. *-un-lft-identity 99.8%

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

      6. add-sqr-sqrt 99.8%

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

      7. fabs-sqr 99.8%

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

      8. add-sqr-sqrt 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. +-lft-identity 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.28:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.00105:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 4: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.28:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.28)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.96)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x (+ x (/ 0.5 x)))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.28) {
		tmp = copysign(log((-0.5 / x)), x);
	} else if (x <= 0.96) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log((x + (x + (0.5 / x)))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.28) {
		tmp = Math.copySign(Math.log((-0.5 / x)), x);
	} else if (x <= 0.96) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(Math.log((x + (x + (0.5 / x)))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.28:
		tmp = math.copysign(math.log((-0.5 / x)), x)
	elif x <= 0.96:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(math.log((x + (x + (0.5 / x)))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.28)
		tmp = copysign(log(Float64(-0.5 / x)), x);
	elseif (x <= 0.96)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(log(Float64(x + Float64(x + Float64(0.5 / x)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.28)
		tmp = sign(x) * abs(log((-0.5 / x)));
	elseif (x <= 0.96)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(log((x + (x + (0.5 / x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.28], N[With[{TMP1 = Abs[N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 0.96], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[(x + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.28000000000000003

   1. Initial program 48.5%

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

      1. +-commutative 48.5%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. unpow1 100.0%

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

      3. sqr-pow 0.0%

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

      4. fabs-sqr 0.0%

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

      5. sqr-pow 3.1%

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

      6. unpow1 3.1%

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

      7. associate-+r- 100.0%

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

      8. mul-1-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. +-inverses 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate-*r/ 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-neg-frac 100.0%

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

      15. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if -1.28000000000000003 < x < 0.95999999999999996

   1. Initial program 9.0%

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

      1. +-commutative 9.0%

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

      2. hypot-1-def 9.0%

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

   3. Simplified 9.0%

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

      1. flip-+ 9.0%

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

      2. div-sub 9.0%

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

      3. pow2 9.0%

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

      4. add-sqr-sqrt 4.1%

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

      5. fabs-sqr 4.1%

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

      6. add-sqr-sqrt 9.0%

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

      7. pow2 9.0%

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

      8. add-sqr-sqrt 4.1%

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

      9. fabs-sqr 4.1%

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

      10. add-sqr-sqrt 8.9%

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

      11. hypot-udef 8.9%

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

      12. hypot-udef 8.9%

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

      13. add-sqr-sqrt 8.9%

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

      14. metadata-eval 8.9%

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

      15. add-sqr-sqrt 4.1%

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

      16. fabs-sqr 4.1%

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

      17. add-sqr-sqrt 9.0%

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

   5. Applied egg-rr 9.0%

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

      1. unpow2 9.0%

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

      2. div-sub 9.0%

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

      3. unpow2 9.0%

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

      4. unpow2 9.0%

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

      5. unpow2 9.0%

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

      6. +-commutative 9.0%

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

      7. associate--r+ 9.0%

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

      8. +-inverses 9.0%

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

      9. metadata-eval 9.0%

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

      10. metadata-eval 9.0%

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

      11. associate-/r* 9.0%

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

      12. neg-mul-1 9.0%

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

      13. sub-neg 9.0%

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

      14. +-commutative 9.0%

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

      15. distribute-neg-in 9.0%

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

      16. remove-double-neg 9.0%

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

      17. sub-neg 9.0%

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

   7. Simplified 9.0%

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

   8. Taylor expanded in x around 0 99.8%

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

   if 0.95999999999999996 < x

   1. Initial program 52.3%

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

      1. +-commutative 52.3%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.9%

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

      1. associate-+r+ 98.9%

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

      2. +-commutative 98.9%

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

      3. +-commutative 98.9%

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

      4. unpow1 98.9%

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

      5. sqr-pow 98.9%

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

      6. fabs-sqr 98.9%

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

      7. sqr-pow 98.9%

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

      8. unpow1 98.9%

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

      9. associate-*r/ 98.9%

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

      10. metadata-eval 98.9%

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

   6. Simplified 98.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.28:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]

Alternative 5: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.28:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.28)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.25)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.28) {
		tmp = copysign(log((-0.5 / x)), x);
	} else if (x <= 1.25) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log((x + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.28) {
		tmp = Math.copySign(Math.log((-0.5 / x)), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(Math.log((x + x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.28:
		tmp = math.copysign(math.log((-0.5 / x)), x)
	elif x <= 1.25:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(math.log((x + x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.28)
		tmp = copysign(log(Float64(-0.5 / x)), x);
	elseif (x <= 1.25)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(log(Float64(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.28)
		tmp = sign(x) * abs(log((-0.5 / x)));
	elseif (x <= 1.25)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.28], N[With[{TMP1 = Abs[N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.28000000000000003

   1. Initial program 48.5%

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

      1. +-commutative 48.5%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. unpow1 100.0%

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

      3. sqr-pow 0.0%

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

      4. fabs-sqr 0.0%

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

      5. sqr-pow 3.1%

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

      6. unpow1 3.1%

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

      7. associate-+r- 100.0%

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

      8. mul-1-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. +-inverses 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate-*r/ 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-neg-frac 100.0%

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

      15. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if -1.28000000000000003 < x < 1.25

   1. Initial program 9.7%

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

      1. +-commutative 9.7%

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

      2. hypot-1-def 9.7%

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

   3. Simplified 9.7%

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

      1. flip-+ 9.7%

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

      2. div-sub 9.7%

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

      3. pow2 9.7%

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

      4. add-sqr-sqrt 4.9%

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

      5. fabs-sqr 4.9%

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

      6. add-sqr-sqrt 9.7%

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

      7. pow2 9.7%

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

      8. add-sqr-sqrt 4.8%

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

      9. fabs-sqr 4.8%

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

      10. add-sqr-sqrt 9.6%

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

      11. hypot-udef 9.6%

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

      12. hypot-udef 9.6%

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

      13. add-sqr-sqrt 9.6%

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

      14. metadata-eval 9.6%

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

      15. add-sqr-sqrt 4.8%

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

      16. fabs-sqr 4.8%

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

      17. add-sqr-sqrt 9.7%

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

   5. Applied egg-rr 9.7%

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

      1. unpow2 9.7%

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

      2. div-sub 9.7%

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

      3. unpow2 9.7%

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

      4. unpow2 9.7%

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

      5. unpow2 9.7%

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

      6. +-commutative 9.7%

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

      7. associate--r+ 9.7%

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

      8. +-inverses 9.7%

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

      9. metadata-eval 9.7%

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

      10. metadata-eval 9.7%

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

      11. associate-/r* 9.7%

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

      12. neg-mul-1 9.7%

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

      13. sub-neg 9.7%

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

      14. +-commutative 9.7%

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

      15. distribute-neg-in 9.7%

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

      16. remove-double-neg 9.7%

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

      17. sub-neg 9.7%

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

   7. Simplified 9.7%

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

   8. Taylor expanded in x around 0 99.2%

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

   if 1.25 < x

   1. Initial program 51.6%

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

      1. +-commutative 51.6%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. unpow1 100.0%

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

      2. sqr-pow 100.0%

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

      3. fabs-sqr 100.0%

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

      4. sqr-pow 100.0%

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

      5. unpow1 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.28:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 6: 81.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -3.2000000000000002

   1. Initial program 48.5%

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

      1. +-commutative 48.5%

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

      2. hypot-1-def 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%

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

      1. mul-1-neg 31.5%

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

   6. Simplified 31.5%

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

   if -3.2000000000000002 < x < 1.25

   1. Initial program 9.7%

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

      1. +-commutative 9.7%

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

      2. hypot-1-def 9.7%

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

   3. Simplified 9.7%

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

   4. Taylor expanded in x around 0 9.9%

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

      1. fma-def 9.9%

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

      2. unpow2 9.9%

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

      3. unpow1 9.9%

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

      4. sqr-pow 4.5%

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

      5. fabs-sqr 4.5%

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

      6. sqr-pow 9.8%

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

      7. unpow1 9.8%

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

      8. log1p-def 98.8%

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

      9. unpow1 98.8%

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

      10. sqr-pow 45.6%

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

      11. fabs-sqr 45.6%

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

      12. sqr-pow 99.0%

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

      13. unpow1 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in x around 0 98.7%

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

      1. unpow2 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in x around 0 98.7%

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

   if 1.25 < x

   1. Initial program 51.6%

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

      1. +-commutative 51.6%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. unpow1 100.0%

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

      2. sqr-pow 100.0%

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

      3. fabs-sqr 100.0%

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

      4. sqr-pow 100.0%

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

      5. unpow1 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 82.2%

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

Alternative 7: 99.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 48.5%

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

      1. +-commutative 48.5%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. unpow1 100.0%

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

      3. sqr-pow 0.0%

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

      4. fabs-sqr 0.0%

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

      5. sqr-pow 3.1%

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

      6. unpow1 3.1%

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

      7. associate-+r- 100.0%

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

      8. mul-1-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. +-inverses 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate-*r/ 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-neg-frac 100.0%

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

      15. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if -1.25 < x < 1.25

   1. Initial program 9.7%

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

      1. +-commutative 9.7%

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

      2. hypot-1-def 9.7%

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

   3. Simplified 9.7%

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

   4. Taylor expanded in x around 0 9.9%

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

      1. fma-def 9.9%

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

      2. unpow2 9.9%

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

      3. unpow1 9.9%

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

      4. sqr-pow 4.5%

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

      5. fabs-sqr 4.5%

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

      6. sqr-pow 9.8%

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

      7. unpow1 9.8%

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

      8. log1p-def 98.8%

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

      9. unpow1 98.8%

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

      10. sqr-pow 45.6%

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

      11. fabs-sqr 45.6%

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

      12. sqr-pow 99.0%

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

      13. unpow1 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in x around 0 98.7%

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

      1. unpow2 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in x around 0 98.7%

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

   if 1.25 < x

   1. Initial program 51.6%

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

      1. +-commutative 51.6%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. unpow1 100.0%

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

      2. sqr-pow 100.0%

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

      3. fabs-sqr 100.0%

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

      4. sqr-pow 100.0%

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

      5. unpow1 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.4%

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

Alternative 8: 64.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.5

   1. Initial program 48.5%

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

      1. +-commutative 48.5%

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

      2. hypot-1-def 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%

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

      1. mul-1-neg 31.5%

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

   6. Simplified 31.5%

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

   if -0.5 < x

   1. Initial program 23.2%

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

      1. +-commutative 23.2%

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

      2. hypot-1-def 38.9%

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

   3. Simplified 38.9%

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

   4. Taylor expanded in x around 0 15.9%

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

      1. log1p-def 76.0%

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

      2. unpow1 76.0%

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

      3. sqr-pow 40.6%

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

      4. fabs-sqr 40.6%

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

      5. sqr-pow 76.0%

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

      6. unpow1 76.0%

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

   6. Simplified 76.0%

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

4. Final simplification 64.9%

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

Alternative 9: 58.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55) (copysign x x) (copysign (log1p x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log1p(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log1p(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.55:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log1p(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55)
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.55], 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 < 1.55000000000000004

   1. Initial program 22.5%

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

      1. +-commutative 22.5%

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

      2. hypot-1-def 39.5%

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

   3. Simplified 39.5%

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

   4. Taylor expanded in x around 0 8.3%

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

      1. fma-def 8.3%

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

      2. unpow2 8.3%

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

      3. unpow1 8.3%

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

      4. sqr-pow 3.0%

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

      5. fabs-sqr 3.0%

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

      6. sqr-pow 8.2%

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

      7. unpow1 8.2%

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

      8. log1p-def 67.8%

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

      9. unpow1 67.8%

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

      10. sqr-pow 30.6%

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

      11. fabs-sqr 30.6%

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

      12. sqr-pow 66.4%

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

      13. unpow1 66.4%

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

   6. Simplified 66.4%

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

   7. Taylor expanded in x around 0 66.1%

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

      1. unpow2 66.1%

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

   9. Simplified 66.1%

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

   10. Taylor expanded in x around 0 67.9%

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

   if 1.55000000000000004 < x

   1. Initial program 51.6%

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

      1. +-commutative 51.6%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 31.7%

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

      1. log1p-def 31.7%

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

      2. unpow1 31.7%

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

      3. sqr-pow 31.7%

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

      4. fabs-sqr 31.7%

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

      5. sqr-pow 31.7%

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

      6. unpow1 31.7%

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

   6. Simplified 31.7%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 10: 52.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.6%

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

   1. +-commutative 29.6%

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

   2. hypot-1-def 54.2%

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

3. Simplified 54.2%

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

4. Taylor expanded in x around 0 7.4%

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

   1. fma-def 7.4%

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

   2. unpow2 7.4%

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

   3. unpow1 7.4%

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

   4. sqr-pow 3.5%

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

   5. fabs-sqr 3.5%

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

   6. sqr-pow 7.4%

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

   7. unpow1 7.4%

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

   8. log1p-def 52.6%

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

   9. unpow1 52.6%

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

   10. sqr-pow 24.4%

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

   11. fabs-sqr 24.4%

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

   12. sqr-pow 51.5%

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

   13. unpow1 51.5%

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

6. Simplified 51.5%

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

7. Taylor expanded in x around 0 51.1%

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

   1. unpow2 51.1%

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

9. Simplified 51.1%

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

10. Taylor expanded in x around 0 52.7%

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

11. Final simplification 52.7%

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

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 2023279 
(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))