Rust f64::asinh

Percentage Accurate: 29.6% → 99.2%

Time: 5.2s

Alternatives: 7

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 29.6% 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: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.0001:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -5.0)
     (copysign (log (/ -0.5 x)) x)
     (if (<= t_0 0.0001)
       (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
       (copysign (log (+ x 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 <= -5.0) {
		tmp = copysign(log((-0.5 / x)), x);
	} else if (t_0 <= 0.0001) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log((x + 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 <= -5.0) {
		tmp = Math.copySign(Math.log((-0.5 / x)), x);
	} else if (t_0 <= 0.0001) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(Math.log((x + 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 <= -5.0:
		tmp = math.copysign(math.log((-0.5 / x)), x)
	elif t_0 <= 0.0001:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(math.log((x + 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 <= -5.0)
		tmp = copysign(log(Float64(-0.5 / x)), x);
	elseif (t_0 <= 0.0001)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(log(Float64(x + 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 <= -5.0)
		tmp = sign(x) * abs(log((-0.5 / x)));
	elseif (t_0 <= 0.0001)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(log((x + 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, -5.0], N[With[{TMP1 = Abs[N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.0001], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + x), $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) 1)))) x) < -5

   1. Initial program 54.5%

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

      1. +-commutative 54.5%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around -inf 98.5%

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

      1. associate--l+ 98.5%

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

      2. unpow1 98.5%

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

      3. sqr-pow 0.0%

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

      4. fabs-sqr 0.0%

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

      5. sqr-pow 4.9%

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

      6. unpow1 4.9%

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

      7. associate-+r- 98.9%

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

      8. mul-1-neg 98.9%

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

      9. sub-neg 98.9%

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

      10. +-inverses 98.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{0} - 0.5 \cdot \frac{1}{x}\right), x\right) \]

      11. neg-sub0 98.9%

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

      12. associate-*r/ 98.9%

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

      13. metadata-eval 98.9%

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

      14. distribute-neg-frac 98.9%

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

      15. metadata-eval 98.9%

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

   6. Simplified 98.9%

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

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < 1.00000000000000005e-4

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

      2. hypot-1-def 6.7%

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

   3. Simplified 6.7%

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

      1. *-un-lft-identity 6.7%

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

      2. log-prod 6.7%

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

      3. metadata-eval 6.7%

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

      4. *-un-lft-identity 6.7%

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

      5. *-un-lft-identity 6.7%

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

      6. add-sqr-sqrt 3.9%

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

      7. fabs-sqr 3.9%

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

      8. add-sqr-sqrt 6.7%

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

   5. Applied egg-rr 6.7%

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

      1. +-lft-identity 6.7%

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

   7. Simplified 6.7%

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

   8. Taylor expanded in x around 0 100.0%

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

   if 1.00000000000000005e-4 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x)

   1. Initial program 53.9%

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

      1. +-commutative 53.9%

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

      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. Taylor expanded in x around inf 100.0%

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

      1. unpow1 100.0%

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

      2. sqr-pow 100.0%

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

      3. fabs-sqr 100.0%

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

      4. sqr-pow 100.0%

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

      5. unpow1 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.0001:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 2: 82.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\mathsf{copysign}\left(\left(x \cdot \left(x + 2\right)\right) \cdot \left(0.5 + x \cdot -0.25\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.65)
   (copysign (log (- x)) x)
   (if (<= x 1.1)
     (copysign (* (* x (+ x 2.0)) (+ 0.5 (* x -0.25))) x)
     (copysign (log (+ x x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.65) {
		tmp = copysign(log(-x), x);
	} else if (x <= 1.1) {
		tmp = copysign(((x * (x + 2.0)) * (0.5 + (x * -0.25))), x);
	} else {
		tmp = copysign(log((x + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.65) {
		tmp = Math.copySign(Math.log(-x), x);
	} else if (x <= 1.1) {
		tmp = Math.copySign(((x * (x + 2.0)) * (0.5 + (x * -0.25))), x);
	} else {
		tmp = Math.copySign(Math.log((x + x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.65:
		tmp = math.copysign(math.log(-x), x)
	elif x <= 1.1:
		tmp = math.copysign(((x * (x + 2.0)) * (0.5 + (x * -0.25))), x)
	else:
		tmp = math.copysign(math.log((x + x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.65)
		tmp = copysign(log(Float64(-x)), x);
	elseif (x <= 1.1)
		tmp = copysign(Float64(Float64(x * Float64(x + 2.0)) * Float64(0.5 + Float64(x * -0.25))), x);
	else
		tmp = copysign(log(Float64(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.65)
		tmp = sign(x) * abs(log(-x));
	elseif (x <= 1.1)
		tmp = sign(x) * abs(((x * (x + 2.0)) * (0.5 + (x * -0.25))));
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.65], N[With[{TMP1 = Abs[N[Log[(-x)], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.1], N[With[{TMP1 = Abs[N[(N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(x * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6499999999999999

   1. Initial program 54.5%

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

      1. +-commutative 54.5%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around -inf 31.2%

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

      1. mul-1-neg 31.2%

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

   6. Simplified 31.2%

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

   if -1.6499999999999999 < x < 1.1000000000000001

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

      2. hypot-1-def 6.7%

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

   3. Simplified 6.7%

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

      1. expm1-log1p-u 6.7%

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

      2. expm1-udef 6.6%

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

      3. log1p-udef 6.6%

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

      4. add-exp-log 6.6%

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

      5. *-un-lft-identity 6.6%

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

      6. *-un-lft-identity 6.6%

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

      7. add-sqr-sqrt 3.9%

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

      8. fabs-sqr 3.9%

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

      9. add-sqr-sqrt 6.7%

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

   5. Applied egg-rr 6.7%

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

   6. Taylor expanded in x around 0 6.5%

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

      1. flip-- 6.5%

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

      2. div-inv 6.5%

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

      3. metadata-eval 6.5%

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

      4. difference-of-sqr-1 6.5%

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

      5. +-commutative 6.5%

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

      6. associate-+l+ 6.5%

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

      7. metadata-eval 6.5%

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

      8. add-exp-log 6.5%

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

      9. log1p-udef 6.5%

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

      10. expm1-udef 99.6%

          \[\leadsto \mathsf{copysign}\left(\left(\left(x + 2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}\right) \cdot \frac{1}{\left(1 + x\right) + 1}, x\right) \]

      11. expm1-log1p-u 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+l+ 99.6%

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

      14. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in x around 0 99.7%

      \[\leadsto \mathsf{copysign}\left(\left(\left(x + 2\right) \cdot x\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot x\right)}, x\right) \]
   10. Step-by-step derivation

       1. *-commutative 99.7%

          \[\leadsto \mathsf{copysign}\left(\left(\left(x + 2\right) \cdot x\right) \cdot \left(0.5 + \color{blue}{x \cdot -0.25}\right), x\right) \]

   11. Simplified 99.7%

       \[\leadsto \mathsf{copysign}\left(\left(\left(x + 2\right) \cdot x\right) \cdot \color{blue}{\left(0.5 + x \cdot -0.25\right)}, x\right) \]

   if 1.1000000000000001 < x

   1. Initial program 53.9%

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

      1. +-commutative 53.9%

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

      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. Taylor expanded in x around inf 100.0%

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

      1. unpow1 100.0%

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

      2. sqr-pow 100.0%

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

      3. fabs-sqr 100.0%

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

      4. sqr-pow 100.0%

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

      5. unpow1 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\mathsf{copysign}\left(\left(x \cdot \left(x + 2\right)\right) \cdot \left(0.5 + x \cdot -0.25\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 3: 99.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\mathsf{copysign}\left(\left(x \cdot \left(x + 2\right)\right) \cdot \left(0.5 + x \cdot -0.25\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.1)
     (copysign (* (* x (+ x 2.0)) (+ 0.5 (* x -0.25))) x)
     (copysign (log (+ x x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = copysign(log((-0.5 / x)), x);
	} else if (x <= 1.1) {
		tmp = copysign(((x * (x + 2.0)) * (0.5 + (x * -0.25))), x);
	} else {
		tmp = copysign(log((x + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = Math.copySign(Math.log((-0.5 / x)), x);
	} else if (x <= 1.1) {
		tmp = Math.copySign(((x * (x + 2.0)) * (0.5 + (x * -0.25))), x);
	} else {
		tmp = Math.copySign(Math.log((x + x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = math.copysign(math.log((-0.5 / x)), x)
	elif x <= 1.1:
		tmp = math.copysign(((x * (x + 2.0)) * (0.5 + (x * -0.25))), x)
	else:
		tmp = math.copysign(math.log((x + x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = copysign(log(Float64(-0.5 / x)), x);
	elseif (x <= 1.1)
		tmp = copysign(Float64(Float64(x * Float64(x + 2.0)) * Float64(0.5 + Float64(x * -0.25))), x);
	else
		tmp = copysign(log(Float64(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = sign(x) * abs(log((-0.5 / x)));
	elseif (x <= 1.1)
		tmp = sign(x) * abs(((x * (x + 2.0)) * (0.5 + (x * -0.25))));
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], 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, 1.1], N[With[{TMP1 = Abs[N[(N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(x * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 54.5%

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

      1. +-commutative 54.5%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around -inf 98.5%

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

      1. associate--l+ 98.5%

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

      2. unpow1 98.5%

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

      3. sqr-pow 0.0%

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

      4. fabs-sqr 0.0%

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

      5. sqr-pow 4.9%

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

      6. unpow1 4.9%

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

      7. associate-+r- 98.9%

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

      8. mul-1-neg 98.9%

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

      9. sub-neg 98.9%

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

      10. +-inverses 98.9%

          \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{0} - 0.5 \cdot \frac{1}{x}\right), x\right) \]

      11. neg-sub0 98.9%

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

      12. associate-*r/ 98.9%

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

      13. metadata-eval 98.9%

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

      14. distribute-neg-frac 98.9%

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

      15. metadata-eval 98.9%

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

   6. Simplified 98.9%

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

   if -1.05000000000000004 < x < 1.1000000000000001

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

      2. hypot-1-def 6.7%

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

   3. Simplified 6.7%

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

      1. expm1-log1p-u 6.7%

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

      2. expm1-udef 6.6%

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

      3. log1p-udef 6.6%

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

      4. add-exp-log 6.6%

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

      5. *-un-lft-identity 6.6%

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

      6. *-un-lft-identity 6.6%

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

      7. add-sqr-sqrt 3.9%

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

      8. fabs-sqr 3.9%

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

      9. add-sqr-sqrt 6.7%

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

   5. Applied egg-rr 6.7%

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

   6. Taylor expanded in x around 0 6.5%

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

      1. flip-- 6.5%

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

      2. div-inv 6.5%

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

      3. metadata-eval 6.5%

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

      4. difference-of-sqr-1 6.5%

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

      5. +-commutative 6.5%

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

      6. associate-+l+ 6.5%

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

      7. metadata-eval 6.5%

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

      8. add-exp-log 6.5%

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

      9. log1p-udef 6.5%

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

      10. expm1-udef 99.6%

          \[\leadsto \mathsf{copysign}\left(\left(\left(x + 2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}\right) \cdot \frac{1}{\left(1 + x\right) + 1}, x\right) \]

      11. expm1-log1p-u 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+l+ 99.6%

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

      14. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in x around 0 99.7%

      \[\leadsto \mathsf{copysign}\left(\left(\left(x + 2\right) \cdot x\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot x\right)}, x\right) \]
   10. Step-by-step derivation

       1. *-commutative 99.7%

          \[\leadsto \mathsf{copysign}\left(\left(\left(x + 2\right) \cdot x\right) \cdot \left(0.5 + \color{blue}{x \cdot -0.25}\right), x\right) \]

   11. Simplified 99.7%

       \[\leadsto \mathsf{copysign}\left(\left(\left(x + 2\right) \cdot x\right) \cdot \color{blue}{\left(0.5 + x \cdot -0.25\right)}, x\right) \]

   if 1.1000000000000001 < x

   1. Initial program 53.9%

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

      1. +-commutative 53.9%

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

      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. Taylor expanded in x around inf 100.0%

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

      1. unpow1 100.0%

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

      2. sqr-pow 100.0%

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

      3. fabs-sqr 100.0%

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

      4. sqr-pow 100.0%

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

      5. unpow1 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\mathsf{copysign}\left(\left(x \cdot \left(x + 2\right)\right) \cdot \left(0.5 + x \cdot -0.25\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 4: 64.5% 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 54.5%

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

      1. +-commutative 54.5%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around -inf 31.2%

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

      1. mul-1-neg 31.2%

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

   6. Simplified 31.2%

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

   if -1 < x

   1. Initial program 22.3%

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

      1. +-commutative 22.3%

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

      2. hypot-1-def 37.6%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in x around 0 14.6%

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

      1. log1p-def 76.6%

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

      2. unpow1 76.6%

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

      3. sqr-pow 45.4%

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

      4. fabs-sqr 45.4%

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

      5. sqr-pow 76.6%

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

      6. unpow1 76.6%

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

   6. Simplified 76.6%

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

4. Final simplification 64.9%

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

Alternative 5: 60.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{copysign}\left(\frac{x \cdot 2}{2 - x}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-7) (copysign (/ (* x 2.0) (- 2.0 x)) x) (copysign (log1p x) x)))

C

double code(double x) {
	double tmp;
	if (x <= -1e-7) {
		tmp = copysign(((x * 2.0) / (2.0 - x)), x);
	} else {
		tmp = copysign(log1p(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1e-7) {
		tmp = Math.copySign(((x * 2.0) / (2.0 - x)), x);
	} else {
		tmp = Math.copySign(Math.log1p(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1e-7:
		tmp = math.copysign(((x * 2.0) / (2.0 - x)), x)
	else:
		tmp = math.copysign(math.log1p(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-7)
		tmp = copysign(Float64(Float64(x * 2.0) / Float64(2.0 - x)), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1e-7], N[With[{TMP1 = Abs[N[(N[(x * 2.0), $MachinePrecision] / N[(2.0 - 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 < -9.9999999999999995e-8

   1. Initial program 54.5%

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

      1. +-commutative 54.5%

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

      2. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

      1. expm1-log1p-u 96.9%

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

      2. expm1-udef 96.9%

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

      3. log1p-udef 96.9%

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

      4. add-exp-log 98.5%

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

      5. *-un-lft-identity 98.5%

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

      6. *-un-lft-identity 98.5%

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

      7. add-sqr-sqrt 0.0%

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

      8. fabs-sqr 0.0%

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

      9. add-sqr-sqrt 5.2%

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

   5. Applied egg-rr 5.2%

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

   6. Taylor expanded in x around 0 5.6%

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

      1. associate--l+ 5.6%

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

      2. flip-+ 5.3%

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

      3. metadata-eval 5.3%

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

      4. sub-neg 5.3%

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

      5. metadata-eval 5.3%

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

      6. sub-neg 5.3%

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

      7. metadata-eval 5.3%

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

      8. sub-neg 5.3%

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

      9. metadata-eval 5.3%

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

   8. Applied egg-rr 5.3%

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

      1. sub-neg 5.3%

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

      2. +-commutative 5.3%

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

      3. distribute-rgt-in 5.3%

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

      4. neg-mul-1 5.3%

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

      5. distribute-neg-in 5.3%

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

      6. remove-double-neg 5.3%

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

      7. associate-+r+ 5.3%

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

      8. +-commutative 5.3%

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

      9. associate-+l+ 5.3%

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

      10. metadata-eval 5.3%

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

      11. +-rgt-identity 5.3%

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

      12. distribute-rgt-neg-in 5.3%

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

      13. +-commutative 5.3%

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

      14. distribute-neg-in 5.3%

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

      15. metadata-eval 5.3%

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

      16. sub-neg 5.3%

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

      17. +-commutative 5.3%

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

      18. associate--r+ 5.3%

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

      19. metadata-eval 5.3%

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

   10. Simplified 5.3%

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

   11. Taylor expanded in x around 0 14.3%

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

       1. *-commutative 14.3%

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

   13. Simplified 14.3%

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

   if -9.9999999999999995e-8 < x

   1. Initial program 22.3%

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

      1. +-commutative 22.3%

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

      2. hypot-1-def 37.6%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in x around 0 14.6%

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

      1. log1p-def 76.6%

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

      2. unpow1 76.6%

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

      3. sqr-pow 45.4%

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

      4. fabs-sqr 45.4%

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

      5. sqr-pow 76.6%

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

      6. unpow1 76.6%

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

   6. Simplified 76.6%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{copysign}\left(\frac{x \cdot 2}{2 - x}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 6: 55.9% accurate, 3.8× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (copysign (/ (* x 2.0) (- 2.0 x)) x))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.6%

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

   1. +-commutative 30.6%

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

   2. hypot-1-def 53.3%

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

3. Simplified 53.3%

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

   1. expm1-log1p-u 52.5%

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

   2. expm1-udef 52.4%

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

   3. log1p-udef 52.5%

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

   4. add-exp-log 53.3%

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

   5. *-un-lft-identity 53.3%

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

   6. *-un-lft-identity 53.3%

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

   7. add-sqr-sqrt 26.5%

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

   8. fabs-sqr 26.5%

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

   9. add-sqr-sqrt 29.2%

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

5. Applied egg-rr 29.2%

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

6. Taylor expanded in x around 0 6.0%

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

   1. associate--l+ 6.0%

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

   2. flip-+ 5.8%

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

   3. metadata-eval 5.8%

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

   4. sub-neg 5.8%

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

   5. metadata-eval 5.8%

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

   6. sub-neg 5.8%

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

   7. metadata-eval 5.8%

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

   8. sub-neg 5.8%

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

   9. metadata-eval 5.8%

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

8. Applied egg-rr 5.8%

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

   1. sub-neg 5.8%

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

   2. +-commutative 5.8%

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

   3. distribute-rgt-in 5.8%

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

   4. neg-mul-1 5.8%

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

   5. distribute-neg-in 5.8%

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

   6. remove-double-neg 5.8%

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

   7. associate-+r+ 12.4%

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

   8. +-commutative 12.4%

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

   9. associate-+l+ 52.0%

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

   10. metadata-eval 52.0%

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

   11. +-rgt-identity 52.0%

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

   12. distribute-rgt-neg-in 52.0%

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

   13. +-commutative 52.0%

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

   14. distribute-neg-in 52.0%

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

   15. metadata-eval 52.0%

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

   16. sub-neg 52.0%

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

   17. +-commutative 52.0%

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

   18. associate--r+ 52.0%

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

   19. metadata-eval 52.0%

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

10. Simplified 52.0%

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

11. Taylor expanded in x around 0 56.3%

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

    1. *-commutative 56.3%

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

13. Simplified 56.3%

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

14. Final simplification 56.3%

    \[\leadsto \mathsf{copysign}\left(\frac{x \cdot 2}{2 - x}, x\right) \]

Alternative 7: 51.9% 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 30.6%

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

   1. +-commutative 30.6%

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

   2. hypot-1-def 53.3%

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

3. Simplified 53.3%

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

   1. *-un-lft-identity 53.3%

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

   2. log-prod 53.3%

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

   3. metadata-eval 53.3%

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

   4. *-un-lft-identity 53.3%

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

   5. *-un-lft-identity 53.3%

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

   6. add-sqr-sqrt 26.6%

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

   7. fabs-sqr 26.6%

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

   8. add-sqr-sqrt 29.3%

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

5. Applied egg-rr 29.3%

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

   1. +-lft-identity 29.3%

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

7. Simplified 29.3%

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

8. Taylor expanded in x around 0 52.2%

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

9. Final simplification 52.2%

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

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 2023201 
(FPCore (x)
  :name "Rust f64::asinh"
  :precision binary64

  :herbie-target
  (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))