Rust f64::asinh

Percentage Accurate: 30.2% → 99.9%

Time: 10.0s

Alternatives: 13

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 30.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -4:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\mathsf{copysign}\left(x + \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, -0.044642857142857144, 0.075\right), -0.16666666666666666\right) \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -4.0)
     (copysign (log (- (hypot 1.0 x) x)) x)
     (if (<= t_0 0.02)
       (copysign
        (+
         x
         (*
          (fma
           (pow x 2.0)
           (fma (pow x 2.0) -0.044642857142857144 0.075)
           -0.16666666666666666)
          (pow x 3.0)))
        x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -4.0) {
		tmp = copysign(log((hypot(1.0, x) - x)), x);
	} else if (t_0 <= 0.02) {
		tmp = copysign((x + (fma(pow(x, 2.0), fma(pow(x, 2.0), -0.044642857142857144, 0.075), -0.16666666666666666) * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -4.0)
		tmp = copysign(log(Float64(hypot(1.0, x) - x)), x);
	elseif (t_0 <= 0.02)
		tmp = copysign(Float64(x + Float64(fma((x ^ 2.0), fma((x ^ 2.0), -0.044642857142857144, 0.075), -0.16666666666666666) * (x ^ 3.0))), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return 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, -4.0], 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, 0.02], N[With[{TMP1 = Abs[N[(x + N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[Power[x, 2.0], $MachinePrecision] * -0.044642857142857144 + 0.075), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -4

   1. Initial program 48.1%

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

      1. add-cube-cbrt 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 4.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]
   5. Step-by-step derivation

      1. rem-cube-cbrt 4.5%

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

      2. +-commutative 4.5%

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

   6. Applied egg-rr 4.5%

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

      1. add-exp-log 4.5%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 4.5%

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

      4. sqr-neg 4.5%

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

      5. +-commutative 4.5%

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

      6. log-rec 4.5%

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

      7. +-commutative 4.5%

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

      8. log-rec 4.5%

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

      9. sqrt-unprod 4.5%

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

      10. add-sqr-sqrt 4.5%

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

      11. add-exp-log 4.5%

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

      12. flip-+ 1.4%

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

      13. associate-/r/ 1.4%

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

   8. Applied egg-rr 1.4%

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

      1. +-commutative 1.4%

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

      2. associate--r+ 46.4%

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

      3. +-inverses 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. remove-double-neg 100.0%

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

      11. sub-neg 100.0%

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

   10. Simplified 100.0%

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

   if -4 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.0200000000000000004

   1. Initial program 7.9%

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

      1. add-cube-cbrt 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 3.1%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 3.1%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 7.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 7.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 7.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 7.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{1 \cdot x + \left({x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right)\right) \cdot x}, x\right) \]

      2. *-lft-identity 100.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x} + \left({x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right)\right) \cdot x, x\right) \]

      3. *-commutative 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{\left(\left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right) \cdot {x}^{2}\right)} \cdot x, x\right) \]

      4. associate-*l* 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{\left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right) \cdot \left({x}^{2} \cdot x\right)}, x\right) \]

      5. fma-neg 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.075 + -0.044642857142857144 \cdot {x}^{2}, -0.16666666666666666\right)} \cdot \left({x}^{2} \cdot x\right), x\right) \]

      6. +-commutative 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \mathsf{fma}\left({x}^{2}, \color{blue}{-0.044642857142857144 \cdot {x}^{2} + 0.075}, -0.16666666666666666\right) \cdot \left({x}^{2} \cdot x\right), x\right) \]

      7. *-commutative 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \mathsf{fma}\left({x}^{2}, \color{blue}{{x}^{2} \cdot -0.044642857142857144} + 0.075, -0.16666666666666666\right) \cdot \left({x}^{2} \cdot x\right), x\right) \]

      8. fma-define 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \mathsf{fma}\left({x}^{2}, \color{blue}{\mathsf{fma}\left({x}^{2}, -0.044642857142857144, 0.075\right)}, -0.16666666666666666\right) \cdot \left({x}^{2} \cdot x\right), x\right) \]

      9. metadata-eval 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, -0.044642857142857144, 0.075\right), \color{blue}{-0.16666666666666666}\right) \cdot \left({x}^{2} \cdot x\right), x\right) \]

      10. unpow2 100.0%

          \[\leadsto \mathsf{copysign}\left(x + \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, -0.044642857142857144, 0.075\right), -0.16666666666666666\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right), x\right) \]

      11. unpow3 100.0%

          \[\leadsto \mathsf{copysign}\left(x + \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, -0.044642857142857144, 0.075\right), -0.16666666666666666\right) \cdot \color{blue}{{x}^{3}}, x\right) \]

   7. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x + \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, -0.044642857142857144, 0.075\right), -0.16666666666666666\right) \cdot {x}^{3}}, x\right) \]

   if 0.0200000000000000004 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 54.9%

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

      1. *-un-lft-identity 54.9%

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

      2. *-commutative 54.9%

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

      3. log-prod 54.9%

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

      4. +-commutative 54.9%

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

      5. hypot-1-def 98.3%

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

      6. add-sqr-sqrt 98.3%

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

      7. fabs-sqr 98.3%

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

      8. add-sqr-sqrt 98.3%

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

      9. metadata-eval 98.3%

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

   4. Applied egg-rr 98.3%

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

      1. +-rgt-identity 98.3%

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

   6. Simplified 98.3%

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

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -4:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + {x}^{2} \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -4.0)
     (copysign (log (- (hypot 1.0 x) x)) x)
     (if (<= t_0 0.02)
       (copysign
        (*
         x
         (+
          1.0
          (*
           (pow x 2.0)
           (-
            (* (pow x 2.0) (+ 0.075 (* (pow x 2.0) -0.044642857142857144)))
            0.16666666666666666))))
        x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -4.0) {
		tmp = copysign(log((hypot(1.0, x) - x)), x);
	} else if (t_0 <= 0.02) {
		tmp = copysign((x * (1.0 + (pow(x, 2.0) * ((pow(x, 2.0) * (0.075 + (pow(x, 2.0) * -0.044642857142857144))) - 0.16666666666666666)))), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -4.0) {
		tmp = Math.copySign(Math.log((Math.hypot(1.0, x) - x)), x);
	} else if (t_0 <= 0.02) {
		tmp = Math.copySign((x * (1.0 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * (0.075 + (Math.pow(x, 2.0) * -0.044642857142857144))) - 0.16666666666666666)))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -4.0:
		tmp = math.copysign(math.log((math.hypot(1.0, x) - x)), x)
	elif t_0 <= 0.02:
		tmp = math.copysign((x * (1.0 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * (0.075 + (math.pow(x, 2.0) * -0.044642857142857144))) - 0.16666666666666666)))), x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -4.0)
		tmp = copysign(log(Float64(hypot(1.0, x) - x)), x);
	elseif (t_0 <= 0.02)
		tmp = copysign(Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * Float64(0.075 + Float64((x ^ 2.0) * -0.044642857142857144))) - 0.16666666666666666)))), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
	tmp = 0.0;
	if (t_0 <= -4.0)
		tmp = sign(x) * abs(log((hypot(1.0, x) - x)));
	elseif (t_0 <= 0.02)
		tmp = sign(x) * abs((x * (1.0 + ((x ^ 2.0) * (((x ^ 2.0) * (0.075 + ((x ^ 2.0) * -0.044642857142857144))) - 0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -4.0], 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, 0.02], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.075 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -4

   1. Initial program 48.1%

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

      1. add-cube-cbrt 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 4.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]
   5. Step-by-step derivation

      1. rem-cube-cbrt 4.5%

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

      2. +-commutative 4.5%

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

   6. Applied egg-rr 4.5%

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

      1. add-exp-log 4.5%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 4.5%

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

      4. sqr-neg 4.5%

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

      5. +-commutative 4.5%

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

      6. log-rec 4.5%

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

      7. +-commutative 4.5%

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

      8. log-rec 4.5%

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

      9. sqrt-unprod 4.5%

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

      10. add-sqr-sqrt 4.5%

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

      11. add-exp-log 4.5%

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

      12. flip-+ 1.4%

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

      13. associate-/r/ 1.4%

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

   8. Applied egg-rr 1.4%

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

      1. +-commutative 1.4%

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

      2. associate--r+ 46.4%

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

      3. +-inverses 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. remove-double-neg 100.0%

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

      11. sub-neg 100.0%

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

   10. Simplified 100.0%

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

   if -4 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.0200000000000000004

   1. Initial program 7.9%

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

      1. add-cube-cbrt 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 3.1%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 3.1%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 7.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 7.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 7.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 7.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right)\right)}, x\right) \]

   if 0.0200000000000000004 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 54.9%

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

      1. *-un-lft-identity 54.9%

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

      2. *-commutative 54.9%

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

      3. log-prod 54.9%

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

      4. +-commutative 54.9%

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

      5. hypot-1-def 98.3%

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

      6. add-sqr-sqrt 98.3%

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

      7. fabs-sqr 98.3%

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

      8. add-sqr-sqrt 98.3%

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

      9. metadata-eval 98.3%

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

   4. Applied egg-rr 98.3%

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

      1. +-rgt-identity 98.3%

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

   6. Simplified 98.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -4:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.02:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + {x}^{2} \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -4:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -4.0)
     (copysign (log (- (hypot 1.0 x) x)) x)
     (if (<= t_0 0.0)
       (copysign
        (+ x (* (pow x 3.0) (- (* (pow x 2.0) 0.075) 0.16666666666666666)))
        x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -4.0) {
		tmp = copysign(log((hypot(1.0, x) - x)), x);
	} else if (t_0 <= 0.0) {
		tmp = copysign((x + (pow(x, 3.0) * ((pow(x, 2.0) * 0.075) - 0.16666666666666666))), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -4.0) {
		tmp = Math.copySign(Math.log((Math.hypot(1.0, x) - x)), x);
	} else if (t_0 <= 0.0) {
		tmp = Math.copySign((x + (Math.pow(x, 3.0) * ((Math.pow(x, 2.0) * 0.075) - 0.16666666666666666))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -4.0:
		tmp = math.copysign(math.log((math.hypot(1.0, x) - x)), x)
	elif t_0 <= 0.0:
		tmp = math.copysign((x + (math.pow(x, 3.0) * ((math.pow(x, 2.0) * 0.075) - 0.16666666666666666))), x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -4.0)
		tmp = copysign(log(Float64(hypot(1.0, x) - x)), x);
	elseif (t_0 <= 0.0)
		tmp = copysign(Float64(x + Float64((x ^ 3.0) * Float64(Float64((x ^ 2.0) * 0.075) - 0.16666666666666666))), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
	tmp = 0.0;
	if (t_0 <= -4.0)
		tmp = sign(x) * abs(log((hypot(1.0, x) - x)));
	elseif (t_0 <= 0.0)
		tmp = sign(x) * abs((x + ((x ^ 3.0) * (((x ^ 2.0) * 0.075) - 0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -4.0], 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, 0.0], N[With[{TMP1 = Abs[N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * 0.075), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -4

   1. Initial program 48.1%

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

      1. add-cube-cbrt 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 4.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]
   5. Step-by-step derivation

      1. rem-cube-cbrt 4.5%

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

      2. +-commutative 4.5%

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

   6. Applied egg-rr 4.5%

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

      1. add-exp-log 4.5%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 4.5%

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

      4. sqr-neg 4.5%

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

      5. +-commutative 4.5%

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

      6. log-rec 4.5%

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

      7. +-commutative 4.5%

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

      8. log-rec 4.5%

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

      9. sqrt-unprod 4.5%

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

      10. add-sqr-sqrt 4.5%

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

      11. add-exp-log 4.5%

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

      12. flip-+ 1.4%

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

      13. associate-/r/ 1.4%

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

   8. Applied egg-rr 1.4%

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

      1. +-commutative 1.4%

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

      2. associate--r+ 46.4%

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

      3. +-inverses 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. remove-double-neg 100.0%

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

      11. sub-neg 100.0%

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

   10. Simplified 100.0%

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

   if -4 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -0.0

   1. Initial program 7.3%

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

      1. add-cube-cbrt 7.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 7.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 7.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 7.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 2.4%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 2.4%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 7.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 7.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 7.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 7.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{1 \cdot x + \left({x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right)\right) \cdot x}, x\right) \]

      2. *-lft-identity 100.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x} + \left({x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right)\right) \cdot x, x\right) \]

      3. *-commutative 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{\left(\left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right) \cdot {x}^{2}\right)} \cdot x, x\right) \]

      4. associate-*l* 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{\left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right) \cdot \left({x}^{2} \cdot x\right)}, x\right) \]

      5. fma-neg 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.075 + -0.044642857142857144 \cdot {x}^{2}, -0.16666666666666666\right)} \cdot \left({x}^{2} \cdot x\right), x\right) \]

      6. +-commutative 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \mathsf{fma}\left({x}^{2}, \color{blue}{-0.044642857142857144 \cdot {x}^{2} + 0.075}, -0.16666666666666666\right) \cdot \left({x}^{2} \cdot x\right), x\right) \]

      7. *-commutative 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \mathsf{fma}\left({x}^{2}, \color{blue}{{x}^{2} \cdot -0.044642857142857144} + 0.075, -0.16666666666666666\right) \cdot \left({x}^{2} \cdot x\right), x\right) \]

      8. fma-define 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \mathsf{fma}\left({x}^{2}, \color{blue}{\mathsf{fma}\left({x}^{2}, -0.044642857142857144, 0.075\right)}, -0.16666666666666666\right) \cdot \left({x}^{2} \cdot x\right), x\right) \]

      9. metadata-eval 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, -0.044642857142857144, 0.075\right), \color{blue}{-0.16666666666666666}\right) \cdot \left({x}^{2} \cdot x\right), x\right) \]

      10. unpow2 100.0%

          \[\leadsto \mathsf{copysign}\left(x + \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, -0.044642857142857144, 0.075\right), -0.16666666666666666\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right), x\right) \]

      11. unpow3 100.0%

          \[\leadsto \mathsf{copysign}\left(x + \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, -0.044642857142857144, 0.075\right), -0.16666666666666666\right) \cdot \color{blue}{{x}^{3}}, x\right) \]

   7. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x + \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, -0.044642857142857144, 0.075\right), -0.16666666666666666\right) \cdot {x}^{3}}, x\right) \]

   8. Taylor expanded in x around 0 99.9%

      \[\leadsto \mathsf{copysign}\left(x + \color{blue}{{x}^{3} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)}, x\right) \]

   if -0.0 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 55.6%

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

      1. *-un-lft-identity 55.6%

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

      2. *-commutative 55.6%

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

      3. log-prod 55.6%

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

      4. +-commutative 55.6%

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

      5. hypot-1-def 98.2%

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

      6. add-sqr-sqrt 98.2%

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

      7. fabs-sqr 98.2%

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

      8. add-sqr-sqrt 98.2%

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

      9. metadata-eval 98.2%

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

   4. Applied egg-rr 98.2%

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

      1. +-rgt-identity 98.2%

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

   6. Simplified 98.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -4:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.005:\\ \;\;\;\;\mathsf{copysign}\left(2 \cdot \log \left({\left(\mathsf{hypot}\left(1, x\right) - x\right)}^{-0.5}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.005)
     (copysign (* 2.0 (log (pow (- (hypot 1.0 x) x) -0.5))) x)
     (if (<= t_0 0.0) (copysign x x) (copysign (log (+ x (hypot 1.0 x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -0.005) {
		tmp = copysign((2.0 * log(pow((hypot(1.0, x) - x), -0.5))), x);
	} else if (t_0 <= 0.0) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -0.005) {
		tmp = Math.copySign((2.0 * Math.log(Math.pow((Math.hypot(1.0, x) - x), -0.5))), x);
	} else if (t_0 <= 0.0) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -0.005:
		tmp = math.copysign((2.0 * math.log(math.pow((math.hypot(1.0, x) - x), -0.5))), x)
	elif t_0 <= 0.0:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -0.005)
		tmp = copysign(Float64(2.0 * log((Float64(hypot(1.0, x) - x) ^ -0.5))), x);
	elseif (t_0 <= 0.0)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
	tmp = 0.0;
	if (t_0 <= -0.005)
		tmp = sign(x) * abs((2.0 * log(((hypot(1.0, x) - x) ^ -0.5))));
	elseif (t_0 <= 0.0)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.005], N[With[{TMP1 = Abs[N[(2.0 * N[Log[N[Power[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -0.0050000000000000001

   1. Initial program 48.6%

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

      1. add-cube-cbrt 48.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 48.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 48.6%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 48.6%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 5.7%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 5.7%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 5.7%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 5.7%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]
   5. Step-by-step derivation

      1. rem-cube-cbrt 5.7%

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

      2. flip-+ 4.1%

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

      3. div-sub 4.1%

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

      4. pow2 4.1%

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

      5. hypot-1-def 4.1%

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

      6. hypot-1-def 4.1%

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

      7. add-sqr-sqrt 4.0%

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

      8. pow2 4.0%

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

   6. Applied egg-rr 4.0%

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

      1. div-sub 4.1%

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

      2. remove-double-neg 4.1%

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

      3. distribute-frac-neg2 4.1%

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

      4. distribute-frac-neg 4.1%

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

      5. +-commutative 4.1%

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

      6. associate--r+ 47.0%

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

      7. +-inverses 99.8%

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

      8. metadata-eval 99.8%

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

      9. metadata-eval 99.8%

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

      10. neg-sub0 99.8%

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

      11. associate-+l- 99.8%

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

      12. neg-sub0 99.8%

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

      13. +-commutative 99.8%

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

      14. sub-neg 99.8%

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

   8. Simplified 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. log-prod 99.9%

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

      3. inv-pow 99.9%

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

      4. sqrt-pow1 99.9%

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

      5. metadata-eval 99.9%

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

      6. inv-pow 99.9%

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

      7. sqrt-pow1 99.9%

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

      8. metadata-eval 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. count-2 99.9%

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

   12. Simplified 99.9%

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

   if -0.0050000000000000001 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -0.0

   1. Initial program 6.6%

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

      1. add-cube-cbrt 6.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 6.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 6.6%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 6.6%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 2.4%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 2.4%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 6.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 6.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 6.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 6.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

   5. Taylor expanded in x around 0 100.0%

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

   if -0.0 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 55.6%

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

      1. *-un-lft-identity 55.6%

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

      2. *-commutative 55.6%

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

      3. log-prod 55.6%

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

      4. +-commutative 55.6%

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

      5. hypot-1-def 98.2%

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

      6. add-sqr-sqrt 98.2%

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

      7. fabs-sqr 98.2%

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

      8. add-sqr-sqrt 98.2%

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

      9. metadata-eval 98.2%

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

   4. Applied egg-rr 98.2%

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

      1. +-rgt-identity 98.2%

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

   6. Simplified 98.2%

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

Alternative 5: 99.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.8e-6)
   (copysign (log (/ -1.0 (- x (hypot 1.0 x)))) x)
   (if (<= x 8e-6) (copysign x x) (copysign (log (+ x (hypot 1.0 x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -5.8e-6) {
		tmp = copysign(log((-1.0 / (x - hypot(1.0, x)))), x);
	} else if (x <= 8e-6) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -5.8e-6) {
		tmp = Math.copySign(Math.log((-1.0 / (x - Math.hypot(1.0, x)))), x);
	} else if (x <= 8e-6) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5.8e-6:
		tmp = math.copysign(math.log((-1.0 / (x - math.hypot(1.0, x)))), x)
	elif x <= 8e-6:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.8e-6)
		tmp = copysign(log(Float64(-1.0 / Float64(x - hypot(1.0, x)))), x);
	elseif (x <= 8e-6)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5.8e-6)
		tmp = sign(x) * abs(log((-1.0 / (x - hypot(1.0, x)))));
	elseif (x <= 8e-6)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5.8e-6], N[With[{TMP1 = Abs[N[Log[N[(-1.0 / N[(x - N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 8e-6], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8000000000000004e-6

   1. Initial program 48.6%

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

      1. +-commutative 48.6%

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

      2. hypot-1-def 99.8%

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

      3. flip-+ 2.6%

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

      4. hypot-1-def 2.6%

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

      5. hypot-1-def 2.6%

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

      6. add-sqr-sqrt 2.6%

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

      7. +-commutative 2.6%

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

      8. hypot-1-def 2.6%

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

      9. +-commutative 2.6%

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

      10. div-sub 2.5%

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

   4. Applied egg-rr 4.0%

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

      1. div-sub 4.1%

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

      2. fma-undefine 4.1%

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

      3. unpow2 4.1%

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

      4. associate--r+ 47.0%

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

      5. +-inverses 99.8%

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

      6. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

   if -5.8000000000000004e-6 < x < 7.99999999999999964e-6

   1. Initial program 6.6%

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

      1. add-cube-cbrt 6.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 6.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 6.6%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 6.6%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 2.4%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 2.4%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 6.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 6.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 6.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 6.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

   5. Taylor expanded in x around 0 100.0%

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

   if 7.99999999999999964e-6 < x

   1. Initial program 55.6%

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

      1. *-un-lft-identity 55.6%

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

      2. *-commutative 55.6%

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

      3. log-prod 55.6%

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

      4. +-commutative 55.6%

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

      5. hypot-1-def 98.2%

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

      6. add-sqr-sqrt 98.2%

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

      7. fabs-sqr 98.2%

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

      8. add-sqr-sqrt 98.2%

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

      9. metadata-eval 98.2%

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

   4. Applied egg-rr 98.2%

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

      1. +-rgt-identity 98.2%

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

   6. Simplified 98.2%

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

Alternative 6: 99.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.5e-6)
   (copysign (log (- (hypot 1.0 x) x)) x)
   (if (<= x 8e-6) (copysign x x) (copysign (log (+ x (hypot 1.0 x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -5.5e-6) {
		tmp = copysign(log((hypot(1.0, x) - x)), x);
	} else if (x <= 8e-6) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -5.5e-6) {
		tmp = Math.copySign(Math.log((Math.hypot(1.0, x) - x)), x);
	} else if (x <= 8e-6) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5.5e-6:
		tmp = math.copysign(math.log((math.hypot(1.0, x) - x)), x)
	elif x <= 8e-6:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.5e-6)
		tmp = copysign(log(Float64(hypot(1.0, x) - x)), x);
	elseif (x <= 8e-6)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5.5e-6)
		tmp = sign(x) * abs(log((hypot(1.0, x) - x)));
	elseif (x <= 8e-6)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5.5e-6], N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 8e-6], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.4999999999999999e-6

   1. Initial program 48.6%

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

      1. add-cube-cbrt 48.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 48.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 48.6%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 48.6%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 5.7%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 5.7%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 5.7%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 5.7%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]
   5. Step-by-step derivation

      1. rem-cube-cbrt 5.7%

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

      2. +-commutative 5.7%

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

   6. Applied egg-rr 5.7%

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

      1. add-exp-log 5.7%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 5.7%

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

      4. sqr-neg 5.7%

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

      5. +-commutative 5.7%

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

      6. log-rec 5.7%

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

      7. +-commutative 5.7%

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

      8. log-rec 5.7%

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

      9. sqrt-unprod 5.7%

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

      10. add-sqr-sqrt 5.7%

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

      11. add-exp-log 5.7%

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

      12. flip-+ 2.7%

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

      13. associate-/r/ 2.7%

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

   8. Applied egg-rr 2.7%

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

      1. +-commutative 2.7%

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

      2. associate--r+ 47.0%

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

      3. +-inverses 99.8%

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

      4. metadata-eval 99.8%

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

      5. metadata-eval 99.8%

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

      6. neg-mul-1 99.8%

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

      7. sub-neg 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. remove-double-neg 99.8%

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

      11. sub-neg 99.8%

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

   10. Simplified 99.8%

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

   if -5.4999999999999999e-6 < x < 7.99999999999999964e-6

   1. Initial program 6.6%

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

      1. add-cube-cbrt 6.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 6.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 6.6%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 6.6%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 2.4%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 2.4%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 6.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 6.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 6.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 6.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

   5. Taylor expanded in x around 0 100.0%

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

   if 7.99999999999999964e-6 < x

   1. Initial program 55.6%

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

      1. *-un-lft-identity 55.6%

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

      2. *-commutative 55.6%

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

      3. log-prod 55.6%

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

      4. +-commutative 55.6%

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

      5. hypot-1-def 98.2%

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

      6. add-sqr-sqrt 98.2%

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

      7. fabs-sqr 98.2%

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

      8. add-sqr-sqrt 98.2%

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

      9. metadata-eval 98.2%

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

   4. Applied egg-rr 98.2%

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

      1. +-rgt-identity 98.2%

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

   6. Simplified 98.2%

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

Alternative 7: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.00086:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.00086)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = copysign(log((-0.5 / x)), x);
	} else if (x <= 0.00086) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.copySign(Math.log((-0.5 / x)), x);
	} else if (x <= 0.00086) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.copysign(math.log((-0.5 / x)), x)
	elif x <= 0.00086:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = copysign(log(Float64(-0.5 / x)), x);
	elseif (x <= 0.00086)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = sign(x) * abs(log((-0.5 / x)));
	elseif (x <= 0.00086)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], 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, 0.00086], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 48.1%

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

      1. add-cube-cbrt 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 4.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

   5. Taylor expanded in x around -inf 99.2%

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

   if -1.25 < x < 8.59999999999999979e-4

   1. Initial program 7.3%

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

      1. add-cube-cbrt 7.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 7.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 7.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 7.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 2.4%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 2.4%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 7.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 7.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 7.2%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 7.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

   5. Taylor expanded in x around 0 99.7%

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

      1. distribute-rgt-in 99.7%

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

      2. *-lft-identity 99.7%

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

      3. associate-*l* 99.7%

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

      4. unpow2 99.7%

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

      5. unpow3 99.7%

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

   7. Simplified 99.7%

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

   if 8.59999999999999979e-4 < x

   1. Initial program 55.6%

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

      1. *-un-lft-identity 55.6%

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

      2. *-commutative 55.6%

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

      3. log-prod 55.6%

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

      4. +-commutative 55.6%

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

      5. hypot-1-def 98.2%

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

      6. add-sqr-sqrt 98.2%

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

      7. fabs-sqr 98.2%

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

      8. add-sqr-sqrt 98.2%

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

      9. metadata-eval 98.2%

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

   4. Applied egg-rr 98.2%

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

      1. +-rgt-identity 98.2%

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

   6. Simplified 98.2%

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

Alternative 8: 99.4% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.25)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (/ 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 <= 1.25) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log((0.5 / x)), x);
	}
	return tmp;
}

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 48.1%

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

      1. add-cube-cbrt 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 4.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

   5. Taylor expanded in x around -inf 99.2%

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

   if -1.25 < x < 1.25

   1. Initial program 7.9%

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

      1. add-cube-cbrt 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 3.1%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 3.1%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 7.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 7.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 7.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 7.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

   5. Taylor expanded in x around 0 99.5%

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

      1. distribute-rgt-in 99.5%

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

      2. *-lft-identity 99.5%

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

      3. associate-*l* 99.5%

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

      4. unpow2 99.5%

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

      5. unpow3 99.5%

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

   7. Simplified 99.5%

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

   if 1.25 < x

   1. Initial program 54.9%

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

      1. add-cube-cbrt 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 98.3%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 98.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]
   5. Step-by-step derivation

      1. rem-cube-cbrt 98.3%

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

      2. +-commutative 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. add-exp-log 98.3%

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

      2. add-sqr-sqrt 97.7%

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

      3. sqrt-unprod 98.3%

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

      4. sqr-neg 98.3%

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

      5. +-commutative 98.3%

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

      6. log-rec 98.3%

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

      7. +-commutative 98.3%

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

      8. log-rec 98.3%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 98.3%

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

      11. add-exp-log 98.3%

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

      12. flip-+ 3.8%

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

      13. associate-/r/ 3.8%

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

   8. Applied egg-rr 3.6%

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

      1. +-commutative 3.6%

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

      2. associate--r+ 5.1%

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

      3. +-inverses 6.6%

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

      4. metadata-eval 6.6%

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

      5. metadata-eval 6.6%

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

      6. neg-mul-1 6.6%

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

      7. sub-neg 6.6%

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

      8. +-commutative 6.6%

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

      9. distribute-neg-in 6.6%

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

      10. remove-double-neg 6.6%

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

      11. sub-neg 6.6%

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

   10. Simplified 6.6%

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

   11. Taylor expanded in x around inf 97.9%

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

Alternative 9: 99.1% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.25) (copysign x x) (copysign (log (/ 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 <= 1.25) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log((0.5 / x)), x);
	}
	return tmp;
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 48.1%

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

      1. add-cube-cbrt 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 4.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

   5. Taylor expanded in x around -inf 99.2%

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

   if -1.25 < x < 1.25

   1. Initial program 7.9%

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

      1. add-cube-cbrt 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 3.1%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 3.1%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 7.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 7.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 7.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 7.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

   5. Taylor expanded in x around 0 99.1%

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

   if 1.25 < x

   1. Initial program 54.9%

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

      1. add-cube-cbrt 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 98.3%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 98.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]
   5. Step-by-step derivation

      1. rem-cube-cbrt 98.3%

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

      2. +-commutative 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. add-exp-log 98.3%

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

      2. add-sqr-sqrt 97.7%

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

      3. sqrt-unprod 98.3%

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

      4. sqr-neg 98.3%

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

      5. +-commutative 98.3%

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

      6. log-rec 98.3%

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

      7. +-commutative 98.3%

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

      8. log-rec 98.3%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 98.3%

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

      11. add-exp-log 98.3%

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

      12. flip-+ 3.8%

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

      13. associate-/r/ 3.8%

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

   8. Applied egg-rr 3.6%

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

      1. +-commutative 3.6%

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

      2. associate--r+ 5.1%

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

      3. +-inverses 6.6%

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

      4. metadata-eval 6.6%

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

      5. metadata-eval 6.6%

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

      6. neg-mul-1 6.6%

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

      7. sub-neg 6.6%

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

      8. +-commutative 6.6%

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

      9. distribute-neg-in 6.6%

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

      10. remove-double-neg 6.6%

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

      11. sub-neg 6.6%

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

   10. Simplified 6.6%

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

   11. Taylor expanded in x around inf 97.9%

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

Alternative 10: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 48.1%

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

      1. add-cube-cbrt 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 48.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 4.5%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 4.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

   5. Taylor expanded in x around -inf 99.2%

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

   if -1.25 < x < 1.25

   1. Initial program 7.9%

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

      1. add-cube-cbrt 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 7.8%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 3.1%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 3.1%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 7.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 7.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 7.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 7.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

   5. Taylor expanded in x around 0 99.1%

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

   if 1.25 < x

   1. Initial program 54.9%

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

      1. add-cube-cbrt 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 98.3%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 98.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

   5. Taylor expanded in x around inf 96.2%

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

      1. *-commutative 96.2%

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

   7. Simplified 96.2%

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

Alternative 11: 74.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 21.9%

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

      1. add-cube-cbrt 21.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 21.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 21.8%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 21.8%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 2.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 2.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 6.7%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 6.7%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 6.7%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 6.7%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

   5. Taylor expanded in x around 0 66.4%

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

   if 1.25 < x

   1. Initial program 54.9%

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

      1. add-cube-cbrt 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 54.9%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 98.3%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 98.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

   5. Taylor expanded in x around inf 96.2%

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

      1. *-commutative 96.2%

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

   7. Simplified 96.2%

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

Alternative 12: 57.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 1.55], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 21.9%

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

      1. add-cube-cbrt 21.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. pow3 21.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

      3. *-un-lft-identity 21.8%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

      4. *-un-lft-identity 21.8%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

      5. add-sqr-sqrt 2.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      6. fabs-sqr 2.0%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      7. add-sqr-sqrt 6.7%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

      8. +-commutative 6.7%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

      9. hypot-1-def 6.7%

         \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

   4. Applied egg-rr 6.7%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

   5. Taylor expanded in x around 0 66.4%

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

   if 1.55000000000000004 < x

   1. Initial program 54.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 31.3%

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

      1. log1p-define 31.3%

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

      2. rem-square-sqrt 31.3%

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

      3. fabs-sqr 31.3%

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

      4. rem-square-sqrt 31.3%

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

   5. Simplified 31.3%

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

Alternative 13: 51.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.4%

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

   1. add-cube-cbrt 29.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}} \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right) \cdot \sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}, x\right) \]

   2. pow3 29.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{\left|x\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right)}, x\right) \]

   3. *-un-lft-identity 29.3%

      \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}}\right)}^{3}\right), x\right) \]

   4. *-un-lft-identity 29.3%

      \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\left|x\right| + \sqrt{x \cdot x + 1}}}\right)}^{3}\right), x\right) \]

   5. add-sqr-sqrt 14.0%

      \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

   6. fabs-sqr 14.0%

      \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

   7. add-sqr-sqrt 17.6%

      \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{\color{blue}{x} + \sqrt{x \cdot x + 1}}\right)}^{3}\right), x\right) \]

   8. +-commutative 17.6%

      \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \sqrt{\color{blue}{1 + x \cdot x}}}\right)}^{3}\right), x\right) \]

   9. hypot-1-def 27.4%

      \[\leadsto \mathsf{copysign}\left(\log \left({\left(\sqrt[3]{x + \color{blue}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right), x\right) \]

4. Applied egg-rr 27.4%

   \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\sqrt[3]{x + \mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}, x\right) \]

5. Taylor expanded in x around 0 52.7%

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

Developer target: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024106 
(FPCore (x)
  :name "Rust f64::asinh"
  :precision binary64

  :alt
  (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)

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