Rust f64::asinh

Percentage Accurate: 29.9% → 99.2%

Time: 5.2s

Alternatives: 9

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: 29.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(x + \left(-0.16666666666666666 \cdot {x}^{3} + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x + x\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 1e-8)
       (copysign
        (+ x (+ (* -0.16666666666666666 (pow x 3.0)) (* 0.075 (pow x 5.0))))
        x)
       (copysign (log (+ (+ x x) (/ 0.5 x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = copysign(-log((hypot(1.0, x) - x)), x);
	} else if (t_0 <= 1e-8) {
		tmp = copysign((x + ((-0.16666666666666666 * pow(x, 3.0)) + (0.075 * pow(x, 5.0)))), x);
	} else {
		tmp = copysign(log(((x + x) + (0.5 / x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = Math.copySign(-Math.log((Math.hypot(1.0, x) - x)), x);
	} else if (t_0 <= 1e-8) {
		tmp = Math.copySign((x + ((-0.16666666666666666 * Math.pow(x, 3.0)) + (0.075 * Math.pow(x, 5.0)))), x);
	} else {
		tmp = Math.copySign(Math.log(((x + x) + (0.5 / x))), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -1.0:
		tmp = math.copysign(-math.log((math.hypot(1.0, x) - x)), x)
	elif t_0 <= 1e-8:
		tmp = math.copysign((x + ((-0.16666666666666666 * math.pow(x, 3.0)) + (0.075 * math.pow(x, 5.0)))), x)
	else:
		tmp = math.copysign(math.log(((x + x) + (0.5 / x))), x)
	return tmp

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = copysign(Float64(-log(Float64(hypot(1.0, x) - x))), x);
	elseif (t_0 <= 1e-8)
		tmp = copysign(Float64(x + Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + Float64(0.075 * (x ^ 5.0)))), x);
	else
		tmp = copysign(log(Float64(Float64(x + x) + Float64(0.5 / 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 <= -1.0)
		tmp = sign(x) * abs(-log((hypot(1.0, x) - x)));
	elseif (t_0 <= 1e-8)
		tmp = sign(x) * abs((x + ((-0.16666666666666666 * (x ^ 3.0)) + (0.075 * (x ^ 5.0)))));
	else
		tmp = sign(x) * abs(log(((x + x) + (0.5 / 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, -1.0], N[With[{TMP1 = Abs[(-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 1e-8], N[With[{TMP1 = Abs[N[(x + N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + 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[(N[(x + x), $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 54.7%

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

      1. +-commutative 54.7%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 4.7%

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

      2. frac-2neg 4.7%

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

      3. log-div 4.7%

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

   5. Applied egg-rr 7.9%

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

      1. sub-neg 7.9%

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

      2. sub-neg 7.9%

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

      3. fma-udef 7.9%

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

      4. unpow2 7.9%

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

      5. associate--r+ 53.2%

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

      6. +-inverses 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-sub0 100.0%

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

      11. sub-neg 100.0%

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

      12. +-commutative 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. remove-double-neg 100.0%

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

      15. sub-neg 100.0%

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

   7. Simplified 100.0%

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

   if -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < 1e-8

   1. Initial program 9.2%

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

      1. +-commutative 9.2%

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

      2. hypot-1-def 9.2%

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

   3. Simplified 9.2%

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

   4. Taylor expanded in x around 0 9.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)\right)}, x\right) \]
   5. Step-by-step derivation

      1. associate-+r+ 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(1 + \left|x\right|\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)}, x\right) \]

      2. +-commutative 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(\left|x\right| + 1\right)} + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      3. rem-square-sqrt 2.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      4. fabs-sqr 2.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      5. rem-square-sqrt 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{x} + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      6. +-commutative 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(1 + x\right)} + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      7. associate-+r+ 9.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)\right)}, x\right) \]

      8. +-commutative 9.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(0.5 \cdot {x}^{2} + -0.125 \cdot {x}^{4}\right)}\right)\right), x\right) \]

      9. associate-+r+ 9.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \color{blue}{\left(\left(x + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)}\right), x\right) \]

      10. rem-square-sqrt 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      11. fabs-sqr 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      12. rem-square-sqrt 9.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\left|\color{blue}{x}\right| + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      13. +-commutative 9.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\left(0.5 \cdot {x}^{2} + \left|x\right|\right)} + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      14. fma-def 9.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \left|x\right|\right)} + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      15. rem-square-sqrt 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      16. fabs-sqr 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      17. rem-square-sqrt 9.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{x}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      18. *-commutative 9.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, x\right) + \color{blue}{{x}^{4} \cdot -0.125}\right)\right), x\right) \]

   6. Simplified 9.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, x\right) + {x}^{4} \cdot -0.125\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0 100.0%

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

   if 1e-8 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x)

   1. Initial program 41.6%

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

      1. +-commutative 41.6%

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

      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%

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

      1. rem-square-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. associate-+r+ 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(x + \left(-0.16666666666666666 \cdot {x}^{3} + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x + x\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.001:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x + x\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.001)
   (copysign (- (log (- (hypot 1.0 x) 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 <= -0.001) {
		tmp = copysign(-log((hypot(1.0, x) - 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 <= -0.001) {
		tmp = Math.copySign(-Math.log((Math.hypot(1.0, x) - 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 <= -0.001:
		tmp = math.copysign(-math.log((math.hypot(1.0, x) - 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 <= -0.001)
		tmp = copysign(Float64(-log(Float64(hypot(1.0, x) - x))), x);
	elseif (x <= 0.96)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(log(Float64(Float64(x + x) + Float64(0.5 / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.001)
		tmp = sign(x) * abs(-log((hypot(1.0, x) - 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, -0.001], N[With[{TMP1 = Abs[(-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 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[(N[(x + x), $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e-3

   1. Initial program 55.3%

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

      1. +-commutative 55.3%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 6.1%

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

      2. frac-2neg 6.1%

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

      3. log-div 6.0%

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

   5. Applied egg-rr 9.3%

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

      1. sub-neg 9.3%

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

      2. sub-neg 9.3%

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

      3. fma-udef 9.3%

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

      4. unpow2 9.3%

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

      5. associate--r+ 53.9%

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

      6. +-inverses 99.9%

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

      7. metadata-eval 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

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

      10. neg-sub0 99.9%

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

      11. sub-neg 99.9%

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

      12. +-commutative 99.9%

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

      13. distribute-neg-in 99.9%

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

      14. remove-double-neg 99.9%

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

      15. sub-neg 99.9%

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

   7. Simplified 99.9%

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

   if -1e-3 < x < 0.95999999999999996

   1. Initial program 8.5%

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

      1. +-commutative 8.5%

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

      2. hypot-1-def 8.5%

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

   3. Simplified 8.5%

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

   4. Taylor expanded in x around 0 8.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)\right)}, x\right) \]
   5. Step-by-step derivation

      1. associate-+r+ 8.5%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(1 + \left|x\right|\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)}, x\right) \]

      2. +-commutative 8.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(\left|x\right| + 1\right)} + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      3. rem-square-sqrt 2.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      4. fabs-sqr 2.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      5. rem-square-sqrt 8.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{x} + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      6. +-commutative 8.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(1 + x\right)} + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      7. associate-+r+ 8.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)\right)}, x\right) \]

      8. +-commutative 8.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(0.5 \cdot {x}^{2} + -0.125 \cdot {x}^{4}\right)}\right)\right), x\right) \]

      9. associate-+r+ 8.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \color{blue}{\left(\left(x + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)}\right), x\right) \]

      10. rem-square-sqrt 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      11. fabs-sqr 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      12. rem-square-sqrt 8.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\left|\color{blue}{x}\right| + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      13. +-commutative 8.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\left(0.5 \cdot {x}^{2} + \left|x\right|\right)} + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      14. fma-def 8.5%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \left|x\right|\right)} + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      15. rem-square-sqrt 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      16. fabs-sqr 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      17. rem-square-sqrt 8.6%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{x}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      18. *-commutative 8.6%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, x\right) + \color{blue}{{x}^{4} \cdot -0.125}\right)\right), x\right) \]

   6. Simplified 8.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, x\right) + {x}^{4} \cdot -0.125\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0 99.9%

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

      1. *-commutative 99.9%

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

   9. Simplified 99.9%

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

   if 0.95999999999999996 < x

   1. Initial program 41.6%

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

      1. +-commutative 41.6%

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

      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%

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

      1. rem-square-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. associate-+r+ 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.001:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x + x\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x + x\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (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 <= -1.25) {
		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 <= -1.25) {
		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 <= -1.25:
		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 <= -1.25)
		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(Float64(x + x) + Float64(0.5 / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = sign(x) * abs(log((-0.5 / 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, -1.25], 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, 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[(N[(x + x), $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 54.7%

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

      1. +-commutative 54.7%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

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

      3. log-prod 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 7.5%

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

      9. metadata-eval 7.5%

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

   5. Applied egg-rr 7.5%

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

      1. +-rgt-identity 7.5%

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

   7. Simplified 7.5%

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

   8. Taylor expanded in x around -inf 97.3%

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

   if -1.25 < x < 0.95999999999999996

   1. Initial program 9.2%

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

      1. +-commutative 9.2%

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

      2. hypot-1-def 9.2%

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

   3. Simplified 9.2%

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

   4. Taylor expanded in x around 0 9.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)\right)}, x\right) \]
   5. Step-by-step derivation

      1. associate-+r+ 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(1 + \left|x\right|\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)}, x\right) \]

      2. +-commutative 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(\left|x\right| + 1\right)} + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      3. rem-square-sqrt 2.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      4. fabs-sqr 2.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      5. rem-square-sqrt 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{x} + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      6. +-commutative 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(1 + x\right)} + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      7. associate-+r+ 9.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)\right)}, x\right) \]

      8. +-commutative 9.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(0.5 \cdot {x}^{2} + -0.125 \cdot {x}^{4}\right)}\right)\right), x\right) \]

      9. associate-+r+ 9.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \color{blue}{\left(\left(x + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)}\right), x\right) \]

      10. rem-square-sqrt 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      11. fabs-sqr 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      12. rem-square-sqrt 9.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\left|\color{blue}{x}\right| + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      13. +-commutative 9.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\left(0.5 \cdot {x}^{2} + \left|x\right|\right)} + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      14. fma-def 9.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \left|x\right|\right)} + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      15. rem-square-sqrt 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      16. fabs-sqr 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      17. rem-square-sqrt 9.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{x}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      18. *-commutative 9.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, x\right) + \color{blue}{{x}^{4} \cdot -0.125}\right)\right), x\right) \]

   6. Simplified 9.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, x\right) + {x}^{4} \cdot -0.125\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0 99.7%

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

      1. *-commutative 99.7%

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

   9. Simplified 99.7%

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

   if 0.95999999999999996 < x

   1. Initial program 41.6%

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

      1. +-commutative 41.6%

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

      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%

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

      1. rem-square-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. associate-+r+ 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x + x\right) + \frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 4: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.26)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) 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 <= 1.26) {
		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 <= -1.25) {
		tmp = Math.copySign(Math.log((-0.5 / x)), x);
	} else if (x <= 1.26) {
		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 <= -1.25:
		tmp = math.copysign(math.log((-0.5 / x)), x)
	elif x <= 1.26:
		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 <= -1.25)
		tmp = copysign(log(Float64(-0.5 / x)), x);
	elseif (x <= 1.26)
		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 <= -1.25)
		tmp = sign(x) * abs(log((-0.5 / x)));
	elseif (x <= 1.26)
		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, -1.25], 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, 1.26], 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 < -1.25

   1. Initial program 54.7%

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

      1. +-commutative 54.7%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

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

      3. log-prod 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 7.5%

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

      9. metadata-eval 7.5%

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

   5. Applied egg-rr 7.5%

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

      1. +-rgt-identity 7.5%

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

   7. Simplified 7.5%

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

   8. Taylor expanded in x around -inf 97.3%

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

   if -1.25 < x < 1.26000000000000001

   1. Initial program 9.2%

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

      1. +-commutative 9.2%

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

      2. hypot-1-def 9.2%

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

   3. Simplified 9.2%

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

   4. Taylor expanded in x around 0 9.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)\right)}, x\right) \]
   5. Step-by-step derivation

      1. associate-+r+ 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(1 + \left|x\right|\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)}, x\right) \]

      2. +-commutative 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(\left|x\right| + 1\right)} + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      3. rem-square-sqrt 2.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      4. fabs-sqr 2.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      5. rem-square-sqrt 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{x} + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      6. +-commutative 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(1 + x\right)} + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      7. associate-+r+ 9.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)\right)}, x\right) \]

      8. +-commutative 9.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(0.5 \cdot {x}^{2} + -0.125 \cdot {x}^{4}\right)}\right)\right), x\right) \]

      9. associate-+r+ 9.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \color{blue}{\left(\left(x + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)}\right), x\right) \]

      10. rem-square-sqrt 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      11. fabs-sqr 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      12. rem-square-sqrt 9.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\left|\color{blue}{x}\right| + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      13. +-commutative 9.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\left(0.5 \cdot {x}^{2} + \left|x\right|\right)} + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      14. fma-def 9.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \left|x\right|\right)} + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      15. rem-square-sqrt 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      16. fabs-sqr 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      17. rem-square-sqrt 9.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{x}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      18. *-commutative 9.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, x\right) + \color{blue}{{x}^{4} \cdot -0.125}\right)\right), x\right) \]

   6. Simplified 9.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, x\right) + {x}^{4} \cdot -0.125\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0 99.7%

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

      1. *-commutative 99.7%

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

   9. Simplified 99.7%

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

   if 1.26000000000000001 < x

   1. Initial program 41.6%

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

      1. +-commutative 41.6%

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

      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 99.7%

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

      1. rem-square-sqrt 99.7%

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

      2. fabs-sqr 99.7%

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

      3. rem-square-sqrt 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 5: 81.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.2)
   (copysign (log (- x)) x)
   (if (<= x 1.26) (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 <= 1.26) {
		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 <= 1.26) {
		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 <= 1.26:
		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 <= 1.26)
		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 <= 1.26)
		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, 1.26], 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 54.0%

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

      1. +-commutative 54.0%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 31.0%

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

      1. mul-1-neg 31.0%

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

   6. Simplified 31.0%

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

   if -3.2000000000000002 < x < 1.26000000000000001

   1. Initial program 9.9%

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

      1. +-commutative 9.9%

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

      2. hypot-1-def 9.9%

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

   3. Simplified 9.9%

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

   4. Taylor expanded in x around 0 9.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)\right)}, x\right) \]
   5. Step-by-step derivation

      1. associate-+r+ 9.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(1 + \left|x\right|\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)}, x\right) \]

      2. +-commutative 9.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(\left|x\right| + 1\right)} + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      3. rem-square-sqrt 2.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      4. fabs-sqr 2.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      5. rem-square-sqrt 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{x} + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      6. +-commutative 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(1 + x\right)} + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      7. associate-+r+ 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)\right)}, x\right) \]

      8. +-commutative 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(0.5 \cdot {x}^{2} + -0.125 \cdot {x}^{4}\right)}\right)\right), x\right) \]

      9. associate-+r+ 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \color{blue}{\left(\left(x + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)}\right), x\right) \]

      10. rem-square-sqrt 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      11. fabs-sqr 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      12. rem-square-sqrt 9.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\left|\color{blue}{x}\right| + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      13. +-commutative 9.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\left(0.5 \cdot {x}^{2} + \left|x\right|\right)} + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      14. fma-def 9.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \left|x\right|\right)} + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      15. rem-square-sqrt 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      16. fabs-sqr 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      17. rem-square-sqrt 9.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{x}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      18. *-commutative 9.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, x\right) + \color{blue}{{x}^{4} \cdot -0.125}\right)\right), x\right) \]

   6. Simplified 9.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, x\right) + {x}^{4} \cdot -0.125\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0 98.4%

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

   if 1.26000000000000001 < x

   1. Initial program 41.6%

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

      1. +-commutative 41.6%

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

      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 99.7%

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

      1. rem-square-sqrt 99.7%

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

      2. fabs-sqr 99.7%

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

      3. rem-square-sqrt 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 6: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.26) (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 <= 1.26) {
		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 <= 1.26) {
		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 <= 1.26:
		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 <= 1.26)
		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 <= 1.26)
		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, 1.26], 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 54.7%

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

      1. +-commutative 54.7%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

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

      3. log-prod 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 7.5%

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

      9. metadata-eval 7.5%

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

   5. Applied egg-rr 7.5%

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

      1. +-rgt-identity 7.5%

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

   7. Simplified 7.5%

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

   8. Taylor expanded in x around -inf 97.3%

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

   if -1.25 < x < 1.26000000000000001

   1. Initial program 9.2%

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

      1. +-commutative 9.2%

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

      2. hypot-1-def 9.2%

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

   3. Simplified 9.2%

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

   4. Taylor expanded in x around 0 9.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)\right)}, x\right) \]
   5. Step-by-step derivation

      1. associate-+r+ 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(1 + \left|x\right|\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)}, x\right) \]

      2. +-commutative 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(\left|x\right| + 1\right)} + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      3. rem-square-sqrt 2.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      4. fabs-sqr 2.9%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      5. rem-square-sqrt 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{x} + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      6. +-commutative 9.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(1 + x\right)} + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      7. associate-+r+ 9.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)\right)}, x\right) \]

      8. +-commutative 9.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(0.5 \cdot {x}^{2} + -0.125 \cdot {x}^{4}\right)}\right)\right), x\right) \]

      9. associate-+r+ 9.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \color{blue}{\left(\left(x + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)}\right), x\right) \]

      10. rem-square-sqrt 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      11. fabs-sqr 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      12. rem-square-sqrt 9.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\left|\color{blue}{x}\right| + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      13. +-commutative 9.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\left(0.5 \cdot {x}^{2} + \left|x\right|\right)} + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      14. fma-def 9.2%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \left|x\right|\right)} + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      15. rem-square-sqrt 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      16. fabs-sqr 2.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      17. rem-square-sqrt 9.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{x}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      18. *-commutative 9.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, x\right) + \color{blue}{{x}^{4} \cdot -0.125}\right)\right), x\right) \]

   6. Simplified 9.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, x\right) + {x}^{4} \cdot -0.125\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0 98.9%

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

   if 1.26000000000000001 < x

   1. Initial program 41.6%

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

      1. +-commutative 41.6%

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

      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 99.7%

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

      1. rem-square-sqrt 99.7%

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

      2. fabs-sqr 99.7%

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

      3. rem-square-sqrt 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 7: 64.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		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 <= -1.0:
		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 <= -1.0)
		tmp = copysign(log(Float64(-x)), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 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 < -1

   1. Initial program 54.7%

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

      1. +-commutative 54.7%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 30.8%

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

      1. mul-1-neg 30.8%

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

   6. Simplified 30.8%

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

   if -1 < x

   1. Initial program 19.7%

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

      1. +-commutative 19.7%

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

      2. hypot-1-def 38.7%

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

   3. Simplified 38.7%

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

   4. Taylor expanded in x around 0 15.7%

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

      1. log1p-def 76.1%

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

      2. rem-square-sqrt 40.8%

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

      3. fabs-sqr 40.8%

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

      4. rem-square-sqrt 76.1%

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

   6. Simplified 76.1%

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

4. Final simplification 65.1%

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

Alternative 8: 58.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   1. Initial program 23.8%

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

      1. +-commutative 23.8%

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

      2. hypot-1-def 38.3%

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

   3. Simplified 38.3%

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

   4. Taylor expanded in x around 0 6.4%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)\right)}, x\right) \]
   5. Step-by-step derivation

      1. associate-+r+ 6.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(1 + \left|x\right|\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)}, x\right) \]

      2. +-commutative 6.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(\left|x\right| + 1\right)} + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      3. rem-square-sqrt 2.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      4. fabs-sqr 2.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      5. rem-square-sqrt 6.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{x} + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      6. +-commutative 6.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(1 + x\right)} + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

      7. associate-+r+ 6.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)\right)}, x\right) \]

      8. +-commutative 6.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(0.5 \cdot {x}^{2} + -0.125 \cdot {x}^{4}\right)}\right)\right), x\right) \]

      9. associate-+r+ 6.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(1 + \color{blue}{\left(\left(x + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)}\right), x\right) \]

      10. rem-square-sqrt 2.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      11. fabs-sqr 2.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      12. rem-square-sqrt 6.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\left|\color{blue}{x}\right| + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      13. +-commutative 6.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\left(0.5 \cdot {x}^{2} + \left|x\right|\right)} + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      14. fma-def 6.4%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \left|x\right|\right)} + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      15. rem-square-sqrt 2.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      16. fabs-sqr 2.0%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      17. rem-square-sqrt 6.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{x}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

      18. *-commutative 6.3%

          \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, x\right) + \color{blue}{{x}^{4} \cdot -0.125}\right)\right), x\right) \]

   6. Simplified 6.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, x\right) + {x}^{4} \cdot -0.125\right)\right)}, x\right) \]

   7. Taylor expanded in x around 0 69.1%

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

   if 1.6000000000000001 < x

   1. Initial program 41.6%

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

      1. +-commutative 41.6%

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

      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. rem-square-sqrt 31.7%

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

      3. fabs-sqr 31.7%

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

      4. rem-square-sqrt 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 59.9%

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

Alternative 9: 51.6% 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 28.2%

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

   1. +-commutative 28.2%

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

   2. hypot-1-def 53.5%

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

3. Simplified 53.5%

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

4. Taylor expanded in x around 0 4.8%

   \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\left|x\right| + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)\right)}, x\right) \]
5. Step-by-step derivation

   1. associate-+r+ 4.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(1 + \left|x\right|\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)}, x\right) \]

   2. +-commutative 4.8%

      \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(\left|x\right| + 1\right)} + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

   3. rem-square-sqrt 1.5%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

   4. fabs-sqr 1.5%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

   5. rem-square-sqrt 4.7%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left(\color{blue}{x} + 1\right) + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

   6. +-commutative 4.7%

      \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\left(1 + x\right)} + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right), x\right) \]

   7. associate-+r+ 4.7%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(x + \left(-0.125 \cdot {x}^{4} + 0.5 \cdot {x}^{2}\right)\right)\right)}, x\right) \]

   8. +-commutative 4.7%

      \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(x + \color{blue}{\left(0.5 \cdot {x}^{2} + -0.125 \cdot {x}^{4}\right)}\right)\right), x\right) \]

   9. associate-+r+ 4.7%

      \[\leadsto \mathsf{copysign}\left(\log \left(1 + \color{blue}{\left(\left(x + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)}\right), x\right) \]

   10. rem-square-sqrt 1.5%

       \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

   11. fabs-sqr 1.5%

       \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|} + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

   12. rem-square-sqrt 4.8%

       \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\left(\left|\color{blue}{x}\right| + 0.5 \cdot {x}^{2}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

   13. +-commutative 4.8%

       \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\left(0.5 \cdot {x}^{2} + \left|x\right|\right)} + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

   14. fma-def 4.8%

       \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \left|x\right|\right)} + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

   15. rem-square-sqrt 1.5%

       \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

   16. fabs-sqr 1.5%

       \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

   17. rem-square-sqrt 4.7%

       \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, \color{blue}{x}\right) + -0.125 \cdot {x}^{4}\right)\right), x\right) \]

   18. *-commutative 4.7%

       \[\leadsto \mathsf{copysign}\left(\log \left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, x\right) + \color{blue}{{x}^{4} \cdot -0.125}\right)\right), x\right) \]

6. Simplified 4.7%

   \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(\mathsf{fma}\left(0.5, {x}^{2}, x\right) + {x}^{4} \cdot -0.125\right)\right)}, x\right) \]

7. Taylor expanded in x around 0 53.4%

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

8. Final simplification 53.4%

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

Developer target: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2023322 
(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))