Rust f64::asinh

Percentage Accurate: 31.0% → 98.9%

Time: 6.1s

Alternatives: 7

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: 31.0% 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: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{copysign}\left(2 \cdot \left(x \cdot 0.5\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -10.0)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.0)
       (copysign (* 2.0 (* x 0.5)) x)
       (copysign (* 2.0 (log (sqrt (+ x (hypot 1.0 x))))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = copysign(-log((hypot(1.0, x) - x)), x);
	} else if (t_0 <= 0.0) {
		tmp = copysign((2.0 * (x * 0.5)), x);
	} else {
		tmp = copysign((2.0 * log(sqrt((x + hypot(1.0, x))))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = Math.copySign(-Math.log((Math.hypot(1.0, x) - x)), x);
	} else if (t_0 <= 0.0) {
		tmp = Math.copySign((2.0 * (x * 0.5)), x);
	} else {
		tmp = Math.copySign((2.0 * Math.log(Math.sqrt((x + Math.hypot(1.0, x))))), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -10.0:
		tmp = math.copysign(-math.log((math.hypot(1.0, x) - x)), x)
	elif t_0 <= 0.0:
		tmp = math.copysign((2.0 * (x * 0.5)), x)
	else:
		tmp = math.copysign((2.0 * math.log(math.sqrt((x + math.hypot(1.0, x))))), x)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = sign(x) * abs(-log((hypot(1.0, x) - x)));
	elseif (t_0 <= 0.0)
		tmp = sign(x) * abs((2.0 * (x * 0.5)));
	else
		tmp = sign(x) * abs((2.0 * log(sqrt((x + hypot(1.0, x))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], N[With[{TMP1 = Abs[(-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[With[{TMP1 = Abs[N[(2.0 * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[(2.0 * N[Log[N[Sqrt[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 58.5%

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

      1. +-commutative 58.5%

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

      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-+ 2.5%

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

      2. frac-2neg 2.5%

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

      3. log-div 2.5%

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

   6. Applied egg-rr 5.9%

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

      1. sub-neg 5.9%

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

      2. sub-neg 5.9%

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

      3. fma-undefine 5.9%

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

      4. unpow2 5.9%

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

      5. associate--r+ 57.1%

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

      6. +-inverses 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-sub0 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate--r- 100.0%

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

      13. neg-sub0 100.0%

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

      14. +-commutative 100.0%

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

      15. 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 -10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.0

   1. Initial program 5.7%

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

      1. +-commutative 5.7%

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

      2. hypot-1-def 5.7%

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

   3. Simplified 5.7%

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

      1. add-sqr-sqrt 5.7%

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

      2. log-prod 5.7%

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

      3. add-sqr-sqrt 2.9%

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

      4. fabs-sqr 2.9%

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

      5. add-sqr-sqrt 5.2%

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

      6. add-sqr-sqrt 2.9%

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

      7. fabs-sqr 2.9%

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

      8. add-sqr-sqrt 5.7%

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

   6. Applied egg-rr 5.7%

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

      1. count-2 5.7%

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

   8. Simplified 5.7%

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

   9. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(2 \cdot \color{blue}{\left(0.5 \cdot x\right)}, x\right) \]

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

   1. Initial program 50.0%

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

      1. +-commutative 50.0%

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

      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. add-sqr-sqrt 99.9%

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

      2. log-prod 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. fabs-sqr 99.9%

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

      5. add-sqr-sqrt 99.9%

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

      6. add-sqr-sqrt 99.9%

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

      7. fabs-sqr 99.9%

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

      8. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. count-2 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0:\\ \;\;\;\;\mathsf{copysign}\left(2 \cdot \left(x \cdot 0.5\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(2 \cdot \left(x \cdot 0.5\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -7.6e-6)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 6.8e-6)
     (copysign (* 2.0 (* x 0.5)) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -7.6e-6) {
		tmp = copysign(-log((hypot(1.0, x) - x)), x);
	} else if (x <= 6.8e-6) {
		tmp = copysign((2.0 * (x * 0.5)), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -7.6e-6) {
		tmp = Math.copySign(-Math.log((Math.hypot(1.0, x) - x)), x);
	} else if (x <= 6.8e-6) {
		tmp = Math.copySign((2.0 * (x * 0.5)), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -7.6e-6:
		tmp = math.copysign(-math.log((math.hypot(1.0, x) - x)), x)
	elif x <= 6.8e-6:
		tmp = math.copysign((2.0 * (x * 0.5)), 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 <= -7.6e-6)
		tmp = copysign(Float64(-log(Float64(hypot(1.0, x) - x))), x);
	elseif (x <= 6.8e-6)
		tmp = copysign(Float64(2.0 * Float64(x * 0.5)), 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 <= -7.6e-6)
		tmp = sign(x) * abs(-log((hypot(1.0, x) - x)));
	elseif (x <= 6.8e-6)
		tmp = sign(x) * abs((2.0 * (x * 0.5)));
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -7.6e-6], N[With[{TMP1 = Abs[(-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 6.8e-6], N[With[{TMP1 = Abs[N[(2.0 * N[(x * 0.5), $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 < -7.6000000000000001e-6

   1. Initial program 58.5%

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

      1. +-commutative 58.5%

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

      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-+ 2.5%

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

      2. frac-2neg 2.5%

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

      3. log-div 2.5%

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

   6. Applied egg-rr 5.9%

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

      1. sub-neg 5.9%

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

      2. sub-neg 5.9%

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

      3. fma-undefine 5.9%

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

      4. unpow2 5.9%

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

      5. associate--r+ 57.1%

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

      6. +-inverses 100.0%

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

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. neg-sub0 100.0%

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

      11. neg-sub0 100.0%

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

      12. associate--r- 100.0%

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

      13. neg-sub0 100.0%

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

      14. +-commutative 100.0%

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

      15. 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 -7.6000000000000001e-6 < x < 6.80000000000000012e-6

   1. Initial program 5.7%

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

      1. +-commutative 5.7%

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

      2. hypot-1-def 5.7%

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

   3. Simplified 5.7%

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

      1. add-sqr-sqrt 5.7%

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

      2. log-prod 5.7%

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

      3. add-sqr-sqrt 2.9%

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

      4. fabs-sqr 2.9%

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

      5. add-sqr-sqrt 5.2%

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

      6. add-sqr-sqrt 2.9%

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

      7. fabs-sqr 2.9%

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

      8. add-sqr-sqrt 5.7%

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

   6. Applied egg-rr 5.7%

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

      1. count-2 5.7%

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

   8. Simplified 5.7%

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

   9. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(2 \cdot \color{blue}{\left(0.5 \cdot x\right)}, x\right) \]

   if 6.80000000000000012e-6 < x

   1. Initial program 50.0%

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

      1. +-commutative 50.0%

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

      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. *-un-lft-identity 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. fabs-sqr 99.9%

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

      6. add-sqr-sqrt 99.9%

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

      7. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. +-rgt-identity 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(2 \cdot \left(x \cdot 0.5\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.18 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.18e-8)
   (copysign (log1p (fabs x)) x)
   (copysign (log (+ x (hypot 1.0 x))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.18e-8) {
		tmp = copysign(log1p(fabs(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 <= 1.18e-8) {
		tmp = Math.copySign(Math.log1p(Math.abs(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 <= 1.18e-8:
		tmp = math.copysign(math.log1p(math.fabs(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 <= 1.18e-8)
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.18e-8], N[With[{TMP1 = Abs[N[Log[1 + N[Abs[x], $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 2 regimes

2. if x < 1.18e-8

   1. Initial program 24.9%

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

      1. +-commutative 24.9%

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

      2. hypot-1-def 40.1%

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

   3. Simplified 40.1%

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

   5. Taylor expanded in x around 0 15.0%

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

      1. log1p-define 74.7%

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

   7. Simplified 74.7%

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

   if 1.18e-8 < x

   1. Initial program 50.0%

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

      1. +-commutative 50.0%

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

      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. *-un-lft-identity 99.9%

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

      2. *-commutative 99.9%

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

      3. log-prod 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. fabs-sqr 99.9%

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

      6. add-sqr-sqrt 99.9%

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

      7. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. +-rgt-identity 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.18 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.2%

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

   1. +-commutative 31.2%

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

   2. hypot-1-def 55.0%

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

3. Simplified 55.0%

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

5. Taylor expanded in x around 0 19.1%

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

   1. log1p-define 63.9%

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

7. Simplified 63.9%

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

8. Final simplification 63.9%

   \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right) \]
9. Add Preprocessing

Alternative 5: 64.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.5) (copysign (- (log (/ -1.0 x))) x) (copysign (log1p x) x)))

C

double code(double x) {
	double tmp;
	if (x <= -0.5) {
		tmp = copysign(-log((-1.0 / x)), x);
	} else {
		tmp = copysign(log1p(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.5) {
		tmp = Math.copySign(-Math.log((-1.0 / x)), x);
	} else {
		tmp = Math.copySign(Math.log1p(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.5:
		tmp = math.copysign(-math.log((-1.0 / x)), x)
	else:
		tmp = math.copysign(math.log1p(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.5)
		tmp = copysign(Float64(-log(Float64(-1.0 / x))), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.5

   1. Initial program 58.5%

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

      1. +-commutative 58.5%

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

      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 31.1%

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

      1. mul-1-neg 31.1%

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

   7. Simplified 31.1%

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

   if -0.5 < x

   1. Initial program 20.9%

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

      1. +-commutative 20.9%

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

      2. hypot-1-def 38.1%

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

   3. Simplified 38.1%

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

   5. Taylor expanded in x around 0 6.4%

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

      1. log1p-define 67.5%

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

   7. Simplified 67.5%

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

   8. Taylor expanded in x around 0 14.6%

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

      1. log1p-define 76.3%

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

      2. rem-square-sqrt 47.2%

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

      3. fabs-sqr 47.2%

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

      4. rem-square-sqrt 76.3%

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

   10. Simplified 76.3%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;\mathsf{copysign}\left(2 \cdot \left(x \cdot 0.5\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.1) (copysign (* 2.0 (* x 0.5)) x) (copysign (log x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 3.1) {
		tmp = copysign((2.0 * (x * 0.5)), x);
	} else {
		tmp = copysign(log(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.1) {
		tmp = Math.copySign((2.0 * (x * 0.5)), x);
	} else {
		tmp = Math.copySign(Math.log(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.1:
		tmp = math.copysign((2.0 * (x * 0.5)), x)
	else:
		tmp = math.copysign(math.log(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.1)
		tmp = copysign(Float64(2.0 * Float64(x * 0.5)), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.1)
		tmp = sign(x) * abs((2.0 * (x * 0.5)));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.1], N[With[{TMP1 = Abs[N[(2.0 * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.10000000000000009

   1. Initial program 25.3%

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

      1. +-commutative 25.3%

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

      2. hypot-1-def 40.3%

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

   3. Simplified 40.3%

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

      1. add-sqr-sqrt 40.3%

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

      2. log-prod 40.3%

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

      3. add-sqr-sqrt 2.4%

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

      4. fabs-sqr 2.4%

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

      5. add-sqr-sqrt 5.1%

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

      6. add-sqr-sqrt 2.4%

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

      7. fabs-sqr 2.4%

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

      8. add-sqr-sqrt 6.0%

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

   6. Applied egg-rr 6.0%

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

      1. count-2 6.0%

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

   8. Simplified 6.0%

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

   9. Taylor expanded in x around 0 65.5%

      \[\leadsto \mathsf{copysign}\left(2 \cdot \color{blue}{\left(0.5 \cdot x\right)}, x\right) \]

   if 3.10000000000000009 < x

   1. Initial program 49.3%

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

      1. +-commutative 49.3%

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

      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 31.6%

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

      1. mul-1-neg 31.6%

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

      2. log-rec 31.6%

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

      3. remove-double-neg 31.6%

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

   7. Simplified 31.6%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;\mathsf{copysign}\left(2 \cdot \left(x \cdot 0.5\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(2 \cdot \left(x \cdot 0.5\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (copysign (* 2.0 (* x 0.5)) x))

C

double code(double x) {
	return copysign((2.0 * (x * 0.5)), x);
}

Java

public static double code(double x) {
	return Math.copySign((2.0 * (x * 0.5)), x);
}

Python

def code(x):
	return math.copysign((2.0 * (x * 0.5)), x)

Julia

function code(x)
	return copysign(Float64(2.0 * Float64(x * 0.5)), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs((2.0 * (x * 0.5)));
end

Wolfram

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

Derivation

1. Initial program 31.2%

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

   1. +-commutative 31.2%

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

   2. hypot-1-def 55.0%

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

3. Simplified 55.0%

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

   1. add-sqr-sqrt 55.0%

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

   2. log-prod 55.0%

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

   3. add-sqr-sqrt 26.4%

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

   4. fabs-sqr 26.4%

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

   5. add-sqr-sqrt 28.4%

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

   6. add-sqr-sqrt 26.4%

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

   7. fabs-sqr 26.4%

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

   8. add-sqr-sqrt 29.1%

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

6. Applied egg-rr 29.1%

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

   1. count-2 29.1%

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

8. Simplified 29.1%

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

9. Taylor expanded in x around 0 50.7%

   \[\leadsto \mathsf{copysign}\left(2 \cdot \color{blue}{\left(0.5 \cdot x\right)}, x\right) \]

10. Final simplification 50.7%

    \[\leadsto \mathsf{copysign}\left(2 \cdot \left(x \cdot 0.5\right), x\right) \]
11. Add Preprocessing

Developer target: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024073 
(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))