Rust f64::asinh

Percentage Accurate: 30.2% → 99.8%

Time: 8.5s

Alternatives: 12

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 30.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.004:\\ \;\;\;\;\mathsf{copysign}\left(x + \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right) \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.5)
     (copysign (log (- (hypot 1.0 x) x)) x)
     (if (<= t_0 0.004)
       (copysign
        (+ x (* (fma 0.075 (* x x) -0.16666666666666666) (pow x 3.0)))
        x)
       (copysign (log (+ 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 <= -0.5) {
		tmp = copysign(log((hypot(1.0, x) - x)), x);
	} else if (t_0 <= 0.004) {
		tmp = copysign((x + (fma(0.075, (x * x), -0.16666666666666666) * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log((x + 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 <= -0.5)
		tmp = copysign(log(Float64(hypot(1.0, x) - x)), x);
	elseif (t_0 <= 0.004)
		tmp = copysign(Float64(x + Float64(fma(0.075, Float64(x * x), -0.16666666666666666) * (x ^ 3.0))), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return 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, -0.5], 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, 0.004], N[With[{TMP1 = Abs[N[(x + N[(N[(0.075 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -0.5

   1. Initial program 50.1%

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

      1. +-commutative 50.1%

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

      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. +-commutative 100.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. add-sqr-sqrt 7.5%

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

   4. Applied egg-rr 7.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 50.1%

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

      3. sqr-neg 50.1%

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

      4. sqrt-unprod 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. sub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   if -0.5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.0040000000000000001

   1. Initial program 8.3%

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

      1. +-commutative 8.3%

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

      2. hypot-1-def 8.3%

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

      3. +-commutative 8.3%

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

      4. add-sqr-sqrt 4.9%

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

      5. fabs-sqr 4.9%

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

      6. add-sqr-sqrt 8.3%

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

   4. Applied egg-rr 8.3%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. *-lft-identity 100.0%

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

      3. unpow2 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

      6. fma-neg 100.0%

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

      7. unpow2 100.0%

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

      8. metadata-eval 100.0%

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

      9. unpow3 100.0%

         \[\leadsto \mathsf{copysign}\left(x + \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right) \cdot \color{blue}{{x}^{3}}, x\right) \]

   7. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x + \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right) \cdot {x}^{3}}, x\right) \]

   if 0.0040000000000000001 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 47.3%

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

      1. +-commutative 47.3%

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

      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. +-commutative 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. fabs-sqr 100.0%

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

      6. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.004:\\ \;\;\;\;\mathsf{copysign}\left(x + \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right) \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -8.5e-6)
   (copysign (log (- (hypot 1.0 x) x)) x)
   (if (<= x 7.2e-6) (copysign x x) (copysign (log (+ x (hypot 1.0 x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -8.5e-6) {
		tmp = copysign(log((hypot(1.0, x) - x)), x);
	} else if (x <= 7.2e-6) {
		tmp = copysign(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 <= -8.5e-6) {
		tmp = Math.copySign(Math.log((Math.hypot(1.0, x) - x)), x);
	} else if (x <= 7.2e-6) {
		tmp = Math.copySign(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 <= -8.5e-6:
		tmp = math.copysign(math.log((math.hypot(1.0, x) - x)), x)
	elif x <= 7.2e-6:
		tmp = math.copysign(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 <= -8.5e-6)
		tmp = copysign(log(Float64(hypot(1.0, x) - x)), x);
	elseif (x <= 7.2e-6)
		tmp = copysign(x, 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 <= -8.5e-6)
		tmp = sign(x) * abs(log((hypot(1.0, x) - x)));
	elseif (x <= 7.2e-6)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -8.5e-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, 7.2e-6], N[With[{TMP1 = Abs[x], 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 < -8.4999999999999999e-6

   1. Initial program 50.1%

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

      1. +-commutative 50.1%

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

      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. +-commutative 100.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. add-sqr-sqrt 7.5%

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

   4. Applied egg-rr 7.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 50.1%

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

      3. sqr-neg 50.1%

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

      4. sqrt-unprod 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. sub-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   if -8.4999999999999999e-6 < x < 7.19999999999999967e-6

   1. Initial program 7.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 7.7%

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

      1. log1p-define 98.9%

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

      2. rem-square-sqrt 49.8%

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

      3. fabs-sqr 49.8%

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

      4. rem-square-sqrt 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0 100.0%

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

   if 7.19999999999999967e-6 < x

   1. Initial program 48.0%

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

      1. +-commutative 48.0%

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

      2. hypot-1-def 99.8%

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

      3. +-commutative 99.8%

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

      4. add-sqr-sqrt 99.8%

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

      5. fabs-sqr 99.8%

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

      6. add-sqr-sqrt 99.8%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.8:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(-x\right) - \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.8)
   (copysign (log (- (- x) (+ x (/ 0.5 x)))) x)
   (if (<= x 7.2e-6) (copysign x x) (copysign (log (+ x (hypot 1.0 x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.8) {
		tmp = copysign(log((-x - (x + (0.5 / x)))), x);
	} else if (x <= 7.2e-6) {
		tmp = copysign(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 <= -0.8) {
		tmp = Math.copySign(Math.log((-x - (x + (0.5 / x)))), x);
	} else if (x <= 7.2e-6) {
		tmp = Math.copySign(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 <= -0.8:
		tmp = math.copysign(math.log((-x - (x + (0.5 / x)))), x)
	elif x <= 7.2e-6:
		tmp = math.copysign(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 <= -0.8)
		tmp = copysign(log(Float64(Float64(-x) - Float64(x + Float64(0.5 / x)))), x);
	elseif (x <= 7.2e-6)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.8)
		tmp = sign(x) * abs(log((-x - (x + (0.5 / x)))));
	elseif (x <= 7.2e-6)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.8], N[With[{TMP1 = Abs[N[Log[N[((-x) - N[(x + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 7.2e-6], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.80000000000000004

   1. Initial program 50.1%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. distribute-rgt-in 98.6%

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

      3. *-lft-identity 98.6%

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

      4. distribute-neg-in 98.6%

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

      5. *-commutative 98.6%

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

      6. mul-1-neg 98.6%

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

      7. distribute-rgt-neg-out 98.6%

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

      8. remove-double-neg 98.6%

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

      9. rem-square-sqrt 0.0%

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

      10. fabs-sqr 0.0%

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

      11. rem-square-sqrt 5.9%

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

      12. associate-*r/ 5.9%

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

      13. metadata-eval 5.9%

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

   5. Simplified 5.9%

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

      1. associate-*r/ 4.8%

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

      2. frac-2neg 4.8%

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

      3. sub-neg 4.8%

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

      4. distribute-neg-frac2 4.8%

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

      5. add-sqr-sqrt 4.8%

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

      6. sqrt-unprod 4.8%

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

      7. sqr-neg 4.8%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 1.8%

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

      10. add-sqr-sqrt 3.5%

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

      11. sqrt-unprod 1.8%

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

      12. sqr-neg 1.8%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 48.7%

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

   7. Applied egg-rr 48.7%

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

      1. distribute-frac-neg 48.7%

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

      2. associate-*l/ 98.6%

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

      3. *-inverses 98.6%

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

      4. *-lft-identity 98.6%

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

   9. Simplified 98.6%

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

   if -0.80000000000000004 < x < 7.19999999999999967e-6

   1. Initial program 7.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 7.7%

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

      1. log1p-define 98.9%

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

      2. rem-square-sqrt 49.8%

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

      3. fabs-sqr 49.8%

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

      4. rem-square-sqrt 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around 0 100.0%

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

   if 7.19999999999999967e-6 < x

   1. Initial program 48.0%

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

      1. +-commutative 48.0%

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

      2. hypot-1-def 99.8%

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

      3. +-commutative 99.8%

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

      4. add-sqr-sqrt 99.8%

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

      5. fabs-sqr 99.8%

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

      6. add-sqr-sqrt 99.8%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.8:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(-x\right) - \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(-x\right) - \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(1 + \left(\frac{x}{x} + \frac{0.5}{x \cdot x}\right)\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.95)
   (copysign (log (- (- x) (+ x (/ 0.5 x)))) x)
   (if (<= x 0.95)
     (copysign (* x (+ 1.0 (* -0.16666666666666666 (pow x 2.0)))) x)
     (copysign (log (* x (+ 1.0 (+ (/ x x) (/ 0.5 (* x x)))))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.95) {
		tmp = copysign(log((-x - (x + (0.5 / x)))), x);
	} else if (x <= 0.95) {
		tmp = copysign((x * (1.0 + (-0.16666666666666666 * pow(x, 2.0)))), x);
	} else {
		tmp = copysign(log((x * (1.0 + ((x / x) + (0.5 / (x * x)))))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.95) {
		tmp = Math.copySign(Math.log((-x - (x + (0.5 / x)))), x);
	} else if (x <= 0.95) {
		tmp = Math.copySign((x * (1.0 + (-0.16666666666666666 * Math.pow(x, 2.0)))), x);
	} else {
		tmp = Math.copySign(Math.log((x * (1.0 + ((x / x) + (0.5 / (x * x)))))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.95:
		tmp = math.copysign(math.log((-x - (x + (0.5 / x)))), x)
	elif x <= 0.95:
		tmp = math.copysign((x * (1.0 + (-0.16666666666666666 * math.pow(x, 2.0)))), x)
	else:
		tmp = math.copysign(math.log((x * (1.0 + ((x / x) + (0.5 / (x * x)))))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.95)
		tmp = copysign(log(Float64(Float64(-x) - Float64(x + Float64(0.5 / x)))), x);
	elseif (x <= 0.95)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 2.0)))), x);
	else
		tmp = copysign(log(Float64(x * Float64(1.0 + Float64(Float64(x / x) + Float64(0.5 / Float64(x * x)))))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.95)
		tmp = sign(x) * abs(log((-x - (x + (0.5 / x)))));
	elseif (x <= 0.95)
		tmp = sign(x) * abs((x * (1.0 + (-0.16666666666666666 * (x ^ 2.0)))));
	else
		tmp = sign(x) * abs(log((x * (1.0 + ((x / x) + (0.5 / (x * x)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.95], N[With[{TMP1 = Abs[N[Log[N[((-x) - N[(x + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 0.95], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x * N[(1.0 + N[(N[(x / x), $MachinePrecision] + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.94999999999999996

   1. Initial program 50.1%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. distribute-rgt-in 98.6%

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

      3. *-lft-identity 98.6%

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

      4. distribute-neg-in 98.6%

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

      5. *-commutative 98.6%

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

      6. mul-1-neg 98.6%

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

      7. distribute-rgt-neg-out 98.6%

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

      8. remove-double-neg 98.6%

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

      9. rem-square-sqrt 0.0%

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

      10. fabs-sqr 0.0%

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

      11. rem-square-sqrt 5.9%

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

      12. associate-*r/ 5.9%

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

      13. metadata-eval 5.9%

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

   5. Simplified 5.9%

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

      1. associate-*r/ 4.8%

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

      2. frac-2neg 4.8%

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

      3. sub-neg 4.8%

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

      4. distribute-neg-frac2 4.8%

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

      5. add-sqr-sqrt 4.8%

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

      6. sqrt-unprod 4.8%

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

      7. sqr-neg 4.8%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 1.8%

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

      10. add-sqr-sqrt 3.5%

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

      11. sqrt-unprod 1.8%

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

      12. sqr-neg 1.8%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 48.7%

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

   7. Applied egg-rr 48.7%

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

      1. distribute-frac-neg 48.7%

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

      2. associate-*l/ 98.6%

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

      3. *-inverses 98.6%

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

      4. *-lft-identity 98.6%

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

   9. Simplified 98.6%

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

   if -0.94999999999999996 < x < 0.94999999999999996

   1. Initial program 8.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 9.2%

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

      1. +-commutative 9.2%

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

      2. fma-define 9.2%

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

      3. unpow2 9.2%

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

      4. associate-/l* 9.2%

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

      5. rem-square-sqrt 5.3%

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

      6. fabs-sqr 5.3%

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

      7. rem-square-sqrt 9.2%

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

      8. log1p-define 99.7%

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

      9. rem-square-sqrt 50.5%

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

      10. fabs-sqr 50.5%

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

      11. rem-square-sqrt 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0 99.9%

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

   if 0.94999999999999996 < x

   1. Initial program 47.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.2%

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

      1. +-commutative 99.2%

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

      2. rem-square-sqrt 99.2%

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

      3. fabs-sqr 99.2%

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

      4. rem-square-sqrt 99.2%

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

      5. unpow2 99.2%

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

   5. Simplified 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(-x\right) - \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(1 + \left(\frac{x}{x} + \frac{0.5}{x \cdot x}\right)\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(-x\right) - \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.95)
   (copysign (log (- (- x) (+ x (/ 0.5 x)))) x)
   (if (<= x 1.25)
     (copysign (* x (+ 1.0 (* -0.16666666666666666 (pow x 2.0)))) x)
     (copysign (log (/ 0.5 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.95) {
		tmp = copysign(log((-x - (x + (0.5 / x)))), x);
	} else if (x <= 1.25) {
		tmp = copysign((x * (1.0 + (-0.16666666666666666 * pow(x, 2.0)))), x);
	} else {
		tmp = copysign(log((0.5 / x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.95) {
		tmp = Math.copySign(Math.log((-x - (x + (0.5 / x)))), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign((x * (1.0 + (-0.16666666666666666 * Math.pow(x, 2.0)))), x);
	} else {
		tmp = Math.copySign(Math.log((0.5 / x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.95:
		tmp = math.copysign(math.log((-x - (x + (0.5 / x)))), x)
	elif x <= 1.25:
		tmp = math.copysign((x * (1.0 + (-0.16666666666666666 * math.pow(x, 2.0)))), x)
	else:
		tmp = math.copysign(math.log((0.5 / x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.95)
		tmp = copysign(log(Float64(Float64(-x) - Float64(x + Float64(0.5 / x)))), x);
	elseif (x <= 1.25)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 2.0)))), x);
	else
		tmp = copysign(log(Float64(0.5 / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.95)
		tmp = sign(x) * abs(log((-x - (x + (0.5 / x)))));
	elseif (x <= 1.25)
		tmp = sign(x) * abs((x * (1.0 + (-0.16666666666666666 * (x ^ 2.0)))));
	else
		tmp = sign(x) * abs(log((0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.95], N[With[{TMP1 = Abs[N[Log[N[((-x) - N[(x + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.94999999999999996

   1. Initial program 50.1%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. distribute-rgt-in 98.6%

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

      3. *-lft-identity 98.6%

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

      4. distribute-neg-in 98.6%

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

      5. *-commutative 98.6%

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

      6. mul-1-neg 98.6%

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

      7. distribute-rgt-neg-out 98.6%

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

      8. remove-double-neg 98.6%

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

      9. rem-square-sqrt 0.0%

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

      10. fabs-sqr 0.0%

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

      11. rem-square-sqrt 5.9%

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

      12. associate-*r/ 5.9%

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

      13. metadata-eval 5.9%

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

   5. Simplified 5.9%

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

      1. associate-*r/ 4.8%

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

      2. frac-2neg 4.8%

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

      3. sub-neg 4.8%

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

      4. distribute-neg-frac2 4.8%

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

      5. add-sqr-sqrt 4.8%

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

      6. sqrt-unprod 4.8%

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

      7. sqr-neg 4.8%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 1.8%

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

      10. add-sqr-sqrt 3.5%

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

      11. sqrt-unprod 1.8%

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

      12. sqr-neg 1.8%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 48.7%

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

   7. Applied egg-rr 48.7%

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

      1. distribute-frac-neg 48.7%

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

      2. associate-*l/ 98.6%

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

      3. *-inverses 98.6%

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

      4. *-lft-identity 98.6%

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

   9. Simplified 98.6%

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

   if -0.94999999999999996 < x < 1.25

   1. Initial program 8.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 9.2%

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

      1. +-commutative 9.2%

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

      2. fma-define 9.2%

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

      3. unpow2 9.2%

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

      4. associate-/l* 9.2%

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

      5. rem-square-sqrt 5.3%

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

      6. fabs-sqr 5.3%

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

      7. rem-square-sqrt 9.2%

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

      8. log1p-define 99.7%

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

      9. rem-square-sqrt 50.5%

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

      10. fabs-sqr 50.5%

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

      11. rem-square-sqrt 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0 99.9%

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

   if 1.25 < x

   1. Initial program 47.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 3.1%

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

      1. mul-1-neg 3.1%

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

      2. distribute-rgt-in 3.1%

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

      3. *-lft-identity 3.1%

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

      4. distribute-neg-in 3.1%

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

      5. *-commutative 3.1%

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

      6. mul-1-neg 3.1%

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

      7. distribute-rgt-neg-out 3.1%

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

      8. remove-double-neg 3.1%

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

      9. rem-square-sqrt 6.6%

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

      10. fabs-sqr 6.6%

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

      11. rem-square-sqrt 3.1%

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

      12. associate-*r/ 3.1%

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

      13. metadata-eval 3.1%

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

   5. Simplified 3.1%

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

      1. add-sqr-sqrt 6.6%

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

      2. sqrt-unprod 3.1%

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

      3. sqr-neg 3.1%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 0.0%

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

      6. cancel-sign-sub-inv 0.0%

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

      7. sub-neg 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 3.7%

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

      10. sqr-neg 3.7%

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

      11. sqrt-unprod 7.2%

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

      12. add-sqr-sqrt 3.7%

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

      13. div-sub 3.7%

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

      14. *-inverses 3.7%

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

      15. associate-/l/ 3.7%

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

   7. Applied egg-rr 3.7%

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

      1. +-commutative 3.7%

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

      2. neg-sub0 3.7%

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

      3. associate-+l- 3.7%

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

      4. sub-neg 3.7%

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

      5. +-commutative 3.7%

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

      6. associate--r+ 3.7%

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

      7. neg-mul-1 3.7%

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

      8. *-commutative 3.7%

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

      9. associate--r+ 3.7%

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

      10. distribute-lft-in 3.7%

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

      11. neg-sub0 3.7%

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

      12. distribute-rgt-neg-in 3.7%

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

      13. associate-+r- 46.2%

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

      14. metadata-eval 46.2%

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

      15. neg-sub0 46.2%

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

      16. remove-double-neg 46.2%

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

      17. associate-/r* 46.2%

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

      18. associate-*r/ 98.9%

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

   9. Simplified 98.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(-x\right) - \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[With[{TMP1 = Abs[N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 49.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-rgt-in 99.8%

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

      3. *-lft-identity 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. *-commutative 99.8%

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

      6. mul-1-neg 99.8%

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

      7. distribute-rgt-neg-out 99.8%

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

      8. remove-double-neg 99.8%

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

      9. rem-square-sqrt 0.0%

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

      10. fabs-sqr 0.0%

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

      11. rem-square-sqrt 5.7%

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

      12. associate-*r/ 5.7%

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

      13. metadata-eval 5.7%

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

   5. Simplified 5.7%

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

   6. Taylor expanded in x around 0 99.0%

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

   if -1.25 < x < 1.25

   1. Initial program 9.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 9.3%

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

      1. +-commutative 9.3%

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

      2. fma-define 9.3%

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

      3. unpow2 9.3%

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

      4. associate-/l* 9.3%

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

      5. rem-square-sqrt 5.2%

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

      6. fabs-sqr 5.2%

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

      7. rem-square-sqrt 9.3%

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

      8. log1p-define 99.1%

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

      9. rem-square-sqrt 50.1%

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

      10. fabs-sqr 50.1%

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

      11. rem-square-sqrt 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in x around 0 99.3%

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

   if 1.25 < x

   1. Initial program 47.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 3.1%

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

      1. mul-1-neg 3.1%

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

      2. distribute-rgt-in 3.1%

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

      3. *-lft-identity 3.1%

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

      4. distribute-neg-in 3.1%

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

      5. *-commutative 3.1%

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

      6. mul-1-neg 3.1%

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

      7. distribute-rgt-neg-out 3.1%

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

      8. remove-double-neg 3.1%

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

      9. rem-square-sqrt 6.6%

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

      10. fabs-sqr 6.6%

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

      11. rem-square-sqrt 3.1%

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

      12. associate-*r/ 3.1%

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

      13. metadata-eval 3.1%

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

   5. Simplified 3.1%

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

      1. add-sqr-sqrt 6.6%

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

      2. sqrt-unprod 3.1%

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

      3. sqr-neg 3.1%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 0.0%

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

      6. cancel-sign-sub-inv 0.0%

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

      7. sub-neg 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 3.7%

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

      10. sqr-neg 3.7%

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

      11. sqrt-unprod 7.2%

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

      12. add-sqr-sqrt 3.7%

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

      13. div-sub 3.7%

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

      14. *-inverses 3.7%

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

      15. associate-/l/ 3.7%

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

   7. Applied egg-rr 3.7%

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

      1. +-commutative 3.7%

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

      2. neg-sub0 3.7%

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

      3. associate-+l- 3.7%

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

      4. sub-neg 3.7%

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

      5. +-commutative 3.7%

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

      6. associate--r+ 3.7%

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

      7. neg-mul-1 3.7%

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

      8. *-commutative 3.7%

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

      9. associate--r+ 3.7%

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

      10. distribute-lft-in 3.7%

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

      11. neg-sub0 3.7%

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

      12. distribute-rgt-neg-in 3.7%

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

      13. associate-+r- 46.2%

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

      14. metadata-eval 46.2%

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

      15. neg-sub0 46.2%

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

      16. remove-double-neg 46.2%

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

      17. associate-/r* 46.2%

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

      18. associate-*r/ 98.9%

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

   9. Simplified 98.9%

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

Alternative 7: 99.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 49.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-rgt-in 99.8%

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

      3. *-lft-identity 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. *-commutative 99.8%

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

      6. mul-1-neg 99.8%

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

      7. distribute-rgt-neg-out 99.8%

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

      8. remove-double-neg 99.8%

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

      9. rem-square-sqrt 0.0%

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

      10. fabs-sqr 0.0%

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

      11. rem-square-sqrt 5.7%

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

      12. associate-*r/ 5.7%

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

      13. metadata-eval 5.7%

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

   5. Simplified 5.7%

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

   6. Taylor expanded in x around 0 99.0%

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

   if -1.25 < x < 1.25

   1. Initial program 9.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 9.3%

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

      1. +-commutative 9.3%

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

      2. fma-define 9.3%

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

      3. unpow2 9.3%

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

      4. associate-/l* 9.3%

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

      5. rem-square-sqrt 5.2%

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

      6. fabs-sqr 5.2%

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

      7. rem-square-sqrt 9.3%

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

      8. log1p-define 99.1%

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

      9. rem-square-sqrt 50.1%

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

      10. fabs-sqr 50.1%

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

      11. rem-square-sqrt 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in x around 0 99.3%

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

      1. distribute-rgt-in 99.3%

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

      2. *-lft-identity 99.3%

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

      3. unpow2 99.3%

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

      4. associate-*l* 99.3%

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

      5. unpow3 99.3%

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

   8. Simplified 99.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x + -0.16666666666666666 \cdot {x}^{3}}, x\right) \]
   9. Step-by-step derivation

      1. +-commutative 99.3%

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

      2. cube-mult 99.3%

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

   10. Applied egg-rr 99.3%

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

   if 1.25 < x

   1. Initial program 47.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 3.1%

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

      1. mul-1-neg 3.1%

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

      2. distribute-rgt-in 3.1%

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

      3. *-lft-identity 3.1%

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

      4. distribute-neg-in 3.1%

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

      5. *-commutative 3.1%

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

      6. mul-1-neg 3.1%

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

      7. distribute-rgt-neg-out 3.1%

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

      8. remove-double-neg 3.1%

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

      9. rem-square-sqrt 6.6%

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

      10. fabs-sqr 6.6%

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

      11. rem-square-sqrt 3.1%

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

      12. associate-*r/ 3.1%

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

      13. metadata-eval 3.1%

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

   5. Simplified 3.1%

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

      1. add-sqr-sqrt 6.6%

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

      2. sqrt-unprod 3.1%

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

      3. sqr-neg 3.1%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 0.0%

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

      6. cancel-sign-sub-inv 0.0%

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

      7. sub-neg 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 3.7%

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

      10. sqr-neg 3.7%

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

      11. sqrt-unprod 7.2%

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

      12. add-sqr-sqrt 3.7%

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

      13. div-sub 3.7%

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

      14. *-inverses 3.7%

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

      15. associate-/l/ 3.7%

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

   7. Applied egg-rr 3.7%

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

      1. +-commutative 3.7%

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

      2. neg-sub0 3.7%

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

      3. associate-+l- 3.7%

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

      4. sub-neg 3.7%

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

      5. +-commutative 3.7%

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

      6. associate--r+ 3.7%

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

      7. neg-mul-1 3.7%

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

      8. *-commutative 3.7%

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

      9. associate--r+ 3.7%

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

      10. distribute-lft-in 3.7%

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

      11. neg-sub0 3.7%

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

      12. distribute-rgt-neg-in 3.7%

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

      13. associate-+r- 46.2%

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

      14. metadata-eval 46.2%

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

      15. neg-sub0 46.2%

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

      16. remove-double-neg 46.2%

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

      17. associate-/r* 46.2%

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

      18. associate-*r/ 98.9%

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

   9. Simplified 98.9%

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

4. Final simplification 99.1%

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

Alternative 8: 81.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.33:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.33) (copysign (log (/ -0.5 x)) x) (copysign (log1p x) x)))

C

double code(double x) {
	double tmp;
	if (x <= -0.33) {
		tmp = copysign(log((-0.5 / x)), x);
	} else {
		tmp = copysign(log1p(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.33) {
		tmp = Math.copySign(Math.log((-0.5 / x)), x);
	} else {
		tmp = Math.copySign(Math.log1p(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.33:
		tmp = math.copysign(math.log((-0.5 / x)), x)
	else:
		tmp = math.copysign(math.log1p(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.33)
		tmp = copysign(log(Float64(-0.5 / x)), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.33], N[With[{TMP1 = Abs[N[Log[N[(-0.5 / 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 < -0.330000000000000016

   1. Initial program 50.1%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. distribute-rgt-in 98.6%

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

      3. *-lft-identity 98.6%

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

      4. distribute-neg-in 98.6%

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

      5. *-commutative 98.6%

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

      6. mul-1-neg 98.6%

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

      7. distribute-rgt-neg-out 98.6%

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

      8. remove-double-neg 98.6%

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

      9. rem-square-sqrt 0.0%

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

      10. fabs-sqr 0.0%

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

      11. rem-square-sqrt 5.9%

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

      12. associate-*r/ 5.9%

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

      13. metadata-eval 5.9%

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

   5. Simplified 5.9%

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

   6. Taylor expanded in x around 0 97.8%

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

   if -0.330000000000000016 < x

   1. Initial program 20.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 14.9%

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

      1. log1p-define 78.3%

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

      2. rem-square-sqrt 44.2%

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

      3. fabs-sqr 44.2%

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

      4. rem-square-sqrt 78.3%

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

   5. Simplified 78.3%

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

Alternative 9: 63.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -0.5) {
		tmp = copysign(log(-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(-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(-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(log(Float64(-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[(-x)], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.5

   1. Initial program 50.1%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 30.9%

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

   if -0.5 < x

   1. Initial program 20.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 14.9%

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

      1. log1p-define 78.3%

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

      2. rem-square-sqrt 44.2%

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

      3. fabs-sqr 44.2%

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

      4. rem-square-sqrt 78.3%

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

   5. Simplified 78.3%

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

4. Final simplification 66.1%

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

Alternative 10: 57.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 22.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 15.6%

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

      1. log1p-define 76.1%

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

      2. rem-square-sqrt 33.2%

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

      3. fabs-sqr 33.2%

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

      4. rem-square-sqrt 65.8%

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

   5. Simplified 65.8%

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

   6. Taylor expanded in x around 0 68.5%

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

   if 1.55000000000000004 < x

   1. Initial program 47.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 31.3%

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

      1. log1p-define 31.3%

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

      2. rem-square-sqrt 31.3%

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

      3. fabs-sqr 31.3%

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

      4. rem-square-sqrt 31.3%

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

   5. Simplified 31.3%

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

Alternative 11: 57.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.2) (copysign x x) (copysign (log x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 3.2) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.2) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.2:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.2)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.2)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.2], N[With[{TMP1 = Abs[x], 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 < 3.2000000000000002

   1. Initial program 22.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 15.6%

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

      1. log1p-define 76.1%

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

      2. rem-square-sqrt 33.2%

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

      3. fabs-sqr 33.2%

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

      4. rem-square-sqrt 65.8%

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

   5. Simplified 65.8%

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

   6. Taylor expanded in x around 0 68.5%

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

   if 3.2000000000000002 < x

   1. Initial program 47.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 31.3%

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

Alternative 12: 50.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.8%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 19.1%

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

   1. log1p-define 66.1%

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

   2. rem-square-sqrt 32.8%

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

   3. fabs-sqr 32.8%

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

   4. rem-square-sqrt 58.1%

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

5. Simplified 58.1%

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

6. Taylor expanded in x around 0 54.5%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]
7. 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 2024097 
(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))