Rust f64::asinh

Percentage Accurate: 30.2% → 100.0%

Time: 10.4s

Alternatives: 9

Speedup: 3.8×

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 30.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0011:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.00084:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0011)
   (copysign (log (- (hypot 1.0 x) x)) x)
   (if (<= x 0.00084)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0011) {
		tmp = copysign(log((hypot(1.0, x) - x)), x);
	} else if (x <= 0.00084) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.0011) {
		tmp = Math.copySign(Math.log((Math.hypot(1.0, x) - x)), x);
	} else if (x <= 0.00084) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.0011:
		tmp = math.copysign(math.log((math.hypot(1.0, x) - x)), x)
	elif x <= 0.00084:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0011)
		tmp = copysign(log(Float64(hypot(1.0, x) - x)), x);
	elseif (x <= 0.00084)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0011)
		tmp = sign(x) * abs(log((hypot(1.0, x) - x)));
	elseif (x <= 0.00084)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.0011], N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 0.00084], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00110000000000000007

   1. Initial program 46.3%

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

      1. +-commutative 46.3%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 2.3%

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

      2. frac-2neg 2.3%

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

      3. log-div 2.3%

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

   6. Applied egg-rr 3.9%

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

      1. neg-sub0 3.9%

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

      2. associate--r- 3.9%

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

      3. neg-sub0 3.9%

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

      4. +-commutative 3.9%

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

      5. fma-undefine 3.9%

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

      6. unpow2 3.9%

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

      7. +-commutative 3.9%

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

      8. associate-+l+ 44.5%

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

      9. sub-neg 44.5%

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

      10. +-inverses 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. neg-sub0 99.9%

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

      14. sub-neg 99.9%

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

      15. distribute-neg-in 99.9%

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

      16. neg-mul-1 99.9%

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

      17. remove-double-neg 99.9%

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

      18. +-commutative 99.9%

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

      19. neg-mul-1 99.9%

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

      20. sub-neg 99.9%

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

   8. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 99.9%

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

      4. sqr-neg 99.9%

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

      5. sqrt-unprod 99.1%

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

      6. add-sqr-sqrt 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. *-lft-identity 99.9%

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

   12. Simplified 99.9%

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

   if -0.00110000000000000007 < x < 8.4000000000000003e-4

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

      2. hypot-1-def 6.7%

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

   3. Simplified 6.7%

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

   5. Taylor expanded in x around 0 7.7%

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

      1. +-commutative 7.7%

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

      2. fma-define 7.7%

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

      3. +-commutative 7.7%

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

      4. rem-square-sqrt 4.6%

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

      5. fabs-sqr 4.6%

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

      6. rem-square-sqrt 7.7%

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

      7. log1p-define 100.0%

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

      8. rem-square-sqrt 54.5%

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

      9. fabs-sqr 54.5%

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

      10. rem-square-sqrt 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

       1. +-commutative 100.0%

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

       2. distribute-rgt-in 100.0%

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

       3. *-un-lft-identity 100.0%

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

       4. *-commutative 100.0%

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

       5. associate-*l* 100.0%

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

       6. unpow2 100.0%

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

       7. pow3 100.0%

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

   12. Applied egg-rr 100.0%

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

   if 8.4000000000000003e-4 < x

   1. Initial program 49.4%

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

      1. +-commutative 49.4%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 49.4%

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

      1. rem-square-sqrt 49.4%

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

      2. fabs-sqr 49.4%

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

      3. rem-square-sqrt 49.4%

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

      4. metadata-eval 49.4%

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

      5. unpow2 49.4%

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

      6. hypot-undefine 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0011:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.00084:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) -5.0)
   (copysign (log (- (hypot 1.0 x) x)) x)
   (copysign (log1p (+ x (+ (hypot 1.0 x) -1.0))) x)))

C

double code(double x) {
	double tmp;
	if (copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x) <= -5.0) {
		tmp = copysign(log((hypot(1.0, x) - x)), x);
	} else {
		tmp = copysign(log1p((x + (hypot(1.0, x) + -1.0))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x) <= -5.0) {
		tmp = Math.copySign(Math.log((Math.hypot(1.0, x) - x)), x);
	} else {
		tmp = Math.copySign(Math.log1p((x + (Math.hypot(1.0, x) + -1.0))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x) <= -5.0:
		tmp = math.copysign(math.log((math.hypot(1.0, x) - x)), x)
	else:
		tmp = math.copysign(math.log1p((x + (math.hypot(1.0, x) + -1.0))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x) <= -5.0)
		tmp = copysign(log(Float64(hypot(1.0, x) - x)), x);
	else
		tmp = copysign(log1p(Float64(x + Float64(hypot(1.0, x) + -1.0))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[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], -5.0], N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + N[(x + N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -5

   1. Initial program 45.6%

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

      1. +-commutative 45.6%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 1.1%

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

      2. frac-2neg 1.1%

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

      3. log-div 1.1%

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

   6. Applied egg-rr 2.7%

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

      1. neg-sub0 2.7%

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

      2. associate--r- 2.7%

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

      3. neg-sub0 2.7%

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

      4. +-commutative 2.7%

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

      5. fma-undefine 2.7%

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

      6. unpow2 2.7%

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

      7. +-commutative 2.7%

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

      8. associate-+l+ 43.8%

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

      9. sub-neg 43.8%

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

      10. +-inverses 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-sub0 100.0%

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

      14. sub-neg 100.0%

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

      15. distribute-neg-in 100.0%

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

      16. neg-mul-1 100.0%

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

      17. remove-double-neg 100.0%

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

      18. +-commutative 100.0%

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

      19. neg-mul-1 100.0%

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

      20. sub-neg 100.0%

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

   8. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 100.0%

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

      4. sqr-neg 100.0%

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

      5. sqrt-unprod 99.2%

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

      6. add-sqr-sqrt 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. *-lft-identity 100.0%

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

   12. Simplified 100.0%

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

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 23.3%

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

      1. +-commutative 23.3%

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

      2. hypot-1-def 42.3%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in x around 0 23.3%

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

      1. rem-square-sqrt 21.1%

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

      2. fabs-sqr 21.1%

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

      3. rem-square-sqrt 23.3%

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

      4. metadata-eval 23.3%

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

      5. unpow2 23.3%

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

      6. hypot-undefine 42.3%

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

   7. Simplified 42.3%

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

      1. log1p-expm1-u 42.3%

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

      2. expm1-undefine 42.3%

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

      3. add-exp-log 42.3%

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

   9. Applied egg-rr 42.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left(x + \mathsf{hypot}\left(1, x\right)\right) - 1\right)}, x\right) \]
   10. Step-by-step derivation

       1. associate--l+ 99.6%

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

   11. Simplified 99.6%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.8 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 9.8e-9)
   (copysign (log1p (- x)) x)
   (copysign (log (+ x (hypot 1.0 x))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 9.8e-9) {
		tmp = copysign(log1p(-x), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 9.8e-9) {
		tmp = Math.copySign(Math.log1p(-x), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 9.8e-9:
		tmp = math.copysign(math.log1p(-x), x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 9.8e-9)
		tmp = copysign(log1p(Float64(-x)), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 9.8e-9], 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 + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.80000000000000007e-9

   1. Initial program 22.2%

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

      1. +-commutative 22.2%

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

      2. hypot-1-def 43.5%

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

   3. Simplified 43.5%

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

      1. flip-+ 4.7%

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

      2. frac-2neg 4.7%

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

      3. log-div 4.7%

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

   6. Applied egg-rr 5.3%

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

      1. neg-sub0 5.3%

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

      2. associate--r- 5.3%

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

      3. neg-sub0 5.3%

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

      4. +-commutative 5.3%

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

      5. fma-undefine 5.3%

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

      6. unpow2 5.3%

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

      7. +-commutative 5.3%

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

      8. associate-+l+ 21.5%

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

      9. sub-neg 21.5%

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

      10. +-inverses 43.5%

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

      11. metadata-eval 43.5%

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

      12. metadata-eval 43.5%

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

      13. neg-sub0 43.5%

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

      14. sub-neg 43.5%

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

      15. distribute-neg-in 43.5%

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

      16. neg-mul-1 43.5%

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

      17. remove-double-neg 43.5%

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

      18. +-commutative 43.5%

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

      19. neg-mul-1 43.5%

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

      20. sub-neg 43.5%

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

   8. Simplified 43.5%

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

   9. Taylor expanded in x around 0 16.3%

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

       1. neg-mul-1 16.3%

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

       2. sub-neg 16.3%

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

   11. Simplified 16.3%

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

       1. *-un-lft-identity 16.3%

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

       2. add-sqr-sqrt 3.6%

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

       3. sqrt-unprod 16.3%

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

       4. sqr-neg 16.3%

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

       5. sqrt-unprod 16.0%

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

       6. add-sqr-sqrt 16.3%

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

       7. sub-neg 16.3%

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

       8. log1p-define 72.6%

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

   13. Applied egg-rr 72.6%

       \[\leadsto \color{blue}{1 \cdot \mathsf{copysign}\left(\mathsf{log1p}\left(-x\right), x\right)} \]
   14. Step-by-step derivation

       1. *-lft-identity 72.6%

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

   15. Simplified 72.6%

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

   if 9.80000000000000007e-9 < x

   1. Initial program 49.6%

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

      1. +-commutative 49.6%

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

      2. hypot-1-def 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 49.6%

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

      1. rem-square-sqrt 49.6%

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

      2. fabs-sqr 49.6%

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

      3. rem-square-sqrt 49.6%

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

      4. metadata-eval 49.6%

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

      5. unpow2 49.6%

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

      6. hypot-undefine 99.4%

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

   7. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.7) (copysign (log1p (- x)) x) (copysign (log (* x 2.0)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.7) {
		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 <= 0.7) {
		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 <= 0.7:
		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 <= 0.7)
		tmp = copysign(log1p(Float64(-x)), x);
	else
		tmp = copysign(log(Float64(x * 2.0)), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.7], 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 < 0.69999999999999996

   1. Initial program 22.7%

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

      1. +-commutative 22.7%

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

      2. hypot-1-def 43.9%

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

   3. Simplified 43.9%

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

      1. flip-+ 5.5%

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

      2. frac-2neg 5.5%

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

      3. log-div 5.5%

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

   6. Applied egg-rr 6.1%

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

      1. neg-sub0 6.1%

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

      2. associate--r- 6.1%

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

      3. neg-sub0 6.1%

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

      4. +-commutative 6.1%

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

      5. fma-undefine 6.1%

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

      6. unpow2 6.1%

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

      7. +-commutative 6.1%

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

      8. associate-+l+ 22.1%

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

      9. sub-neg 22.1%

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

      10. +-inverses 43.9%

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

      11. metadata-eval 43.9%

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

      12. metadata-eval 43.9%

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

      13. neg-sub0 43.9%

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

      14. sub-neg 43.9%

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

      15. distribute-neg-in 43.9%

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

      16. neg-mul-1 43.9%

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

      17. remove-double-neg 43.9%

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

      18. +-commutative 43.9%

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

      19. neg-mul-1 43.9%

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

      20. sub-neg 43.9%

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

   8. Simplified 43.9%

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

   9. Taylor expanded in x around 0 16.6%

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

       1. neg-mul-1 16.6%

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

       2. sub-neg 16.6%

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

   11. Simplified 16.6%

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

       1. *-un-lft-identity 16.6%

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

       2. add-sqr-sqrt 4.0%

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

       3. sqrt-unprod 16.6%

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

       4. sqr-neg 16.6%

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

       5. sqrt-unprod 15.9%

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

       6. add-sqr-sqrt 16.6%

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

       7. sub-neg 16.6%

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

       8. log1p-define 72.3%

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

   13. Applied egg-rr 72.3%

       \[\leadsto \color{blue}{1 \cdot \mathsf{copysign}\left(\mathsf{log1p}\left(-x\right), x\right)} \]
   14. Step-by-step derivation

       1. *-lft-identity 72.3%

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

   15. Simplified 72.3%

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

   if 0.69999999999999996 < x

   1. Initial program 48.7%

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

      1. +-commutative 48.7%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.5%

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

      1. +-commutative 99.5%

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

      2. rem-square-sqrt 99.5%

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

      3. fabs-sqr 99.5%

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

      4. rem-square-sqrt 99.5%

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

      5. *-inverses 99.5%

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

      6. metadata-eval 99.5%

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

   7. Simplified 99.5%

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

Alternative 5: 64.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.02:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.02) (copysign (log1p (- x)) x) (copysign (log1p x) x)))

C

double code(double x) {
	double tmp;
	if (x <= -0.02) {
		tmp = copysign(log1p(-x), x);
	} else {
		tmp = copysign(log1p(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.02) {
		tmp = Math.copySign(Math.log1p(-x), x);
	} else {
		tmp = Math.copySign(Math.log1p(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.02:
		tmp = math.copysign(math.log1p(-x), x)
	else:
		tmp = math.copysign(math.log1p(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.02)
		tmp = copysign(log1p(Float64(-x)), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.02], 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[1 + x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0200000000000000004

   1. Initial program 46.3%

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

      1. +-commutative 46.3%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 2.3%

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

      2. frac-2neg 2.3%

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

      3. log-div 2.3%

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

   6. Applied egg-rr 3.9%

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

      1. neg-sub0 3.9%

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

      2. associate--r- 3.9%

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

      3. neg-sub0 3.9%

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

      4. +-commutative 3.9%

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

      5. fma-undefine 3.9%

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

      6. unpow2 3.9%

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

      7. +-commutative 3.9%

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

      8. associate-+l+ 44.5%

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

      9. sub-neg 44.5%

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

      10. +-inverses 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. neg-sub0 99.9%

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

      14. sub-neg 99.9%

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

      15. distribute-neg-in 99.9%

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

      16. neg-mul-1 99.9%

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

      17. remove-double-neg 99.9%

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

      18. +-commutative 99.9%

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

      19. neg-mul-1 99.9%

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

      20. sub-neg 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0 31.5%

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

       1. neg-mul-1 31.5%

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

       2. sub-neg 31.5%

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

   11. Simplified 31.5%

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

       1. *-un-lft-identity 31.5%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 31.5%

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

       4. sqr-neg 31.5%

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

       5. sqrt-unprod 31.5%

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

       6. add-sqr-sqrt 31.5%

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

       7. sub-neg 31.5%

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

       8. log1p-define 31.5%

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

   13. Applied egg-rr 31.5%

       \[\leadsto \color{blue}{1 \cdot \mathsf{copysign}\left(\mathsf{log1p}\left(-x\right), x\right)} \]
   14. Step-by-step derivation

       1. *-lft-identity 31.5%

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

   15. Simplified 31.5%

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

   if -0.0200000000000000004 < x

   1. Initial program 22.9%

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

      1. +-commutative 22.9%

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

      2. hypot-1-def 42.1%

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

   3. Simplified 42.1%

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

      1. flip-+ 5.3%

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

      2. frac-2neg 5.3%

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

      3. log-div 5.3%

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

   6. Applied egg-rr 5.2%

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

      1. neg-sub0 5.2%

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

      2. associate--r- 5.2%

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

      3. neg-sub0 5.2%

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

      4. +-commutative 5.2%

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

      5. fma-undefine 5.2%

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

      6. unpow2 5.2%

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

      7. +-commutative 5.2%

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

      8. associate-+l+ 5.8%

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

      9. sub-neg 5.8%

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

      10. +-inverses 6.4%

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

      11. metadata-eval 6.4%

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

      12. metadata-eval 6.4%

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

      13. neg-sub0 6.4%

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

      14. sub-neg 6.4%

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

      15. distribute-neg-in 6.4%

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

      16. neg-mul-1 6.4%

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

      17. remove-double-neg 6.4%

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

      18. +-commutative 6.4%

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

      19. neg-mul-1 6.4%

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

      20. sub-neg 6.4%

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

   8. Simplified 6.4%

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

   9. Taylor expanded in x around 0 4.3%

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

       1. neg-mul-1 4.3%

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

       2. sub-neg 4.3%

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

   11. Simplified 4.3%

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

       1. *-un-lft-identity 4.3%

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

       2. add-sqr-sqrt 4.2%

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

       3. sqrt-unprod 4.3%

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

       4. sqr-neg 4.3%

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

       5. sqrt-unprod 3.6%

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

       6. add-sqr-sqrt 4.3%

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

       7. sub-neg 4.3%

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

       8. log1p-define 61.9%

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

   13. Applied egg-rr 61.9%

       \[\leadsto \color{blue}{1 \cdot \mathsf{copysign}\left(\mathsf{log1p}\left(-x\right), x\right)} \]
   14. Step-by-step derivation

       1. *-lft-identity 61.9%

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

   15. Simplified 61.9%

       \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{log1p}\left(-x\right), x\right)} \]
   16. Step-by-step derivation

       1. *-un-lft-identity 61.9%

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

       2. neg-mul-1 61.9%

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

       3. neg-mul-1 61.9%

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

       4. add-sqr-sqrt 27.8%

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

       5. sqrt-unprod 34.9%

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

       6. sqr-neg 34.9%

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

       7. sqrt-unprod 45.4%

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

       8. add-sqr-sqrt 73.7%

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

   17. Applied egg-rr 73.7%

       \[\leadsto \color{blue}{1 \cdot \mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)} \]
   18. Step-by-step derivation

       1. *-lft-identity 73.7%

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

   19. Simplified 73.7%

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

Alternative 6: 57.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (copysign (log1p x) x))

C

double code(double x) {
	return copysign(log1p(x), x);
}

Java

public static double code(double x) {
	return Math.copySign(Math.log1p(x), x);
}

Python

def code(x):
	return math.copysign(math.log1p(x), x)

Julia

function code(x)
	return copysign(log1p(x), x)
end

Wolfram

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

Derivation

1. Initial program 29.6%

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

   1. +-commutative 29.6%

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

   2. hypot-1-def 58.8%

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

3. Simplified 58.8%

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

   1. flip-+ 4.4%

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

   2. frac-2neg 4.4%

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

   3. log-div 4.4%

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

6. Applied egg-rr 4.9%

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

   1. neg-sub0 4.9%

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

   2. associate--r- 4.9%

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

   3. neg-sub0 4.9%

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

   4. +-commutative 4.9%

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

   5. fma-undefine 4.9%

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

   6. unpow2 4.9%

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

   7. +-commutative 4.9%

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

   8. associate-+l+ 17.0%

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

   9. sub-neg 17.0%

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

   10. +-inverses 33.4%

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

   11. metadata-eval 33.4%

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

   12. metadata-eval 33.4%

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

   13. neg-sub0 33.4%

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

   14. sub-neg 33.4%

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

   15. distribute-neg-in 33.4%

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

   16. neg-mul-1 33.4%

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

   17. remove-double-neg 33.4%

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

   18. +-commutative 33.4%

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

   19. neg-mul-1 33.4%

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

   20. sub-neg 33.4%

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

8. Simplified 33.4%

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

9. Taylor expanded in x around 0 12.2%

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

    1. neg-mul-1 12.2%

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

    2. sub-neg 12.2%

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

11. Simplified 12.2%

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

    1. *-un-lft-identity 12.2%

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

    2. add-sqr-sqrt 3.0%

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

    3. sqrt-unprod 12.2%

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

    4. sqr-neg 12.2%

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

    5. sqrt-unprod 11.7%

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

    6. add-sqr-sqrt 12.2%

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

    7. sub-neg 12.2%

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

    8. log1p-define 53.1%

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

13. Applied egg-rr 53.1%

    \[\leadsto \color{blue}{1 \cdot \mathsf{copysign}\left(\mathsf{log1p}\left(-x\right), x\right)} \]
14. Step-by-step derivation

    1. *-lft-identity 53.1%

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

15. Simplified 53.1%

    \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{log1p}\left(-x\right), x\right)} \]
16. Step-by-step derivation

    1. *-un-lft-identity 53.1%

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

    2. neg-mul-1 53.1%

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

    3. neg-mul-1 53.1%

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

    4. add-sqr-sqrt 28.9%

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

    5. sqrt-unprod 29.1%

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

    6. sqr-neg 29.1%

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

    7. sqrt-unprod 32.3%

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

    8. add-sqr-sqrt 52.5%

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

17. Applied egg-rr 52.5%

    \[\leadsto \color{blue}{1 \cdot \mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)} \]
18. Step-by-step derivation

    1. *-lft-identity 52.5%

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

19. Simplified 52.5%

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

Alternative 7: 52.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x))

C

double code(double x) {
	return copysign((x * (1.0 + ((x * x) * -0.16666666666666666))), x);
}

Java

public static double code(double x) {
	return Math.copySign((x * (1.0 + ((x * x) * -0.16666666666666666))), x);
}

Python

def code(x):
	return math.copysign((x * (1.0 + ((x * x) * -0.16666666666666666))), x)

Julia

function code(x)
	return copysign(Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs((x * (1.0 + ((x * x) * -0.16666666666666666))));
end

Wolfram

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

Derivation

1. Initial program 29.6%

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

   1. +-commutative 29.6%

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

   2. hypot-1-def 58.8%

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

3. Simplified 58.8%

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

5. Taylor expanded in x around 0 6.3%

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

   1. +-commutative 6.3%

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

   2. fma-define 6.3%

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

   3. +-commutative 6.3%

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

   4. rem-square-sqrt 3.4%

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

   5. fabs-sqr 3.4%

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

   6. rem-square-sqrt 6.3%

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

   7. log1p-define 47.0%

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

   8. rem-square-sqrt 25.5%

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

   9. fabs-sqr 25.5%

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

   10. rem-square-sqrt 45.7%

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

7. Simplified 45.7%

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

8. Taylor expanded in x around 0 46.6%

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

   1. *-commutative 46.6%

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

10. Simplified 46.6%

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

    1. unpow2 46.6%

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

12. Applied egg-rr 46.6%

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

Alternative 8: 51.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(x \cdot \left(1 - x \cdot 0.5\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (copysign (* x (- 1.0 (* x 0.5))) x))

C

double code(double x) {
	return copysign((x * (1.0 - (x * 0.5))), x);
}

Java

public static double code(double x) {
	return Math.copySign((x * (1.0 - (x * 0.5))), x);
}

Python

def code(x):
	return math.copysign((x * (1.0 - (x * 0.5))), x)

Julia

function code(x)
	return copysign(Float64(x * Float64(1.0 - Float64(x * 0.5))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs((x * (1.0 - (x * 0.5))));
end

Wolfram

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

Derivation

1. Initial program 29.6%

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

   1. +-commutative 29.6%

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

   2. hypot-1-def 58.8%

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

3. Simplified 58.8%

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

   1. flip-+ 4.4%

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

   2. frac-2neg 4.4%

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

   3. log-div 4.4%

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

6. Applied egg-rr 4.9%

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

   1. neg-sub0 4.9%

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

   2. associate--r- 4.9%

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

   3. neg-sub0 4.9%

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

   4. +-commutative 4.9%

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

   5. fma-undefine 4.9%

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

   6. unpow2 4.9%

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

   7. +-commutative 4.9%

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

   8. associate-+l+ 17.0%

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

   9. sub-neg 17.0%

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

   10. +-inverses 33.4%

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

   11. metadata-eval 33.4%

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

   12. metadata-eval 33.4%

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

   13. neg-sub0 33.4%

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

   14. sub-neg 33.4%

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

   15. distribute-neg-in 33.4%

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

   16. neg-mul-1 33.4%

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

   17. remove-double-neg 33.4%

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

   18. +-commutative 33.4%

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

   19. neg-mul-1 33.4%

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

   20. sub-neg 33.4%

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

8. Simplified 33.4%

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

9. Taylor expanded in x around 0 12.2%

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

    1. neg-mul-1 12.2%

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

    2. sub-neg 12.2%

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

11. Simplified 12.2%

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

12. Taylor expanded in x around 0 46.3%

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

    1. *-commutative 46.3%

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

14. Simplified 46.3%

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

    1. add-sqr-sqrt 25.1%

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

    2. sqrt-unprod 46.3%

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

    3. sqr-neg 46.3%

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

    4. sqrt-unprod 21.2%

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

    5. add-sqr-sqrt 46.3%

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

    6. cancel-sign-sub-inv 46.3%

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

16. Applied egg-rr 46.3%

    \[\leadsto \mathsf{copysign}\left(x \cdot \color{blue}{\left(1 - x \cdot 0.5\right)}, x\right) \]
17. Add Preprocessing

Alternative 9: 51.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(x \cdot \left(1 + x \cdot 0.5\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (copysign (* x (+ 1.0 (* x 0.5))) x))

C

double code(double x) {
	return copysign((x * (1.0 + (x * 0.5))), x);
}

Java

public static double code(double x) {
	return Math.copySign((x * (1.0 + (x * 0.5))), x);
}

Python

def code(x):
	return math.copysign((x * (1.0 + (x * 0.5))), x)

Julia

function code(x)
	return copysign(Float64(x * Float64(1.0 + Float64(x * 0.5))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs((x * (1.0 + (x * 0.5))));
end

Wolfram

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

Derivation

1. Initial program 29.6%

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

   1. +-commutative 29.6%

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

   2. hypot-1-def 58.8%

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

3. Simplified 58.8%

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

   1. flip-+ 4.4%

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

   2. frac-2neg 4.4%

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

   3. log-div 4.4%

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

6. Applied egg-rr 4.9%

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

   1. neg-sub0 4.9%

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

   2. associate--r- 4.9%

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

   3. neg-sub0 4.9%

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

   4. +-commutative 4.9%

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

   5. fma-undefine 4.9%

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

   6. unpow2 4.9%

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

   7. +-commutative 4.9%

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

   8. associate-+l+ 17.0%

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

   9. sub-neg 17.0%

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

   10. +-inverses 33.4%

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

   11. metadata-eval 33.4%

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

   12. metadata-eval 33.4%

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

   13. neg-sub0 33.4%

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

   14. sub-neg 33.4%

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

   15. distribute-neg-in 33.4%

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

   16. neg-mul-1 33.4%

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

   17. remove-double-neg 33.4%

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

   18. +-commutative 33.4%

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

   19. neg-mul-1 33.4%

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

   20. sub-neg 33.4%

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

8. Simplified 33.4%

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

9. Taylor expanded in x around 0 12.2%

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

    1. neg-mul-1 12.2%

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

    2. sub-neg 12.2%

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

11. Simplified 12.2%

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

12. Taylor expanded in x around 0 46.3%

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

    1. *-commutative 46.3%

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

14. Simplified 46.3%

    \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot 0.5\right)}, x\right) \]
15. 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 2024179 
(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))