Rust f64::asinh

Percentage Accurate: 29.7% → 99.9%

Time: 10.3s

Alternatives: 12

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 29.7% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + 1\right)}^{2}\\ t_1 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(0.001388888888888889, {x}^{2} \cdot \left(\frac{45}{x + 1} + \left(\frac{45}{t\_0} + \frac{30}{{\left(x + 1\right)}^{3}}\right)\right), \frac{-0.125}{x + 1} + \frac{-0.125}{t\_0}\right), \frac{0.5}{x + 1}\right), \mathsf{log1p}\left(x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (+ x 1.0) 2.0))
        (t_1 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_1 -1.0)
     (copysign (log (- (hypot 1.0 x) x)) x)
     (if (<= t_1 0.0002)
       (copysign
        (fma
         (pow x 2.0)
         (fma
          (pow x 2.0)
          (fma
           0.001388888888888889
           (*
            (pow x 2.0)
            (+
             (/ 45.0 (+ x 1.0))
             (+ (/ 45.0 t_0) (/ 30.0 (pow (+ x 1.0) 3.0)))))
           (+ (/ -0.125 (+ x 1.0)) (/ -0.125 t_0)))
          (/ 0.5 (+ x 1.0)))
         (log1p x))
        x)
       (copysign (log (+ (fabs x) (hypot 1.0 x))) x)))))

C

double code(double x) {
	double t_0 = pow((x + 1.0), 2.0);
	double t_1 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = copysign(log((hypot(1.0, x) - x)), x);
	} else if (t_1 <= 0.0002) {
		tmp = copysign(fma(pow(x, 2.0), fma(pow(x, 2.0), fma(0.001388888888888889, (pow(x, 2.0) * ((45.0 / (x + 1.0)) + ((45.0 / t_0) + (30.0 / pow((x + 1.0), 3.0))))), ((-0.125 / (x + 1.0)) + (-0.125 / t_0))), (0.5 / (x + 1.0))), log1p(x)), x);
	} else {
		tmp = copysign(log((fabs(x) + hypot(1.0, x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x + 1.0) ^ 2.0
	t_1 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = copysign(log(Float64(hypot(1.0, x) - x)), x);
	elseif (t_1 <= 0.0002)
		tmp = copysign(fma((x ^ 2.0), fma((x ^ 2.0), fma(0.001388888888888889, Float64((x ^ 2.0) * Float64(Float64(45.0 / Float64(x + 1.0)) + Float64(Float64(45.0 / t_0) + Float64(30.0 / (Float64(x + 1.0) ^ 3.0))))), Float64(Float64(-0.125 / Float64(x + 1.0)) + Float64(-0.125 / t_0))), Float64(0.5 / Float64(x + 1.0))), log1p(x)), x);
	else
		tmp = copysign(log(Float64(abs(x) + hypot(1.0, x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(x + 1.0), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = 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$1, -1.0], N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$1, 0.0002], N[With[{TMP1 = Abs[N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.001388888888888889 * N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(45.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(45.0 / t$95$0), $MachinePrecision] + N[(30.0 / N[Power[N[(x + 1.0), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.125 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(-0.125 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]]

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) < -1

   1. Initial program 39.9%

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

      1. flip-+ 2.3%

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

      2. clear-num 2.3%

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

      3. log-div 2.3%

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

      4. metadata-eval 2.3%

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

      5. +-commutative 2.3%

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

      6. hypot-1-def 2.3%

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

      7. add-sqr-sqrt 0.0%

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

      8. fabs-sqr 0.0%

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

      9. add-sqr-sqrt 3.4%

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

   4. Applied egg-rr 3.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 3.4%

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

      2. div-sub 3.4%

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

      3. *-rgt-identity 3.4%

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

      4. associate-/l* 3.4%

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

      5. fmm-def 3.4%

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

      6. fma-undefine 3.4%

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

      7. unpow2 3.4%

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

      8. associate--r+ 3.4%

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

      9. +-inverses 3.4%

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

      10. metadata-eval 3.4%

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

      11. metadata-eval 3.4%

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

      12. *-rgt-identity 3.4%

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

      13. associate-/l* 3.4%

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

      14. fma-undefine 3.4%

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

      15. unpow2 3.4%

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

      16. associate--r+ 37.9%

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

   6. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 100.0%

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

      4. sqr-neg 100.0%

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

      5. sqrt-unprod 99.1%

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

      6. add-sqr-sqrt 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   10. Simplified 100.0%

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

   if -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 2.0000000000000001e-4

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0 9.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + {x}^{2} \cdot \left({x}^{2} \cdot \left(-0.041666666666666664 \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right) + 0.001388888888888889 \cdot \left({x}^{2} \cdot \left(45 \cdot \frac{1}{1 + \left|x\right|} + \left(45 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}} + 30 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{3}}\right)\right)\right)\right) + 0.5 \cdot \frac{1}{1 + \left|x\right|}\right)}, x\right) \]

   5. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(0.001388888888888889, {x}^{2} \cdot \left(\frac{45}{x + 1} + \left(\frac{45}{{\left(x + 1\right)}^{2}} + \frac{30}{{\left(x + 1\right)}^{3}}\right)\right), \frac{-0.125}{x + 1} + \frac{-0.125}{{\left(x + 1\right)}^{2}}\right), \frac{0.5}{x + 1}\right), \mathsf{log1p}\left(x\right)\right)}, x\right) \]

   if 2.0000000000000001e-4 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 68.9%

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

      1. +-commutative 68.9%

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

      2. hypot-1-def 99.7%

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

   3. Simplified 99.7%

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

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.0002:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (- (hypot 1.0 x) x)) x)
     (if (<= t_0 0.0002)
       (copysign
        (*
         x
         (+
          1.0
          (*
           (pow x 2.0)
           (-
            (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144)))
            0.16666666666666666))))
        x)
       (copysign (log (+ (fabs 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 <= -1.0) {
		tmp = copysign(log((hypot(1.0, x) - x)), x);
	} else if (t_0 <= 0.0002) {
		tmp = copysign((x * (1.0 + (pow(x, 2.0) * (((x * x) * (0.075 + ((x * x) * -0.044642857142857144))) - 0.16666666666666666)))), x);
	} else {
		tmp = copysign(log((fabs(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 <= -1.0) {
		tmp = Math.copySign(Math.log((Math.hypot(1.0, x) - x)), x);
	} else if (t_0 <= 0.0002) {
		tmp = Math.copySign((x * (1.0 + (Math.pow(x, 2.0) * (((x * x) * (0.075 + ((x * x) * -0.044642857142857144))) - 0.16666666666666666)))), x);
	} else {
		tmp = Math.copySign(Math.log((Math.abs(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 <= -1.0:
		tmp = math.copysign(math.log((math.hypot(1.0, x) - x)), x)
	elif t_0 <= 0.0002:
		tmp = math.copysign((x * (1.0 + (math.pow(x, 2.0) * (((x * x) * (0.075 + ((x * x) * -0.044642857142857144))) - 0.16666666666666666)))), x)
	else:
		tmp = math.copysign(math.log((math.fabs(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 <= -1.0)
		tmp = copysign(log(Float64(hypot(1.0, x) - x)), x);
	elseif (t_0 <= 0.0002)
		tmp = copysign(Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(Float64(x * x) * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144))) - 0.16666666666666666)))), x);
	else
		tmp = copysign(log(Float64(abs(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 <= -1.0)
		tmp = sign(x) * abs(log((hypot(1.0, x) - x)));
	elseif (t_0 <= 0.0002)
		tmp = sign(x) * abs((x * (1.0 + ((x ^ 2.0) * (((x * x) * (0.075 + ((x * x) * -0.044642857142857144))) - 0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log((abs(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, -1.0], N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.0002], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(0.075 + N[(N[(x * x), $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[(N[Abs[x], $MachinePrecision] + 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) < -1

   1. Initial program 39.9%

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

      1. flip-+ 2.3%

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

      2. clear-num 2.3%

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

      3. log-div 2.3%

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

      4. metadata-eval 2.3%

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

      5. +-commutative 2.3%

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

      6. hypot-1-def 2.3%

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

      7. add-sqr-sqrt 0.0%

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

      8. fabs-sqr 0.0%

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

      9. add-sqr-sqrt 3.4%

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

   4. Applied egg-rr 3.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 3.4%

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

      2. div-sub 3.4%

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

      3. *-rgt-identity 3.4%

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

      4. associate-/l* 3.4%

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

      5. fmm-def 3.4%

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

      6. fma-undefine 3.4%

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

      7. unpow2 3.4%

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

      8. associate--r+ 3.4%

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

      9. +-inverses 3.4%

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

      10. metadata-eval 3.4%

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

      11. metadata-eval 3.4%

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

      12. *-rgt-identity 3.4%

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

      13. associate-/l* 3.4%

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

      14. fma-undefine 3.4%

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

      15. unpow2 3.4%

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

      16. associate--r+ 37.9%

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

   6. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 100.0%

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

      4. sqr-neg 100.0%

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

      5. sqrt-unprod 99.1%

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

      6. add-sqr-sqrt 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   10. Simplified 100.0%

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

   if -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 2.0000000000000001e-4

   1. Initial program 8.1%

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

      1. flip-+ 8.0%

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

      2. clear-num 8.0%

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

      3. log-div 7.9%

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

      4. metadata-eval 7.9%

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

      5. +-commutative 7.9%

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

      6. hypot-1-def 7.9%

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

      7. add-sqr-sqrt 5.3%

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

      8. fabs-sqr 5.3%

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

      9. add-sqr-sqrt 7.9%

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

   4. Applied egg-rr 8.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 8.0%

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

      2. div-sub 7.9%

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

      3. *-rgt-identity 7.9%

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

      4. associate-/l* 7.9%

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

      5. fmm-def 7.9%

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

      6. fma-undefine 7.9%

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

      7. unpow2 7.9%

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

      8. associate--r+ 7.9%

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

      9. +-inverses 7.9%

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

      10. metadata-eval 7.9%

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

      11. metadata-eval 7.9%

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

      12. *-rgt-identity 7.9%

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

      13. associate-/l* 7.9%

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

      14. fma-undefine 7.9%

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

      15. unpow2 7.9%

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

      16. associate--r+ 8.0%

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

   6. Simplified 8.0%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. unpow2 100.0%

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

   11. Applied egg-rr 100.0%

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

   if 2.0000000000000001e-4 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 68.9%

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

      1. +-commutative 68.9%

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

      2. hypot-1-def 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.0002:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.022:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.0245:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.022)
   (copysign (log (- (hypot 1.0 x) x)) x)
   (if (<= x 0.0245)
     (copysign
      (*
       x
       (+
        1.0
        (*
         (pow x 2.0)
         (-
          (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144)))
          0.16666666666666666))))
      x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.022) {
		tmp = copysign(log((hypot(1.0, x) - x)), x);
	} else if (x <= 0.0245) {
		tmp = copysign((x * (1.0 + (pow(x, 2.0) * (((x * x) * (0.075 + ((x * x) * -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 tmp;
	if (x <= -0.022) {
		tmp = Math.copySign(Math.log((Math.hypot(1.0, x) - x)), x);
	} else if (x <= 0.0245) {
		tmp = Math.copySign((x * (1.0 + (Math.pow(x, 2.0) * (((x * x) * (0.075 + ((x * x) * -0.044642857142857144))) - 0.16666666666666666)))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.022:
		tmp = math.copysign(math.log((math.hypot(1.0, x) - x)), x)
	elif x <= 0.0245:
		tmp = math.copysign((x * (1.0 + (math.pow(x, 2.0) * (((x * x) * (0.075 + ((x * x) * -0.044642857142857144))) - 0.16666666666666666)))), 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 <= -0.022)
		tmp = copysign(log(Float64(hypot(1.0, x) - x)), x);
	elseif (x <= 0.0245)
		tmp = copysign(Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(Float64(x * x) * Float64(0.075 + Float64(Float64(x * x) * -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)
	tmp = 0.0;
	if (x <= -0.022)
		tmp = sign(x) * abs(log((hypot(1.0, x) - x)));
	elseif (x <= 0.0245)
		tmp = sign(x) * abs((x * (1.0 + ((x ^ 2.0) * (((x * x) * (0.075 + ((x * x) * -0.044642857142857144))) - 0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.022], N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 0.0245], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(0.075 + N[(N[(x * x), $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 x < -0.021999999999999999

   1. Initial program 39.9%

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

      1. flip-+ 2.3%

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

      2. clear-num 2.3%

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

      3. log-div 2.3%

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

      4. metadata-eval 2.3%

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

      5. +-commutative 2.3%

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

      6. hypot-1-def 2.3%

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

      7. add-sqr-sqrt 0.0%

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

      8. fabs-sqr 0.0%

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

      9. add-sqr-sqrt 3.4%

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

   4. Applied egg-rr 3.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 3.4%

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

      2. div-sub 3.4%

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

      3. *-rgt-identity 3.4%

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

      4. associate-/l* 3.4%

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

      5. fmm-def 3.4%

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

      6. fma-undefine 3.4%

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

      7. unpow2 3.4%

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

      8. associate--r+ 3.4%

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

      9. +-inverses 3.4%

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

      10. metadata-eval 3.4%

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

      11. metadata-eval 3.4%

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

      12. *-rgt-identity 3.4%

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

      13. associate-/l* 3.4%

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

      14. fma-undefine 3.4%

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

      15. unpow2 3.4%

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

      16. associate--r+ 37.9%

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

   6. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 100.0%

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

      4. sqr-neg 100.0%

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

      5. sqrt-unprod 99.1%

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

      6. add-sqr-sqrt 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   10. Simplified 100.0%

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

   if -0.021999999999999999 < x < 0.024500000000000001

   1. Initial program 8.1%

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

      1. flip-+ 8.0%

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

      2. clear-num 8.0%

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

      3. log-div 7.9%

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

      4. metadata-eval 7.9%

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

      5. +-commutative 7.9%

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

      6. hypot-1-def 7.9%

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

      7. add-sqr-sqrt 5.3%

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

      8. fabs-sqr 5.3%

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

      9. add-sqr-sqrt 7.9%

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

   4. Applied egg-rr 8.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 8.0%

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

      2. div-sub 7.9%

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

      3. *-rgt-identity 7.9%

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

      4. associate-/l* 7.9%

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

      5. fmm-def 7.9%

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

      6. fma-undefine 7.9%

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

      7. unpow2 7.9%

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

      8. associate--r+ 7.9%

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

      9. +-inverses 7.9%

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

      10. metadata-eval 7.9%

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

      11. metadata-eval 7.9%

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

      12. *-rgt-identity 7.9%

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

      13. associate-/l* 7.9%

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

      14. fma-undefine 7.9%

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

      15. unpow2 7.9%

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

      16. associate--r+ 8.0%

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

   6. Simplified 8.0%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. unpow2 100.0%

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

   11. Applied egg-rr 100.0%

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

   if 0.024500000000000001 < x

   1. Initial program 68.9%

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

   3. Taylor expanded in x around 0 68.9%

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

      1. rem-square-sqrt 68.9%

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

      2. fabs-sqr 68.9%

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

      3. metadata-eval 68.9%

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

      4. unpow2 68.9%

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

      5. hypot-undefine 99.7%

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

      6. rem-square-sqrt 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.022:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.0245:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.0245:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 0.0245)
     (copysign
      (*
       x
       (+
        1.0
        (*
         (pow x 2.0)
         (-
          (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144)))
          0.16666666666666666))))
      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 <= 0.0245) {
		tmp = copysign((x * (1.0 + (pow(x, 2.0) * (((x * x) * (0.075 + ((x * x) * -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 tmp;
	if (x <= -1.25) {
		tmp = Math.copySign(Math.log((-0.5 / x)), x);
	} else if (x <= 0.0245) {
		tmp = Math.copySign((x * (1.0 + (Math.pow(x, 2.0) * (((x * x) * (0.075 + ((x * x) * -0.044642857142857144))) - 0.16666666666666666)))), 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 <= 0.0245:
		tmp = math.copysign((x * (1.0 + (math.pow(x, 2.0) * (((x * x) * (0.075 + ((x * x) * -0.044642857142857144))) - 0.16666666666666666)))), 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 <= 0.0245)
		tmp = copysign(Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(Float64(x * x) * Float64(0.075 + Float64(Float64(x * x) * -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)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = sign(x) * abs(log((-0.5 / x)));
	elseif (x <= 0.0245)
		tmp = sign(x) * abs((x * (1.0 + ((x ^ 2.0) * (((x * x) * (0.075 + ((x * x) * -0.044642857142857144))) - 0.16666666666666666)))));
	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, 0.0245], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(0.075 + N[(N[(x * x), $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 x < -1.25

   1. Initial program 39.9%

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

   3. Taylor expanded in x around 0 39.9%

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

      1. rem-square-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. metadata-eval 0.0%

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

      4. unpow2 0.0%

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

      5. hypot-undefine 0.0%

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

      6. rem-square-sqrt 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in x around -inf 98.7%

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

   if -1.25 < x < 0.024500000000000001

   1. Initial program 8.1%

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

      1. flip-+ 8.0%

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

      2. clear-num 8.0%

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

      3. log-div 7.9%

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

      4. metadata-eval 7.9%

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

      5. +-commutative 7.9%

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

      6. hypot-1-def 7.9%

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

      7. add-sqr-sqrt 5.3%

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

      8. fabs-sqr 5.3%

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

      9. add-sqr-sqrt 7.9%

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

   4. Applied egg-rr 8.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 8.0%

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

      2. div-sub 7.9%

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

      3. *-rgt-identity 7.9%

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

      4. associate-/l* 7.9%

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

      5. fmm-def 7.9%

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

      6. fma-undefine 7.9%

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

      7. unpow2 7.9%

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

      8. associate--r+ 7.9%

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

      9. +-inverses 7.9%

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

      10. metadata-eval 7.9%

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

      11. metadata-eval 7.9%

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

      12. *-rgt-identity 7.9%

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

      13. associate-/l* 7.9%

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

      14. fma-undefine 7.9%

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

      15. unpow2 7.9%

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

      16. associate--r+ 8.0%

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

   6. Simplified 8.0%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. unpow2 100.0%

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

   11. Applied egg-rr 100.0%

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

   if 0.024500000000000001 < x

   1. Initial program 68.9%

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

   3. Taylor expanded in x around 0 68.9%

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

      1. rem-square-sqrt 68.9%

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

      2. fabs-sqr 68.9%

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

      3. metadata-eval 68.9%

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

      4. unpow2 68.9%

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

      5. hypot-undefine 99.7%

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

      6. rem-square-sqrt 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.0245:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.25)
     (copysign
      (*
       x
       (+
        1.0
        (*
         (pow x 2.0)
         (-
          (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144)))
          0.16666666666666666))))
      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 * (1.0 + (pow(x, 2.0) * (((x * x) * (0.075 + ((x * x) * -0.044642857142857144))) - 0.16666666666666666)))), 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 * (1.0 + (Math.pow(x, 2.0) * (((x * x) * (0.075 + ((x * x) * -0.044642857142857144))) - 0.16666666666666666)))), 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 * (1.0 + (math.pow(x, 2.0) * (((x * x) * (0.075 + ((x * x) * -0.044642857142857144))) - 0.16666666666666666)))), 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(Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(Float64(x * x) * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144))) - 0.16666666666666666)))), 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 * (1.0 + ((x ^ 2.0) * (((x * x) * (0.075 + ((x * x) * -0.044642857142857144))) - 0.16666666666666666)))));
	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[N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(0.075 + N[(N[(x * x), $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 * 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 39.9%

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

   3. Taylor expanded in x around 0 39.9%

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

      1. rem-square-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. metadata-eval 0.0%

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

      4. unpow2 0.0%

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

      5. hypot-undefine 0.0%

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

      6. rem-square-sqrt 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in x around -inf 98.7%

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

   if -1.25 < x < 1.25

   1. Initial program 10.6%

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

      1. flip-+ 10.6%

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

      2. clear-num 10.6%

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

      3. log-div 10.5%

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

      4. metadata-eval 10.5%

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

      5. +-commutative 10.5%

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

      6. hypot-1-def 10.5%

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

      7. add-sqr-sqrt 7.9%

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

      8. fabs-sqr 7.9%

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

      9. add-sqr-sqrt 10.5%

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

   4. Applied egg-rr 10.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 10.6%

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

      2. div-sub 10.5%

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

      3. *-rgt-identity 10.5%

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

      4. associate-/l* 10.5%

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

      5. fmm-def 10.5%

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

      6. fma-undefine 10.5%

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

      7. unpow2 10.5%

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

      8. associate--r+ 10.5%

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

      9. +-inverses 10.5%

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

      10. metadata-eval 10.5%

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

      11. metadata-eval 10.5%

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

      12. *-rgt-identity 10.5%

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

      13. associate-/l* 10.5%

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

      14. fma-undefine 10.5%

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

      15. unpow2 10.5%

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

      16. associate--r+ 10.6%

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

   6. Simplified 10.6%

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

   7. Taylor expanded in x around 0 99.2%

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

      1. unpow2 99.2%

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

   9. Applied egg-rr 99.2%

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

       1. unpow2 99.2%

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

   11. Applied egg-rr 99.2%

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

   if 1.25 < x

   1. Initial program 67.2%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. rem-square-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. *-inverses 100.0%

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

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.25)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) 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 + (-0.16666666666666666 * pow(x, 3.0))), 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 + (-0.16666666666666666 * Math.pow(x, 3.0))), 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 + (-0.16666666666666666 * math.pow(x, 3.0))), 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(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), 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 + (-0.16666666666666666 * (x ^ 3.0))));
	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[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 * 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 39.9%

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

   3. Taylor expanded in x around 0 39.9%

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

      1. rem-square-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. metadata-eval 0.0%

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

      4. unpow2 0.0%

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

      5. hypot-undefine 0.0%

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

      6. rem-square-sqrt 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in x around -inf 98.7%

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

   if -1.25 < x < 1.25

   1. Initial program 10.6%

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

      1. flip-+ 10.6%

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

      2. clear-num 10.6%

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

      3. log-div 10.5%

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

      4. metadata-eval 10.5%

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

      5. +-commutative 10.5%

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

      6. hypot-1-def 10.5%

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

      7. add-sqr-sqrt 7.9%

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

      8. fabs-sqr 7.9%

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

      9. add-sqr-sqrt 10.5%

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

   4. Applied egg-rr 10.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 10.6%

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

      2. div-sub 10.5%

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

      3. *-rgt-identity 10.5%

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

      4. associate-/l* 10.5%

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

      5. fmm-def 10.5%

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

      6. fma-undefine 10.5%

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

      7. unpow2 10.5%

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

      8. associate--r+ 10.5%

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

      9. +-inverses 10.5%

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

      10. metadata-eval 10.5%

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

      11. metadata-eval 10.5%

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

      12. *-rgt-identity 10.5%

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

      13. associate-/l* 10.5%

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

      14. fma-undefine 10.5%

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

      15. unpow2 10.5%

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

      16. associate--r+ 10.6%

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

   6. Simplified 10.6%

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

   7. Taylor expanded in x around 0 98.2%

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

      1. distribute-rgt-in 98.2%

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

      2. *-lft-identity 98.2%

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

      3. associate-*l* 98.2%

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

      4. unpow2 98.2%

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

      5. unpow3 98.2%

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

   9. Simplified 98.2%

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

   if 1.25 < x

   1. Initial program 67.2%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. rem-square-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. *-inverses 100.0%

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

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

Alternative 7: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.25)
     (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) 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 * (1.0 + ((x * x) * -0.16666666666666666))), 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 * (1.0 + ((x * x) * -0.16666666666666666))), 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 * (1.0 + ((x * x) * -0.16666666666666666))), 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(Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666))), 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 * (1.0 + ((x * x) * -0.16666666666666666))));
	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[N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], 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 39.9%

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

   3. Taylor expanded in x around 0 39.9%

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

      1. rem-square-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. metadata-eval 0.0%

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

      4. unpow2 0.0%

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

      5. hypot-undefine 0.0%

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

      6. rem-square-sqrt 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in x around -inf 98.7%

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

   if -1.25 < x < 1.25

   1. Initial program 10.6%

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

      1. flip-+ 10.6%

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

      2. clear-num 10.6%

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

      3. log-div 10.5%

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

      4. metadata-eval 10.5%

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

      5. +-commutative 10.5%

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

      6. hypot-1-def 10.5%

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

      7. add-sqr-sqrt 7.9%

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

      8. fabs-sqr 7.9%

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

      9. add-sqr-sqrt 10.5%

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

   4. Applied egg-rr 10.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 10.6%

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

      2. div-sub 10.5%

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

      3. *-rgt-identity 10.5%

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

      4. associate-/l* 10.5%

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

      5. fmm-def 10.5%

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

      6. fma-undefine 10.5%

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

      7. unpow2 10.5%

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

      8. associate--r+ 10.5%

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

      9. +-inverses 10.5%

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

      10. metadata-eval 10.5%

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

      11. metadata-eval 10.5%

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

      12. *-rgt-identity 10.5%

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

      13. associate-/l* 10.5%

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

      14. fma-undefine 10.5%

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

      15. unpow2 10.5%

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

      16. associate--r+ 10.6%

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

   6. Simplified 10.6%

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

   7. Taylor expanded in x around 0 98.2%

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

      1. *-commutative 98.2%

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

   9. Simplified 98.2%

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

       1. unpow2 99.2%

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

   11. Applied egg-rr 98.2%

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

   if 1.25 < x

   1. Initial program 67.2%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. rem-square-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. *-inverses 100.0%

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

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

Alternative 8: 82.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.5)
   (copysign (log (- x)) x)
   (if (<= x 1.25)
     (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
     (copysign (log (* x 2.0)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -3.5) {
		tmp = copysign(log(-x), x);
	} else if (x <= 1.25) {
		tmp = copysign((x * (1.0 + ((x * x) * -0.16666666666666666))), x);
	} else {
		tmp = copysign(log((x * 2.0)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.5) {
		tmp = Math.copySign(Math.log(-x), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign((x * (1.0 + ((x * x) * -0.16666666666666666))), x);
	} else {
		tmp = Math.copySign(Math.log((x * 2.0)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.5:
		tmp = math.copysign(math.log(-x), x)
	elif x <= 1.25:
		tmp = math.copysign((x * (1.0 + ((x * x) * -0.16666666666666666))), x)
	else:
		tmp = math.copysign(math.log((x * 2.0)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.5)
		tmp = copysign(log(Float64(-x)), x);
	elseif (x <= 1.25)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666))), 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 <= -3.5)
		tmp = sign(x) * abs(log(-x));
	elseif (x <= 1.25)
		tmp = sign(x) * abs((x * (1.0 + ((x * x) * -0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.5], N[With[{TMP1 = Abs[N[Log[(-x)], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], 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 < -3.5

   1. Initial program 38.8%

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

   3. Taylor expanded in x around -inf 31.9%

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

      1. neg-mul-1 31.9%

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

   5. Simplified 31.9%

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

   if -3.5 < x < 1.25

   1. Initial program 11.3%

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

      1. flip-+ 11.2%

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

      2. clear-num 11.2%

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

      3. log-div 11.2%

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

      4. metadata-eval 11.2%

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

      5. +-commutative 11.2%

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

      6. hypot-1-def 11.1%

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

      7. add-sqr-sqrt 7.9%

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

      8. fabs-sqr 7.9%

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

      9. add-sqr-sqrt 11.1%

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

   4. Applied egg-rr 11.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 11.2%

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

      2. div-sub 11.2%

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

      3. *-rgt-identity 11.2%

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

      4. associate-/l* 11.2%

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

      5. fmm-def 11.2%

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

      6. fma-undefine 11.2%

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

      7. unpow2 11.2%

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

      8. associate--r+ 11.2%

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

      9. +-inverses 11.2%

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

      10. metadata-eval 11.2%

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

      11. metadata-eval 11.2%

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

      12. *-rgt-identity 11.2%

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

      13. associate-/l* 11.2%

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

      14. fma-undefine 11.2%

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

      15. unpow2 11.2%

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

      16. associate--r+ 11.3%

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

   6. Simplified 11.3%

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

   7. Taylor expanded in x around 0 97.7%

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

      1. *-commutative 97.7%

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

   9. Simplified 97.7%

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

       1. unpow2 98.6%

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

   11. Applied egg-rr 97.7%

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

   if 1.25 < x

   1. Initial program 67.2%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. rem-square-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. *-inverses 100.0%

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

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

Alternative 9: 65.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 39.9%

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

   3. Taylor expanded in x around -inf 31.7%

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

      1. neg-mul-1 31.7%

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

   5. Simplified 31.7%

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

   if -1 < x

   1. Initial program 28.3%

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

   3. Taylor expanded in x around 0 14.9%

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

      1. log1p-define 75.7%

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

      2. rem-square-sqrt 51.0%

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

      3. fabs-sqr 51.0%

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

      4. rem-square-sqrt 75.7%

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

   5. Simplified 75.7%

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

Alternative 10: 59.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55)
   (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
   (copysign (log1p x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = copysign((x * (1.0 + ((x * x) * -0.16666666666666666))), 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 * (1.0 + ((x * x) * -0.16666666666666666))), 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 * (1.0 + ((x * x) * -0.16666666666666666))), x)
	else:
		tmp = math.copysign(math.log1p(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666))), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.55], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 19.4%

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

      1. flip-+ 8.1%

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

      2. clear-num 8.1%

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

      3. log-div 8.1%

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

      4. metadata-eval 8.1%

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

      5. +-commutative 8.1%

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

      6. hypot-1-def 8.0%

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

      7. add-sqr-sqrt 5.6%

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

      8. fabs-sqr 5.6%

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

      9. add-sqr-sqrt 8.4%

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

   4. Applied egg-rr 8.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 8.4%

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

      2. div-sub 8.4%

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

      3. *-rgt-identity 8.4%

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

      4. associate-/l* 8.4%

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

      5. fmm-def 8.4%

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

      6. fma-undefine 8.4%

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

      7. unpow2 8.4%

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

      8. associate--r+ 8.4%

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

      9. +-inverses 8.4%

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

      10. metadata-eval 8.4%

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

      11. metadata-eval 8.4%

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

      12. *-rgt-identity 8.4%

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

      13. associate-/l* 8.4%

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

      14. fma-undefine 8.4%

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

      15. unpow2 8.4%

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

      16. associate--r+ 18.8%

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

   6. Simplified 37.3%

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

   7. Taylor expanded in x around 0 70.0%

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

      1. *-commutative 70.0%

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

   9. Simplified 70.0%

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

       1. unpow2 70.6%

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

   11. Applied egg-rr 70.0%

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

   if 1.55000000000000004 < x

   1. Initial program 67.2%

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

   3. Taylor expanded in x around 0 30.9%

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

      1. log1p-define 30.9%

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

      2. rem-square-sqrt 30.9%

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

      3. fabs-sqr 30.9%

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

      4. rem-square-sqrt 30.9%

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

   5. Simplified 30.9%

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

Alternative 11: 59.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.4)
   (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
   (copysign (log x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 3.4) {
		tmp = copysign((x * (1.0 + ((x * x) * -0.16666666666666666))), x);
	} else {
		tmp = copysign(log(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.4) {
		tmp = Math.copySign((x * (1.0 + ((x * x) * -0.16666666666666666))), x);
	} else {
		tmp = Math.copySign(Math.log(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.4:
		tmp = math.copysign((x * (1.0 + ((x * x) * -0.16666666666666666))), x)
	else:
		tmp = math.copysign(math.log(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.4)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666))), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.4)
		tmp = sign(x) * abs((x * (1.0 + ((x * x) * -0.16666666666666666))));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.4], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.39999999999999991

   1. Initial program 19.4%

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

      1. flip-+ 8.1%

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

      2. clear-num 8.1%

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

      3. log-div 8.1%

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

      4. metadata-eval 8.1%

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

      5. +-commutative 8.1%

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

      6. hypot-1-def 8.0%

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

      7. add-sqr-sqrt 5.6%

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

      8. fabs-sqr 5.6%

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

      9. add-sqr-sqrt 8.4%

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

   4. Applied egg-rr 8.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-sub0 8.4%

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

      2. div-sub 8.4%

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

      3. *-rgt-identity 8.4%

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

      4. associate-/l* 8.4%

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

      5. fmm-def 8.4%

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

      6. fma-undefine 8.4%

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

      7. unpow2 8.4%

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

      8. associate--r+ 8.4%

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

      9. +-inverses 8.4%

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

      10. metadata-eval 8.4%

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

      11. metadata-eval 8.4%

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

      12. *-rgt-identity 8.4%

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

      13. associate-/l* 8.4%

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

      14. fma-undefine 8.4%

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

      15. unpow2 8.4%

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

      16. associate--r+ 18.8%

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

   6. Simplified 37.3%

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

   7. Taylor expanded in x around 0 70.0%

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

      1. *-commutative 70.0%

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

   9. Simplified 70.0%

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

       1. unpow2 70.6%

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

   11. Applied egg-rr 70.0%

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

   if 3.39999999999999991 < x

   1. Initial program 67.2%

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

   3. Taylor expanded in x around inf 30.9%

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

      1. mul-1-neg 30.9%

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

      2. log-rec 30.9%

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

      3. remove-double-neg 30.9%

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

   5. Simplified 30.9%

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

Alternative 12: 52.7% 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 31.0%

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

   1. flip-+ 6.1%

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

   2. clear-num 6.1%

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

   3. log-div 6.1%

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

   4. metadata-eval 6.1%

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

   5. +-commutative 6.1%

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

   6. hypot-1-def 6.1%

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

   7. add-sqr-sqrt 4.3%

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

   8. fabs-sqr 4.3%

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

   9. add-sqr-sqrt 6.4%

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

4. Applied egg-rr 6.4%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)}, x\right) \]
5. Step-by-step derivation

   1. neg-sub0 6.4%

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

   2. div-sub 6.4%

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

   3. *-rgt-identity 6.4%

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

   4. associate-/l* 6.4%

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

   5. fmm-def 6.4%

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

   6. fma-undefine 6.4%

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

   7. unpow2 6.4%

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

   8. associate--r+ 6.4%

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

   9. +-inverses 6.4%

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

   10. metadata-eval 6.4%

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

   11. metadata-eval 6.4%

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

   12. *-rgt-identity 6.4%

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

   13. associate-/l* 6.4%

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

   14. fma-undefine 6.4%

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

   15. unpow2 6.4%

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

   16. associate--r+ 14.7%

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

6. Simplified 29.1%

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

7. Taylor expanded in x around 0 54.2%

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

Developer Target 1: 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 2024181 
(FPCore (x)
  :name "Rust f64::asinh"
  :precision binary64

  :alt
  (! :herbie-platform default (let* ((ax (fabs x)) (ix (/ 1 ax))) (copysign (log1p (+ ax (/ ax (+ (hypot 1 ix) ix)))) x)))

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