Rust f64::asinh

Percentage Accurate: 30.6% → 98.7%

Time: 6.9s

Alternatives: 8

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 30.6% 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: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -4:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-10}:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -4.0)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 1e-10)
       (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
       (copysign (log (* x 2.0)) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -4.0) {
		tmp = copysign(-log((hypot(1.0, x) - x)), x);
	} else if (t_0 <= 1e-10) {
		tmp = copysign((x + (pow(x, 3.0) * -0.16666666666666666)), x);
	} else {
		tmp = copysign(log((x * 2.0)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -4.0) {
		tmp = Math.copySign(-Math.log((Math.hypot(1.0, x) - x)), x);
	} else if (t_0 <= 1e-10) {
		tmp = Math.copySign((x + (Math.pow(x, 3.0) * -0.16666666666666666)), x);
	} else {
		tmp = Math.copySign(Math.log((x * 2.0)), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -4.0:
		tmp = math.copysign(-math.log((math.hypot(1.0, x) - x)), x)
	elif t_0 <= 1e-10:
		tmp = math.copysign((x + (math.pow(x, 3.0) * -0.16666666666666666)), x)
	else:
		tmp = math.copysign(math.log((x * 2.0)), x)
	return tmp

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -4.0)
		tmp = copysign(Float64(-log(Float64(hypot(1.0, x) - x))), x);
	elseif (t_0 <= 1e-10)
		tmp = copysign(Float64(x + Float64((x ^ 3.0) * -0.16666666666666666)), x);
	else
		tmp = copysign(log(Float64(x * 2.0)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
	tmp = 0.0;
	if (t_0 <= -4.0)
		tmp = sign(x) * abs(-log((hypot(1.0, x) - x)));
	elseif (t_0 <= 1e-10)
		tmp = sign(x) * abs((x + ((x ^ 3.0) * -0.16666666666666666)));
	else
		tmp = sign(x) * abs(log((x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -4.0], N[With[{TMP1 = Abs[(-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 1e-10], N[With[{TMP1 = Abs[N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 61.6%

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

      1. +-commutative 61.6%

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

      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-+ 3.6%

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

      2. frac-2neg 3.6%

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

      3. log-div 3.6%

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

   6. Applied egg-rr 6.5%

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

      1. sub-neg 6.5%

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

      2. fma-undefine 6.5%

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

      3. unpow2 6.5%

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

      4. distribute-neg-in 6.5%

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

      5. metadata-eval 6.5%

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

      6. associate-+r+ 60.3%

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

      7. sub-neg 60.3%

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

      8. +-inverses 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. neg-sub0 100.0%

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

      13. neg-sub0 100.0%

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

      14. associate--r- 100.0%

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

      15. neg-sub0 100.0%

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

      16. +-commutative 100.0%

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

      17. 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%

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

   if -4 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 1.00000000000000004e-10

   1. Initial program 5.9%

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

      1. +-commutative 5.9%

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

      2. hypot-1-def 5.9%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in x around 0 5.9%

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

      1. rem-square-sqrt 2.9%

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

      2. fabs-sqr 2.9%

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

      3. rem-square-sqrt 6.0%

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

      4. metadata-eval 6.0%

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

      5. unpow2 6.0%

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

      6. hypot-undefine 6.0%

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

   7. Simplified 6.0%

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

   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. distribute-rgt-in 100.0%

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

      2. *-lft-identity 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. unpow2 100.0%

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

      7. cube-mult 100.0%

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

   10. Simplified 100.0%

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

   if 1.00000000000000004e-10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 40.3%

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

      1. +-commutative 40.3%

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

      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 100.0%

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

      1. rem-square-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. *-inverses 100.0%

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

      5. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

Alternative 2: 99.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[With[{TMP1 = Abs[N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 61.6%

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

      1. +-commutative 61.6%

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

      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. Taylor expanded in x around 0 61.6%

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

      1. rem-square-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. rem-square-sqrt 6.6%

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

      4. metadata-eval 6.6%

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

      5. unpow2 6.6%

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

      6. hypot-undefine 6.6%

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

   7. Simplified 6.6%

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

   8. Taylor expanded in x around -inf 97.9%

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

   if -1.25 < x < 1.25

   1. Initial program 5.9%

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

      1. +-commutative 5.9%

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

      2. hypot-1-def 5.9%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in x around 0 5.9%

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

      1. rem-square-sqrt 2.9%

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

      2. fabs-sqr 2.9%

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

      3. rem-square-sqrt 6.0%

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

      4. metadata-eval 6.0%

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

      5. unpow2 6.0%

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

      6. hypot-undefine 6.0%

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

   7. Simplified 6.0%

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

   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. distribute-rgt-in 100.0%

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

      2. *-lft-identity 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. unpow2 100.0%

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

      7. cube-mult 100.0%

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

   10. Simplified 100.0%

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

   if 1.25 < x

   1. Initial program 40.3%

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

      1. +-commutative 40.3%

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

      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 100.0%

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

      1. rem-square-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. *-inverses 100.0%

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

      5. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

Alternative 3: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 61.6%

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

      1. +-commutative 61.6%

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

      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. Taylor expanded in x around 0 61.6%

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

      1. rem-square-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. rem-square-sqrt 6.6%

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

      4. metadata-eval 6.6%

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

      5. unpow2 6.6%

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

      6. hypot-undefine 6.6%

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

   7. Simplified 6.6%

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

   8. Taylor expanded in x around -inf 97.9%

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

   if -1.25 < x < 1.25

   1. Initial program 5.9%

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

      1. +-commutative 5.9%

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

      2. hypot-1-def 5.9%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in x around 0 5.9%

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

      1. rem-square-sqrt 2.9%

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

      2. fabs-sqr 2.9%

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

      3. rem-square-sqrt 5.9%

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

   7. Simplified 5.9%

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

   8. Taylor expanded in x around 0 100.0%

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

   if 1.25 < x

   1. Initial program 40.3%

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

      1. +-commutative 40.3%

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

      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 100.0%

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

      1. rem-square-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. *-inverses 100.0%

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

      5. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

Alternative 4: 81.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 0.72], N[With[{TMP1 = Abs[N[Log[1 + (-x)], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.71999999999999997

   1. Initial program 23.8%

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

      1. +-commutative 23.8%

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

      2. hypot-1-def 36.2%

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

   3. Simplified 36.2%

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

   5. Taylor expanded in x around 0 13.9%

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

      1. rem-square-sqrt 2.0%

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

      2. fabs-sqr 2.0%

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

      3. rem-square-sqrt 4.0%

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

   7. Simplified 4.0%

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

      1. add-sqr-sqrt 2.0%

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

      2. sqrt-unprod 10.1%

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

      3. sqr-neg 10.1%

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

      4. sqrt-unprod 12.0%

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

      5. add-sqr-sqrt 14.0%

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

      6. sub-neg 14.0%

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

   9. Applied egg-rr 14.0%

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

   10. Taylor expanded in x around -inf 14.0%

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

       1. log1p-define 77.4%

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

       2. neg-mul-1 77.4%

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

   12. Simplified 77.4%

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

   if 0.71999999999999997 < x

   1. Initial program 40.3%

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

      1. +-commutative 40.3%

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

      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 100.0%

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

      1. rem-square-sqrt 100.0%

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

      2. fabs-sqr 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. *-inverses 100.0%

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

      5. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

Alternative 5: 65.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-8) (copysign (log1p (- x)) x) (copysign (log1p x) x)))

C

double code(double x) {
	double tmp;
	if (x <= -5e-8) {
		tmp = copysign(log1p(-x), x);
	} else {
		tmp = copysign(log1p(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -5e-8) {
		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 <= -5e-8:
		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 <= -5e-8)
		tmp = copysign(log1p(Float64(-x)), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-8], 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 < -4.9999999999999998e-8

   1. Initial program 61.6%

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

      1. +-commutative 61.6%

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

      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 31.4%

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

      1. rem-square-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. rem-square-sqrt 0.9%

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

   7. Simplified 0.9%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 19.7%

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

      3. sqr-neg 19.7%

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

      4. sqrt-unprod 31.4%

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

      5. add-sqr-sqrt 31.4%

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

      6. sub-neg 31.4%

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

   9. Applied egg-rr 31.4%

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

   10. Taylor expanded in x around -inf 31.4%

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

       1. log1p-define 31.4%

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

       2. neg-mul-1 31.4%

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

   12. Simplified 31.4%

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

   if -4.9999999999999998e-8 < x

   1. Initial program 16.4%

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

      1. +-commutative 16.4%

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

      2. hypot-1-def 35.0%

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

   3. Simplified 35.0%

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

   5. Taylor expanded in x around 0 13.8%

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

      1. log1p-define 78.6%

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

      2. rem-square-sqrt 43.4%

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

      3. fabs-sqr 43.4%

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

      4. rem-square-sqrt 78.6%

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

   7. Simplified 78.6%

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

Alternative 6: 65.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 61.6%

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

      1. +-commutative 61.6%

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

      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. Taylor expanded in x around -inf 31.0%

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

      1. mul-1-neg 31.0%

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

   7. Simplified 31.0%

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

   if -1 < x

   1. Initial program 16.6%

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

      1. +-commutative 16.6%

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

      2. hypot-1-def 35.2%

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

   3. Simplified 35.2%

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

   5. Taylor expanded in x around 0 14.0%

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

      1. log1p-define 78.5%

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

      2. rem-square-sqrt 43.2%

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

      3. fabs-sqr 43.2%

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

      4. rem-square-sqrt 78.5%

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

   7. Simplified 78.5%

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

Alternative 7: 59.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 23.8%

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

      1. +-commutative 23.8%

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

      2. hypot-1-def 36.2%

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

   3. Simplified 36.2%

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

   5. Taylor expanded in x around 0 13.9%

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

      1. rem-square-sqrt 2.0%

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

      2. fabs-sqr 2.0%

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

      3. rem-square-sqrt 4.0%

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

   7. Simplified 4.0%

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

   8. Taylor expanded in x around 0 69.8%

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

   if 1.55000000000000004 < x

   1. Initial program 40.3%

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

      1. +-commutative 40.3%

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

      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 0 32.0%

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

      1. log1p-define 32.0%

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

      2. rem-square-sqrt 32.0%

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

      3. fabs-sqr 32.0%

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

      4. rem-square-sqrt 32.0%

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

   7. Simplified 32.0%

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

Alternative 8: 52.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.7%

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

   1. +-commutative 27.7%

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

   2. hypot-1-def 51.1%

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

3. Simplified 51.1%

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

5. Taylor expanded in x around 0 18.2%

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

   1. rem-square-sqrt 9.0%

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

   2. fabs-sqr 9.0%

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

   3. rem-square-sqrt 10.5%

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

7. Simplified 10.5%

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

8. Taylor expanded in x around 0 54.5%

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

Developer target: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024110 
(FPCore (x)
  :name "Rust f64::asinh"
  :precision binary64

  :alt
  (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)

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