Rust f64::asinh

Percentage Accurate: 30.4% → 99.3%

Time: 8.5s

Alternatives: 10

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 30.4% 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(-1 + \left(1 - \log \left(x \cdot -2\right)\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + {x}^{2} \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-1 + \left(1 - \log \left(\frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -10.0)
     (copysign (+ -1.0 (- 1.0 (log (* x -2.0)))) x)
     (if (<= t_0 0.02)
       (copysign
        (*
         x
         (+
          1.0
          (*
           (pow x 2.0)
           (-
            (* (pow x 2.0) (+ 0.075 (* (pow x 2.0) -0.044642857142857144)))
            0.16666666666666666))))
        x)
       (copysign (+ -1.0 (- 1.0 (log (/ 0.5 x)))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = copysign((-1.0 + (1.0 - log((x * -2.0)))), x);
	} else if (t_0 <= 0.02) {
		tmp = copysign((x * (1.0 + (pow(x, 2.0) * ((pow(x, 2.0) * (0.075 + (pow(x, 2.0) * -0.044642857142857144))) - 0.16666666666666666)))), x);
	} else {
		tmp = copysign((-1.0 + (1.0 - log((0.5 / x)))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = Math.copySign((-1.0 + (1.0 - Math.log((x * -2.0)))), x);
	} else if (t_0 <= 0.02) {
		tmp = Math.copySign((x * (1.0 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * (0.075 + (Math.pow(x, 2.0) * -0.044642857142857144))) - 0.16666666666666666)))), x);
	} else {
		tmp = Math.copySign((-1.0 + (1.0 - Math.log((0.5 / x)))), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -10.0:
		tmp = math.copysign((-1.0 + (1.0 - math.log((x * -2.0)))), x)
	elif t_0 <= 0.02:
		tmp = math.copysign((x * (1.0 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * (0.075 + (math.pow(x, 2.0) * -0.044642857142857144))) - 0.16666666666666666)))), x)
	else:
		tmp = math.copysign((-1.0 + (1.0 - math.log((0.5 / x)))), x)
	return tmp

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = copysign(Float64(-1.0 + Float64(1.0 - log(Float64(x * -2.0)))), x);
	elseif (t_0 <= 0.02)
		tmp = copysign(Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * Float64(0.075 + Float64((x ^ 2.0) * -0.044642857142857144))) - 0.16666666666666666)))), x);
	else
		tmp = copysign(Float64(-1.0 + Float64(1.0 - log(Float64(0.5 / x)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = sign(x) * abs((-1.0 + (1.0 - log((x * -2.0)))));
	elseif (t_0 <= 0.02)
		tmp = sign(x) * abs((x * (1.0 + ((x ^ 2.0) * (((x ^ 2.0) * (0.075 + ((x ^ 2.0) * -0.044642857142857144))) - 0.16666666666666666)))));
	else
		tmp = sign(x) * abs((-1.0 + (1.0 - log((0.5 / x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], N[With[{TMP1 = Abs[N[(-1.0 + N[(1.0 - N[Log[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.02], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.075 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[(-1.0 + N[(1.0 - N[Log[N[(0.5 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -10

   1. Initial program 37.2%

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

      1. +-commutative 37.2%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 0.0%

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

      2. div-sub 0.0%

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

      3. pow2 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. fabs-sqr 0.0%

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

      9. add-sqr-sqrt 0.0%

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

   6. Applied egg-rr 1.1%

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

      1. div-sub 1.1%

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

      2. fma-undefine 1.1%

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

      3. unpow2 1.1%

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

      4. associate--r+ 35.2%

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

      5. sub-neg 35.2%

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

      6. +-inverses 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-undefine 0.0%

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

   10. Applied egg-rr 0.0%

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

       1. sub-neg 0.0%

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

       2. metadata-eval 0.0%

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

       3. +-commutative 0.0%

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

       4. log1p-undefine 0.0%

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

       5. rem-exp-log 100.0%

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

       6. metadata-eval 100.0%

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

       7. associate-/r* 100.0%

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

       8. *-commutative 100.0%

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

       9. log-rec 100.0%

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

       10. unsub-neg 100.0%

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

       11. *-commutative 100.0%

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

       12. neg-mul-1 100.0%

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

       13. sub-neg 100.0%

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

       14. distribute-neg-in 100.0%

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

       15. remove-double-neg 100.0%

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

       16. hypot-undefine 37.2%

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

       17. metadata-eval 37.2%

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

       18. unpow2 37.2%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around -inf 100.0%

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

       1. *-commutative 100.0%

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

   15. Simplified 100.0%

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

   if -10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.0200000000000000004

   1. Initial program 8.8%

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

      1. +-commutative 8.8%

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

      2. hypot-1-def 8.8%

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

   3. Simplified 8.8%

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

      1. flip-+ 8.7%

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

      2. div-sub 8.7%

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

      3. pow2 8.7%

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

      4. add-sqr-sqrt 4.1%

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

      5. fabs-sqr 4.1%

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

      6. add-sqr-sqrt 8.7%

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

      7. add-sqr-sqrt 4.1%

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

      8. fabs-sqr 4.1%

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

      9. add-sqr-sqrt 8.3%

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

   6. Applied egg-rr 8.8%

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

      1. div-sub 8.8%

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

      2. fma-undefine 8.8%

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

      3. unpow2 8.8%

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

      4. associate--r+ 8.8%

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

      5. sub-neg 8.8%

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

      6. +-inverses 8.8%

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

      7. metadata-eval 8.8%

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

      8. metadata-eval 8.8%

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

   8. Simplified 8.8%

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

   9. Taylor expanded in x around 0 100.0%

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

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

   1. Initial program 51.6%

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

      1. +-commutative 51.6%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

      1. flip-+ 1.5%

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

      2. div-sub 1.5%

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

      3. pow2 1.5%

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

      4. add-sqr-sqrt 1.4%

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

      5. fabs-sqr 1.4%

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

      6. add-sqr-sqrt 1.5%

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

      7. add-sqr-sqrt 1.3%

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

      8. fabs-sqr 1.3%

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

      9. add-sqr-sqrt 1.5%

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

   6. Applied egg-rr 1.5%

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

      1. div-sub 1.5%

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

      2. fma-undefine 1.5%

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

      3. unpow2 1.5%

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

      4. associate--r+ 1.5%

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

      5. sub-neg 1.5%

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

      6. +-inverses 1.5%

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

      7. metadata-eval 1.5%

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

      8. metadata-eval 1.5%

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

   8. Simplified 1.5%

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

      1. expm1-log1p-u 1.5%

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

      2. expm1-undefine 1.5%

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

   10. Applied egg-rr 1.5%

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

       1. sub-neg 1.5%

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

       2. metadata-eval 1.5%

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

       3. +-commutative 1.5%

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

       4. log1p-undefine 1.5%

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

       5. rem-exp-log 1.5%

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

       6. metadata-eval 1.5%

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

       7. associate-/r* 1.5%

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

       8. *-commutative 1.5%

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

       9. log-rec 4.6%

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

       10. unsub-neg 4.6%

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

       11. *-commutative 4.6%

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

       12. neg-mul-1 4.6%

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

       13. sub-neg 4.6%

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

       14. distribute-neg-in 4.6%

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

       15. remove-double-neg 4.6%

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

       16. hypot-undefine 4.6%

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

       17. metadata-eval 4.6%

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

       18. unpow2 4.6%

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

   12. Simplified 4.6%

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

   13. Taylor expanded in x around inf 98.9%

       \[\leadsto \mathsf{copysign}\left(-1 + \left(1 - \log \color{blue}{\left(\frac{0.5}{x}\right)}\right), x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(-1 + \left(1 - \log \left(x \cdot -2\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.02:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + {x}^{2} \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-1 + \left(1 - \log \left(\frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(-1 + \left(1 - \log \left(x \cdot -2\right)\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot 0.075 - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-1 + \left(1 - \log \left(\frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -10.0)
     (copysign (+ -1.0 (- 1.0 (log (* x -2.0)))) x)
     (if (<= t_0 0.02)
       (copysign
        (* x (+ 1.0 (* (pow x 2.0) (- (* (* x x) 0.075) 0.16666666666666666))))
        x)
       (copysign (+ -1.0 (- 1.0 (log (/ 0.5 x)))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = copysign((-1.0 + (1.0 - log((x * -2.0)))), x);
	} else if (t_0 <= 0.02) {
		tmp = copysign((x * (1.0 + (pow(x, 2.0) * (((x * x) * 0.075) - 0.16666666666666666)))), x);
	} else {
		tmp = copysign((-1.0 + (1.0 - log((0.5 / x)))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = Math.copySign((-1.0 + (1.0 - Math.log((x * -2.0)))), x);
	} else if (t_0 <= 0.02) {
		tmp = Math.copySign((x * (1.0 + (Math.pow(x, 2.0) * (((x * x) * 0.075) - 0.16666666666666666)))), x);
	} else {
		tmp = Math.copySign((-1.0 + (1.0 - Math.log((0.5 / x)))), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -10.0:
		tmp = math.copysign((-1.0 + (1.0 - math.log((x * -2.0)))), x)
	elif t_0 <= 0.02:
		tmp = math.copysign((x * (1.0 + (math.pow(x, 2.0) * (((x * x) * 0.075) - 0.16666666666666666)))), x)
	else:
		tmp = math.copysign((-1.0 + (1.0 - math.log((0.5 / x)))), x)
	return tmp

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = copysign(Float64(-1.0 + Float64(1.0 - log(Float64(x * -2.0)))), x);
	elseif (t_0 <= 0.02)
		tmp = copysign(Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(Float64(x * x) * 0.075) - 0.16666666666666666)))), x);
	else
		tmp = copysign(Float64(-1.0 + Float64(1.0 - log(Float64(0.5 / x)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = sign(x) * abs((-1.0 + (1.0 - log((x * -2.0)))));
	elseif (t_0 <= 0.02)
		tmp = sign(x) * abs((x * (1.0 + ((x ^ 2.0) * (((x * x) * 0.075) - 0.16666666666666666)))));
	else
		tmp = sign(x) * abs((-1.0 + (1.0 - log((0.5 / x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], N[With[{TMP1 = Abs[N[(-1.0 + N[(1.0 - N[Log[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.02], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.075), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[(-1.0 + N[(1.0 - N[Log[N[(0.5 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -10

   1. Initial program 37.2%

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

      1. +-commutative 37.2%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 0.0%

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

      2. div-sub 0.0%

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

      3. pow2 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. fabs-sqr 0.0%

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

      9. add-sqr-sqrt 0.0%

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

   6. Applied egg-rr 1.1%

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

      1. div-sub 1.1%

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

      2. fma-undefine 1.1%

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

      3. unpow2 1.1%

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

      4. associate--r+ 35.2%

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

      5. sub-neg 35.2%

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

      6. +-inverses 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-undefine 0.0%

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

   10. Applied egg-rr 0.0%

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

       1. sub-neg 0.0%

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

       2. metadata-eval 0.0%

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

       3. +-commutative 0.0%

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

       4. log1p-undefine 0.0%

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

       5. rem-exp-log 100.0%

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

       6. metadata-eval 100.0%

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

       7. associate-/r* 100.0%

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

       8. *-commutative 100.0%

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

       9. log-rec 100.0%

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

       10. unsub-neg 100.0%

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

       11. *-commutative 100.0%

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

       12. neg-mul-1 100.0%

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

       13. sub-neg 100.0%

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

       14. distribute-neg-in 100.0%

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

       15. remove-double-neg 100.0%

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

       16. hypot-undefine 37.2%

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

       17. metadata-eval 37.2%

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

       18. unpow2 37.2%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around -inf 100.0%

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

       1. *-commutative 100.0%

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

   15. Simplified 100.0%

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

   if -10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.0200000000000000004

   1. Initial program 8.8%

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

      1. +-commutative 8.8%

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

      2. hypot-1-def 8.8%

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

   3. Simplified 8.8%

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

      1. flip-+ 8.7%

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

      2. div-sub 8.7%

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

      3. pow2 8.7%

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

      4. add-sqr-sqrt 4.1%

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

      5. fabs-sqr 4.1%

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

      6. add-sqr-sqrt 8.7%

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

      7. add-sqr-sqrt 4.1%

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

      8. fabs-sqr 4.1%

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

      9. add-sqr-sqrt 8.3%

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

   6. Applied egg-rr 8.8%

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

      1. div-sub 8.8%

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

      2. fma-undefine 8.8%

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

      3. unpow2 8.8%

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

      4. associate--r+ 8.8%

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

      5. sub-neg 8.8%

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

      6. +-inverses 8.8%

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

      7. metadata-eval 8.8%

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

      8. metadata-eval 8.8%

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

   8. Simplified 8.8%

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

   9. Taylor expanded in x around 0 99.8%

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

       1. unpow2 99.8%

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

   11. Applied egg-rr 99.8%

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

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

   1. Initial program 51.6%

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

      1. +-commutative 51.6%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

      1. flip-+ 1.5%

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

      2. div-sub 1.5%

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

      3. pow2 1.5%

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

      4. add-sqr-sqrt 1.4%

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

      5. fabs-sqr 1.4%

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

      6. add-sqr-sqrt 1.5%

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

      7. add-sqr-sqrt 1.3%

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

      8. fabs-sqr 1.3%

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

      9. add-sqr-sqrt 1.5%

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

   6. Applied egg-rr 1.5%

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

      1. div-sub 1.5%

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

      2. fma-undefine 1.5%

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

      3. unpow2 1.5%

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

      4. associate--r+ 1.5%

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

      5. sub-neg 1.5%

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

      6. +-inverses 1.5%

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

      7. metadata-eval 1.5%

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

      8. metadata-eval 1.5%

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

   8. Simplified 1.5%

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

      1. expm1-log1p-u 1.5%

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

      2. expm1-undefine 1.5%

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

   10. Applied egg-rr 1.5%

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

       1. sub-neg 1.5%

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

       2. metadata-eval 1.5%

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

       3. +-commutative 1.5%

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

       4. log1p-undefine 1.5%

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

       5. rem-exp-log 1.5%

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

       6. metadata-eval 1.5%

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

       7. associate-/r* 1.5%

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

       8. *-commutative 1.5%

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

       9. log-rec 4.6%

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

       10. unsub-neg 4.6%

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

       11. *-commutative 4.6%

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

       12. neg-mul-1 4.6%

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

       13. sub-neg 4.6%

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

       14. distribute-neg-in 4.6%

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

       15. remove-double-neg 4.6%

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

       16. hypot-undefine 4.6%

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

       17. metadata-eval 4.6%

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

       18. unpow2 4.6%

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

   12. Simplified 4.6%

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

   13. Taylor expanded in x around inf 98.9%

       \[\leadsto \mathsf{copysign}\left(-1 + \left(1 - \log \color{blue}{\left(\frac{0.5}{x}\right)}\right), x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(-1 + \left(1 - \log \left(x \cdot -2\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.02:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + {x}^{2} \cdot \left(\left(x \cdot x\right) \cdot 0.075 - 0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-1 + \left(1 - \log \left(\frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(-1 + \left(1 - \log \left(x \cdot -2\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-1 + \left(1 - \log \left(\frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (+ -1.0 (- 1.0 (log (* x -2.0)))) x)
   (if (<= x 1.25)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (+ -1.0 (- 1.0 (log (/ 0.5 x)))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = copysign((-1.0 + (1.0 - log((x * -2.0)))), x);
	} else if (x <= 1.25) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign((-1.0 + (1.0 - log((0.5 / x)))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.copySign((-1.0 + (1.0 - Math.log((x * -2.0)))), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign((-1.0 + (1.0 - Math.log((0.5 / x)))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.copysign((-1.0 + (1.0 - math.log((x * -2.0)))), x)
	elif x <= 1.25:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign((-1.0 + (1.0 - math.log((0.5 / x)))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = copysign(Float64(-1.0 + Float64(1.0 - log(Float64(x * -2.0)))), x);
	elseif (x <= 1.25)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(Float64(-1.0 + Float64(1.0 - log(Float64(0.5 / x)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = sign(x) * abs((-1.0 + (1.0 - log((x * -2.0)))));
	elseif (x <= 1.25)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs((-1.0 + (1.0 - log((0.5 / x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[With[{TMP1 = Abs[N[(-1.0 + N[(1.0 - N[Log[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]), $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[(-1.0 + N[(1.0 - N[Log[N[(0.5 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 37.2%

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

      1. +-commutative 37.2%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 0.0%

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

      2. div-sub 0.0%

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

      3. pow2 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. fabs-sqr 0.0%

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

      9. add-sqr-sqrt 0.0%

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

   6. Applied egg-rr 1.1%

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

      1. div-sub 1.1%

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

      2. fma-undefine 1.1%

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

      3. unpow2 1.1%

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

      4. associate--r+ 35.2%

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

      5. sub-neg 35.2%

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

      6. +-inverses 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-undefine 0.0%

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

   10. Applied egg-rr 0.0%

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

       1. sub-neg 0.0%

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

       2. metadata-eval 0.0%

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

       3. +-commutative 0.0%

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

       4. log1p-undefine 0.0%

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

       5. rem-exp-log 100.0%

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

       6. metadata-eval 100.0%

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

       7. associate-/r* 100.0%

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

       8. *-commutative 100.0%

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

       9. log-rec 100.0%

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

       10. unsub-neg 100.0%

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

       11. *-commutative 100.0%

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

       12. neg-mul-1 100.0%

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

       13. sub-neg 100.0%

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

       14. distribute-neg-in 100.0%

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

       15. remove-double-neg 100.0%

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

       16. hypot-undefine 37.2%

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

       17. metadata-eval 37.2%

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

       18. unpow2 37.2%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around -inf 100.0%

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

       1. *-commutative 100.0%

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

   15. Simplified 100.0%

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

   if -1.25 < x < 1.25

   1. Initial program 8.8%

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

      1. +-commutative 8.8%

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

      2. hypot-1-def 8.8%

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

   3. Simplified 8.8%

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

      1. flip-+ 8.7%

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

      2. div-sub 8.7%

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

      3. pow2 8.7%

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

      4. add-sqr-sqrt 4.1%

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

      5. fabs-sqr 4.1%

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

      6. add-sqr-sqrt 8.7%

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

      7. add-sqr-sqrt 4.1%

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

      8. fabs-sqr 4.1%

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

      9. add-sqr-sqrt 8.3%

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

   6. Applied egg-rr 8.8%

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

      1. div-sub 8.8%

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

      2. fma-undefine 8.8%

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

      3. unpow2 8.8%

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

      4. associate--r+ 8.8%

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

      5. sub-neg 8.8%

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

      6. +-inverses 8.8%

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

      7. metadata-eval 8.8%

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

      8. metadata-eval 8.8%

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

   8. Simplified 8.8%

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

   9. Taylor expanded in x around 0 99.5%

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

       1. distribute-rgt-in 99.5%

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

       2. *-lft-identity 99.5%

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

       3. associate-*l* 99.5%

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

       4. unpow2 99.5%

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

       5. unpow3 99.5%

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

   11. Simplified 99.5%

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

   if 1.25 < x

   1. Initial program 51.6%

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

      1. +-commutative 51.6%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

      1. flip-+ 1.5%

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

      2. div-sub 1.5%

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

      3. pow2 1.5%

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

      4. add-sqr-sqrt 1.4%

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

      5. fabs-sqr 1.4%

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

      6. add-sqr-sqrt 1.5%

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

      7. add-sqr-sqrt 1.3%

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

      8. fabs-sqr 1.3%

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

      9. add-sqr-sqrt 1.5%

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

   6. Applied egg-rr 1.5%

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

      1. div-sub 1.5%

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

      2. fma-undefine 1.5%

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

      3. unpow2 1.5%

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

      4. associate--r+ 1.5%

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

      5. sub-neg 1.5%

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

      6. +-inverses 1.5%

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

      7. metadata-eval 1.5%

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

      8. metadata-eval 1.5%

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

   8. Simplified 1.5%

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

      1. expm1-log1p-u 1.5%

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

      2. expm1-undefine 1.5%

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

   10. Applied egg-rr 1.5%

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

       1. sub-neg 1.5%

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

       2. metadata-eval 1.5%

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

       3. +-commutative 1.5%

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

       4. log1p-undefine 1.5%

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

       5. rem-exp-log 1.5%

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

       6. metadata-eval 1.5%

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

       7. associate-/r* 1.5%

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

       8. *-commutative 1.5%

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

       9. log-rec 4.6%

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

       10. unsub-neg 4.6%

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

       11. *-commutative 4.6%

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

       12. neg-mul-1 4.6%

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

       13. sub-neg 4.6%

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

       14. distribute-neg-in 4.6%

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

       15. remove-double-neg 4.6%

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

       16. hypot-undefine 4.6%

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

       17. metadata-eval 4.6%

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

       18. unpow2 4.6%

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

   12. Simplified 4.6%

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

   13. Taylor expanded in x around inf 98.9%

       \[\leadsto \mathsf{copysign}\left(-1 + \left(1 - \log \color{blue}{\left(\frac{0.5}{x}\right)}\right), x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(-1 + \left(1 - \log \left(x \cdot -2\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (+ -1.0 (- 1.0 (log (* x -2.0)))) 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((-1.0 + (1.0 - log((x * -2.0)))), 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((-1.0 + (1.0 - Math.log((x * -2.0)))), 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((-1.0 + (1.0 - math.log((x * -2.0)))), 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(Float64(-1.0 + Float64(1.0 - log(Float64(x * -2.0)))), 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((-1.0 + (1.0 - log((x * -2.0)))));
	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[(-1.0 + N[(1.0 - N[Log[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]), $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 37.2%

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

      1. +-commutative 37.2%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 0.0%

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

      2. div-sub 0.0%

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

      3. pow2 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. fabs-sqr 0.0%

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

      9. add-sqr-sqrt 0.0%

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

   6. Applied egg-rr 1.1%

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

      1. div-sub 1.1%

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

      2. fma-undefine 1.1%

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

      3. unpow2 1.1%

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

      4. associate--r+ 35.2%

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

      5. sub-neg 35.2%

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

      6. +-inverses 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-undefine 0.0%

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

   10. Applied egg-rr 0.0%

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

       1. sub-neg 0.0%

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

       2. metadata-eval 0.0%

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

       3. +-commutative 0.0%

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

       4. log1p-undefine 0.0%

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

       5. rem-exp-log 100.0%

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

       6. metadata-eval 100.0%

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

       7. associate-/r* 100.0%

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

       8. *-commutative 100.0%

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

       9. log-rec 100.0%

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

       10. unsub-neg 100.0%

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

       11. *-commutative 100.0%

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

       12. neg-mul-1 100.0%

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

       13. sub-neg 100.0%

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

       14. distribute-neg-in 100.0%

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

       15. remove-double-neg 100.0%

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

       16. hypot-undefine 37.2%

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

       17. metadata-eval 37.2%

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

       18. unpow2 37.2%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around -inf 100.0%

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

       1. *-commutative 100.0%

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

   15. Simplified 100.0%

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

   if -1.25 < x < 1.25

   1. Initial program 8.8%

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

      1. +-commutative 8.8%

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

      2. hypot-1-def 8.8%

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

   3. Simplified 8.8%

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

      1. flip-+ 8.7%

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

      2. div-sub 8.7%

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

      3. pow2 8.7%

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

      4. add-sqr-sqrt 4.1%

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

      5. fabs-sqr 4.1%

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

      6. add-sqr-sqrt 8.7%

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

      7. add-sqr-sqrt 4.1%

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

      8. fabs-sqr 4.1%

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

      9. add-sqr-sqrt 8.3%

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

   6. Applied egg-rr 8.8%

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

      1. div-sub 8.8%

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

      2. fma-undefine 8.8%

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

      3. unpow2 8.8%

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

      4. associate--r+ 8.8%

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

      5. sub-neg 8.8%

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

      6. +-inverses 8.8%

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

      7. metadata-eval 8.8%

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

      8. metadata-eval 8.8%

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

   8. Simplified 8.8%

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

   9. Taylor expanded in x around 0 99.5%

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

       1. distribute-rgt-in 99.5%

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

       2. *-lft-identity 99.5%

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

       3. associate-*l* 99.5%

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

       4. unpow2 99.5%

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

       5. unpow3 99.5%

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

   11. Simplified 99.5%

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

   if 1.25 < x

   1. Initial program 51.6%

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

      1. +-commutative 51.6%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in x around inf 97.4%

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

      1. rem-square-sqrt 97.4%

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

      2. fabs-sqr 97.4%

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

      3. rem-square-sqrt 97.4%

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

      4. *-inverses 97.4%

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

      5. metadata-eval 97.4%

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

   7. Simplified 97.4%

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

Alternative 5: 81.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.15:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.15)
   (copysign (log (- x)) x)
   (if (<= x 1.25) (copysign x x) (copysign (log (* x 2.0)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -3.15) {
		tmp = copysign(log(-x), x);
	} else if (x <= 1.25) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log((x * 2.0)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.15) {
		tmp = Math.copySign(Math.log(-x), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log((x * 2.0)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.15:
		tmp = math.copysign(math.log(-x), x)
	elif x <= 1.25:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log((x * 2.0)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.15)
		tmp = copysign(log(Float64(-x)), x);
	elseif (x <= 1.25)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float64(x * 2.0)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.15)
		tmp = sign(x) * abs(log(-x));
	elseif (x <= 1.25)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.15], N[With[{TMP1 = Abs[N[Log[(-x)], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.14999999999999991

   1. Initial program 37.2%

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

      1. +-commutative 37.2%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around -inf 31.9%

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

      1. mul-1-neg 31.9%

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

   7. Simplified 31.9%

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

   if -3.14999999999999991 < x < 1.25

   1. Initial program 8.8%

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

      1. +-commutative 8.8%

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

      2. hypot-1-def 8.8%

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

   3. Simplified 8.8%

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

      1. flip-+ 8.7%

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

      2. div-sub 8.7%

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

      3. pow2 8.7%

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

      4. add-sqr-sqrt 4.1%

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

      5. fabs-sqr 4.1%

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

      6. add-sqr-sqrt 8.7%

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

      7. add-sqr-sqrt 4.1%

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

      8. fabs-sqr 4.1%

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

      9. add-sqr-sqrt 8.3%

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

   6. Applied egg-rr 8.8%

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

      1. div-sub 8.8%

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

      2. fma-undefine 8.8%

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

      3. unpow2 8.8%

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

      4. associate--r+ 8.8%

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

      5. sub-neg 8.8%

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

      6. +-inverses 8.8%

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

      7. metadata-eval 8.8%

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

      8. metadata-eval 8.8%

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

   8. Simplified 8.8%

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

   9. Taylor expanded in x around 0 99.2%

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

   if 1.25 < x

   1. Initial program 51.6%

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

      1. +-commutative 51.6%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in x around inf 97.4%

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

      1. rem-square-sqrt 97.4%

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

      2. fabs-sqr 97.4%

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

      3. rem-square-sqrt 97.4%

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

      4. *-inverses 97.4%

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

      5. metadata-eval 97.4%

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

   7. Simplified 97.4%

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

Alternative 6: 81.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;\mathsf{copysign}\left(-\mathsf{log1p}\left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.72) (copysign (- (log1p (- x))) x) (copysign (log (* x 2.0)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.72) {
		tmp = copysign(-log1p(-x), x);
	} else {
		tmp = copysign(log((x * 2.0)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.72) {
		tmp = Math.copySign(-Math.log1p(-x), x);
	} else {
		tmp = Math.copySign(Math.log((x * 2.0)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.72:
		tmp = math.copysign(-math.log1p(-x), x)
	else:
		tmp = math.copysign(math.log((x * 2.0)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.72)
		tmp = copysign(Float64(-log1p(Float64(-x))), x);
	else
		tmp = copysign(log(Float64(x * 2.0)), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.72], N[With[{TMP1 = Abs[(-N[Log[1 + (-x)], $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 2 regimes

2. if x < 0.71999999999999997

   1. Initial program 16.8%

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

      1. +-commutative 16.8%

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

      2. hypot-1-def 34.4%

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

   3. Simplified 34.4%

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

      1. flip-+ 6.3%

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

      2. div-sub 6.3%

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

      3. pow2 6.3%

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

      4. add-sqr-sqrt 3.0%

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

      5. fabs-sqr 3.0%

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

      6. add-sqr-sqrt 6.3%

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

      7. add-sqr-sqrt 3.0%

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

      8. fabs-sqr 3.0%

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

      9. add-sqr-sqrt 6.0%

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

   6. Applied egg-rr 6.6%

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

      1. div-sub 6.6%

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

      2. fma-undefine 6.6%

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

      3. unpow2 6.6%

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

      4. associate--r+ 16.2%

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

      5. sub-neg 16.2%

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

      6. +-inverses 34.4%

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

      7. metadata-eval 34.4%

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

      8. metadata-eval 34.4%

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

   8. Simplified 34.4%

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

   9. Taylor expanded in x around 0 14.6%

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

   10. Taylor expanded in x around -inf 14.6%

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

       1. log-rec 14.6%

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

       2. neg-mul-1 14.6%

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

       3. log1p-define 79.2%

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

   12. Simplified 79.2%

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

   if 0.71999999999999997 < x

   1. Initial program 51.6%

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

      1. +-commutative 51.6%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in x around inf 97.4%

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

      1. rem-square-sqrt 97.4%

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

      2. fabs-sqr 97.4%

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

      3. rem-square-sqrt 97.4%

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

      4. *-inverses 97.4%

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

      5. metadata-eval 97.4%

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

   7. Simplified 97.4%

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

Alternative 7: 64.6% 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 37.2%

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

      1. +-commutative 37.2%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around -inf 31.9%

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

      1. mul-1-neg 31.9%

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

   7. Simplified 31.9%

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

   if -1 < x

   1. Initial program 22.3%

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

      1. +-commutative 22.3%

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

      2. hypot-1-def 37.2%

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

   3. Simplified 37.2%

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

   5. Taylor expanded in x around 0 15.3%

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

      1. log1p-define 76.7%

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

      2. rem-square-sqrt 46.3%

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

      3. fabs-sqr 46.3%

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

      4. rem-square-sqrt 76.7%

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

   7. Simplified 76.7%

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

Alternative 8: 60.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = copysign((-1.0 + (1.0 / (x + 1.0))), 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((-1.0 + (1.0 / (x + 1.0))), x);
	} else {
		tmp = Math.copySign(Math.log1p(x), x);
	}
	return tmp;
}

Python

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[With[{TMP1 = Abs[N[(-1.0 + N[(1.0 / N[(x + 1.0), $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

   1. Initial program 37.2%

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

      1. +-commutative 37.2%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 0.0%

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

      2. div-sub 0.0%

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

      3. pow2 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. fabs-sqr 0.0%

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

      9. add-sqr-sqrt 0.0%

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

   6. Applied egg-rr 1.1%

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

      1. div-sub 1.1%

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

      2. fma-undefine 1.1%

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

      3. unpow2 1.1%

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

      4. associate--r+ 35.2%

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

      5. sub-neg 35.2%

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

      6. +-inverses 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-undefine 0.0%

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

   10. Applied egg-rr 0.0%

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

       1. sub-neg 0.0%

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

       2. metadata-eval 0.0%

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

       3. +-commutative 0.0%

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

       4. log1p-undefine 0.0%

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

       5. rem-exp-log 100.0%

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

       6. metadata-eval 100.0%

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

       7. associate-/r* 100.0%

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

       8. *-commutative 100.0%

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

       9. log-rec 100.0%

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

       10. unsub-neg 100.0%

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

       11. *-commutative 100.0%

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

       12. neg-mul-1 100.0%

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

       13. sub-neg 100.0%

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

       14. distribute-neg-in 100.0%

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

       15. remove-double-neg 100.0%

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

       16. hypot-undefine 37.2%

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

       17. metadata-eval 37.2%

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

       18. unpow2 37.2%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around 0 4.9%

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

       1. neg-mul-1 4.9%

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

   15. Simplified 4.9%

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

       1. sub-neg 4.9%

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

       2. neg-sub0 4.9%

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

       3. metadata-eval 4.9%

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

       4. associate-+r- 4.9%

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

       5. associate-+r- 4.9%

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

       6. metadata-eval 4.9%

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

       7. neg-sub0 4.9%

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

       8. remove-double-neg 4.9%

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

       9. add-sqr-sqrt 0.0%

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

       10. sqrt-prod 4.5%

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

       11. sqr-neg 4.5%

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

       12. sqrt-unprod 4.9%

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

       13. add-sqr-sqrt 4.9%

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

       14. rem-exp-log 4.9%

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

       15. log1p-undefine 4.9%

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

       16. add-sqr-sqrt 4.9%

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

       17. sqrt-unprod 4.9%

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

       18. sqr-neg 4.9%

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

       19. sqrt-unprod 0.0%

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

       20. add-sqr-sqrt 14.0%

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

       21. exp-neg 14.0%

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

       22. log1p-undefine 14.0%

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

   17. Applied egg-rr 14.0%

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

       1. +-commutative 14.0%

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

   19. Simplified 14.0%

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

   if -1 < x

   1. Initial program 22.3%

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

      1. +-commutative 22.3%

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

      2. hypot-1-def 37.2%

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

   3. Simplified 37.2%

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

   5. Taylor expanded in x around 0 15.3%

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

      1. log1p-define 76.7%

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

      2. rem-square-sqrt 46.3%

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

      3. fabs-sqr 46.3%

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

      4. rem-square-sqrt 76.7%

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

   7. Simplified 76.7%

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

Alternative 9: 56.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 8\right):\\ \;\;\;\;\mathsf{copysign}\left(-1 + \frac{1}{x + 1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -2.0) (not (<= x 8.0)))
   (copysign (+ -1.0 (/ 1.0 (+ x 1.0))) x)
   (copysign x x)))

C

double code(double x) {
	double tmp;
	if ((x <= -2.0) || !(x <= 8.0)) {
		tmp = copysign((-1.0 + (1.0 / (x + 1.0))), x);
	} else {
		tmp = copysign(x, x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x <= -2.0) || !(x <= 8.0)) {
		tmp = Math.copySign((-1.0 + (1.0 / (x + 1.0))), x);
	} else {
		tmp = Math.copySign(x, x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -2.0) or not (x <= 8.0):
		tmp = math.copysign((-1.0 + (1.0 / (x + 1.0))), x)
	else:
		tmp = math.copysign(x, x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -2.0) || !(x <= 8.0))
		tmp = copysign(Float64(-1.0 + Float64(1.0 / Float64(x + 1.0))), x);
	else
		tmp = copysign(x, x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -2.0) || ~((x <= 8.0)))
		tmp = sign(x) * abs((-1.0 + (1.0 / (x + 1.0))));
	else
		tmp = sign(x) * abs(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -2.0], N[Not[LessEqual[x, 8.0]], $MachinePrecision]], N[With[{TMP1 = Abs[N[(-1.0 + N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2 or 8 < x

   1. Initial program 44.5%

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

      1. +-commutative 44.5%

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

      2. hypot-1-def 99.2%

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

   3. Simplified 99.2%

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

      1. flip-+ 0.0%

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

      2. div-sub 0.0%

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

      3. pow2 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. fabs-sqr 0.0%

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

      9. add-sqr-sqrt 0.0%

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

   6. Applied egg-rr 0.5%

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

      1. div-sub 0.5%

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

      2. fma-undefine 0.5%

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

      3. unpow2 0.5%

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

      4. associate--r+ 16.2%

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

      5. sub-neg 16.2%

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

      6. +-inverses 46.2%

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

      7. metadata-eval 46.2%

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

      8. metadata-eval 46.2%

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

   8. Simplified 46.2%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-undefine 0.0%

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

   10. Applied egg-rr 0.0%

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

       1. sub-neg 0.0%

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

       2. metadata-eval 0.0%

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

       3. +-commutative 0.0%

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

       4. log1p-undefine 0.0%

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

       5. rem-exp-log 46.2%

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

       6. metadata-eval 46.2%

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

       7. associate-/r* 46.2%

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

       8. *-commutative 46.2%

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

       9. log-rec 47.8%

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

       10. unsub-neg 47.8%

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

       11. *-commutative 47.8%

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

       12. neg-mul-1 47.8%

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

       13. sub-neg 47.8%

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

       14. distribute-neg-in 47.8%

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

       15. remove-double-neg 47.8%

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

       16. hypot-undefine 18.9%

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

       17. metadata-eval 18.9%

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

       18. unpow2 18.9%

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

   12. Simplified 47.8%

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

   13. Taylor expanded in x around 0 5.2%

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

       1. neg-mul-1 5.2%

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

   15. Simplified 5.2%

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

       1. sub-neg 5.2%

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

       2. neg-sub0 5.2%

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

       3. metadata-eval 5.2%

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

       4. associate-+r- 5.2%

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

       5. associate-+r- 5.2%

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

       6. metadata-eval 5.2%

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

       7. neg-sub0 5.2%

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

       8. remove-double-neg 5.2%

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

       9. add-sqr-sqrt 2.9%

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

       10. sqrt-prod 4.8%

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

       11. sqr-neg 4.8%

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

       12. sqrt-unprod 2.3%

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

       13. add-sqr-sqrt 5.2%

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

       14. rem-exp-log 2.3%

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

       15. log1p-undefine 2.3%

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

       16. add-sqr-sqrt 2.3%

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

       17. sqrt-unprod 2.3%

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

       18. sqr-neg 2.3%

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

       19. sqrt-unprod 0.0%

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

       20. add-sqr-sqrt 6.5%

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

       21. exp-neg 6.5%

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

       22. log1p-undefine 6.5%

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

   17. Applied egg-rr 14.0%

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

       1. +-commutative 14.0%

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

   19. Simplified 14.0%

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

   if -2 < x < 8

   1. Initial program 9.4%

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

      1. +-commutative 9.4%

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

      2. hypot-1-def 9.4%

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

   3. Simplified 9.4%

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

      1. flip-+ 9.4%

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

      2. div-sub 9.4%

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

      3. pow2 9.4%

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

      4. add-sqr-sqrt 4.8%

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

      5. fabs-sqr 4.8%

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

      6. add-sqr-sqrt 9.4%

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

      7. add-sqr-sqrt 4.7%

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

      8. fabs-sqr 4.7%

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

      9. add-sqr-sqrt 9.0%

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

   6. Applied egg-rr 9.4%

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

      1. div-sub 9.4%

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

      2. fma-undefine 9.4%

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

      3. unpow2 9.4%

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

      4. associate--r+ 9.4%

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

      5. sub-neg 9.4%

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

      6. +-inverses 9.4%

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

      7. metadata-eval 9.4%

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

      8. metadata-eval 9.4%

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

   8. Simplified 9.4%

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

   9. Taylor expanded in x around 0 98.6%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 8\right):\\ \;\;\;\;\mathsf{copysign}\left(-1 + \frac{1}{x + 1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.0% 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 25.5%

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

   1. +-commutative 25.5%

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

   2. hypot-1-def 50.4%

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

3. Simplified 50.4%

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

   1. flip-+ 5.1%

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

   2. div-sub 5.1%

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

   3. pow2 5.1%

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

   4. add-sqr-sqrt 2.6%

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

   5. fabs-sqr 2.6%

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

   6. add-sqr-sqrt 5.1%

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

   7. add-sqr-sqrt 2.6%

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

   8. fabs-sqr 2.6%

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

   9. add-sqr-sqrt 4.9%

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

6. Applied egg-rr 5.3%

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

   1. div-sub 5.3%

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

   2. fma-undefine 5.3%

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

   3. unpow2 5.3%

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

   4. associate--r+ 12.5%

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

   5. sub-neg 12.5%

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

   6. +-inverses 26.2%

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

   7. metadata-eval 26.2%

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

   8. metadata-eval 26.2%

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

8. Simplified 26.2%

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

9. Taylor expanded in x around 0 55.9%

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

Developer Target 1: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024132 
(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))