Rust f64::asinh

Percentage Accurate: 29.7% → 99.7%

Time: 10.5s

Alternatives: 11

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 29.7% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{copysign}\left(x + \left(-0.16666666666666666 \cdot {x}^{3} + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -0.02)
     (copysign (log (+ (fabs x) (hypot 1.0 x))) x)
     (if (<= t_0 2e-13)
       (copysign
        (+ x (+ (* -0.16666666666666666 (pow x 3.0)) (* 0.075 (pow x 5.0))))
        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 <= -0.02) {
		tmp = copysign(log((fabs(x) + hypot(1.0, x))), x);
	} else if (t_0 <= 2e-13) {
		tmp = copysign((x + ((-0.16666666666666666 * pow(x, 3.0)) + (0.075 * pow(x, 5.0)))), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = Math.copySign(Math.log((Math.abs(x) + Math.hypot(1.0, x))), x);
	} else if (t_0 <= 2e-13) {
		tmp = Math.copySign((x + ((-0.16666666666666666 * Math.pow(x, 3.0)) + (0.075 * Math.pow(x, 5.0)))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -0.02:
		tmp = math.copysign(math.log((math.fabs(x) + math.hypot(1.0, x))), x)
	elif t_0 <= 2e-13:
		tmp = math.copysign((x + ((-0.16666666666666666 * math.pow(x, 3.0)) + (0.075 * math.pow(x, 5.0)))), x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = copysign(log(Float64(abs(x) + hypot(1.0, x))), x);
	elseif (t_0 <= 2e-13)
		tmp = copysign(Float64(x + Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + Float64(0.075 * (x ^ 5.0)))), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = sign(x) * abs(log((abs(x) + hypot(1.0, x))));
	elseif (t_0 <= 2e-13)
		tmp = sign(x) * abs((x + ((-0.16666666666666666 * (x ^ 3.0)) + (0.075 * (x ^ 5.0)))));
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 2e-13], N[With[{TMP1 = Abs[N[(x + N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.075 * N[Power[x, 5.0], $MachinePrecision]), $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) 1)))) x) < -0.0200000000000000004

   1. Initial program 51.5%

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

      1. +-commutative 51.5%

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

      2. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

   if -0.0200000000000000004 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < 2.0000000000000001e-13

   1. Initial program 7.0%

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

      1. +-commutative 7.0%

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

      2. hypot-1-def 7.0%

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

      3. add0 7.0%

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

      4. +-commutative 7.0%

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

      5. hypot-1-def 7.0%

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

      6. +-commutative 7.0%

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

      7. +-commutative 7.0%

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

      8. add-sqr-sqrt 2.7%

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

      9. fabs-sqr 2.7%

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

      10. add-sqr-sqrt 7.2%

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

      11. +-commutative 7.2%

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

      12. hypot-1-def 7.2%

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

   4. Applied egg-rr 7.2%

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

      1. add0 7.2%

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

   6. Simplified 7.2%

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

   7. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x + \left(-0.16666666666666666 \cdot {x}^{3} + 0.075 \cdot {x}^{5}\right)}, x\right) \]

   if 2.0000000000000001e-13 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x)

   1. Initial program 54.8%

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

      1. +-commutative 54.8%

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

      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. add0 100.0%

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

      4. +-commutative 100.0%

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

      5. hypot-1-def 54.8%

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

      6. +-commutative 54.8%

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

      7. +-commutative 54.8%

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

      8. add-sqr-sqrt 54.8%

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

      9. fabs-sqr 54.8%

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

      10. add-sqr-sqrt 54.8%

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

      11. +-commutative 54.8%

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

      12. hypot-1-def 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. add0 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.02:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{copysign}\left(x + \left(-0.16666666666666666 \cdot {x}^{3} + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32:\\ \;\;\;\;\mathsf{copysign}\left(\left(1 + \log \left(\frac{-0.5}{x}\right)\right) + -1, x\right)\\ \mathbf{elif}\;x \leq 0.0078:\\ \;\;\;\;\mathsf{copysign}\left(x + \left(-0.16666666666666666 \cdot {x}^{3} + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.32)
   (copysign (+ (+ 1.0 (log (/ -0.5 x))) -1.0) x)
   (if (<= x 0.0078)
     (copysign
      (+ x (+ (* -0.16666666666666666 (pow x 3.0)) (* 0.075 (pow x 5.0))))
      x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.32) {
		tmp = copysign(((1.0 + log((-0.5 / x))) + -1.0), x);
	} else if (x <= 0.0078) {
		tmp = copysign((x + ((-0.16666666666666666 * pow(x, 3.0)) + (0.075 * pow(x, 5.0)))), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.32) {
		tmp = Math.copySign(((1.0 + Math.log((-0.5 / x))) + -1.0), x);
	} else if (x <= 0.0078) {
		tmp = Math.copySign((x + ((-0.16666666666666666 * Math.pow(x, 3.0)) + (0.075 * Math.pow(x, 5.0)))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.32:
		tmp = math.copysign(((1.0 + math.log((-0.5 / x))) + -1.0), x)
	elif x <= 0.0078:
		tmp = math.copysign((x + ((-0.16666666666666666 * math.pow(x, 3.0)) + (0.075 * math.pow(x, 5.0)))), x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.32)
		tmp = copysign(Float64(Float64(1.0 + log(Float64(-0.5 / x))) + -1.0), x);
	elseif (x <= 0.0078)
		tmp = copysign(Float64(x + Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + Float64(0.075 * (x ^ 5.0)))), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.32)
		tmp = sign(x) * abs(((1.0 + log((-0.5 / x))) + -1.0));
	elseif (x <= 0.0078)
		tmp = sign(x) * abs((x + ((-0.16666666666666666 * (x ^ 3.0)) + (0.075 * (x ^ 5.0)))));
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.32], N[With[{TMP1 = Abs[N[(N[(1.0 + N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 0.0078], N[With[{TMP1 = Abs[N[(x + N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.075 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.32000000000000006

   1. Initial program 50.7%

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

      1. +-commutative 50.7%

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

      2. hypot-1-def 100.0%

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

      3. add0 100.0%

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

      4. +-commutative 100.0%

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

      5. hypot-1-def 50.7%

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

      6. +-commutative 50.7%

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

      7. +-commutative 50.7%

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

      8. add-sqr-sqrt 0.0%

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

      9. fabs-sqr 0.0%

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

      10. add-sqr-sqrt 4.9%

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

      11. +-commutative 4.9%

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

      12. hypot-1-def 4.9%

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

   4. Applied egg-rr 4.9%

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

      1. add0 4.9%

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

   6. Simplified 4.9%

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

      1. expm1-log1p-u 0.0%

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

      2. log1p-define 0.0%

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

      3. expm1-undefine 0.0%

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

      4. add-exp-log 4.9%

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

   8. Applied egg-rr 4.9%

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

   9. Taylor expanded in x around -inf 98.7%

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

   if -1.32000000000000006 < x < 0.0077999999999999996

   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. Step-by-step derivation

      1. +-commutative 7.7%

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

      2. hypot-1-def 7.7%

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

      3. add0 7.7%

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

      4. +-commutative 7.7%

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

      5. hypot-1-def 7.7%

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

      6. +-commutative 7.7%

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

      7. +-commutative 7.7%

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

      8. add-sqr-sqrt 2.6%

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

      9. fabs-sqr 2.6%

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

      10. add-sqr-sqrt 7.8%

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

      11. +-commutative 7.8%

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

      12. hypot-1-def 7.9%

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

   4. Applied egg-rr 7.9%

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

      1. add0 7.9%

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

   6. Simplified 7.9%

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

   7. Taylor expanded in x around 0 99.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x + \left(-0.16666666666666666 \cdot {x}^{3} + 0.075 \cdot {x}^{5}\right)}, x\right) \]

   if 0.0077999999999999996 < x

   1. Initial program 54.8%

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

      1. +-commutative 54.8%

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

      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. add0 100.0%

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

      4. +-commutative 100.0%

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

      5. hypot-1-def 54.8%

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

      6. +-commutative 54.8%

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

      7. +-commutative 54.8%

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

      8. add-sqr-sqrt 54.8%

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

      9. fabs-sqr 54.8%

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

      10. add-sqr-sqrt 54.8%

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

      11. +-commutative 54.8%

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

      12. hypot-1-def 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. add0 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32:\\ \;\;\;\;\mathsf{copysign}\left(\left(1 + \log \left(\frac{-0.5}{x}\right)\right) + -1, x\right)\\ \mathbf{elif}\;x \leq 0.0078:\\ \;\;\;\;\mathsf{copysign}\left(x + \left(-0.16666666666666666 \cdot {x}^{3} + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\left(1 + \log \left(\frac{-0.5}{x}\right)\right) + -1, x\right)\\ \mathbf{elif}\;x \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (+ (+ 1.0 (log (/ -0.5 x))) -1.0) x)
   (if (<= x 0.001)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = copysign(((1.0 + log((-0.5 / x))) + -1.0), x);
	} else if (x <= 0.001) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.copySign(((1.0 + Math.log((-0.5 / x))) + -1.0), x);
	} else if (x <= 0.001) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.copysign(((1.0 + math.log((-0.5 / x))) + -1.0), x)
	elif x <= 0.001:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = copysign(Float64(Float64(1.0 + log(Float64(-0.5 / x))) + -1.0), x);
	elseif (x <= 0.001)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = sign(x) * abs(((1.0 + log((-0.5 / x))) + -1.0));
	elseif (x <= 0.001)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[With[{TMP1 = Abs[N[(N[(1.0 + N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 0.001], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 50.7%

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

      1. +-commutative 50.7%

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

      2. hypot-1-def 100.0%

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

      3. add0 100.0%

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

      4. +-commutative 100.0%

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

      5. hypot-1-def 50.7%

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

      6. +-commutative 50.7%

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

      7. +-commutative 50.7%

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

      8. add-sqr-sqrt 0.0%

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

      9. fabs-sqr 0.0%

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

      10. add-sqr-sqrt 4.9%

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

      11. +-commutative 4.9%

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

      12. hypot-1-def 4.9%

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

   4. Applied egg-rr 4.9%

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

      1. add0 4.9%

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

   6. Simplified 4.9%

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

      1. expm1-log1p-u 0.0%

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

      2. log1p-define 0.0%

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

      3. expm1-undefine 0.0%

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

      4. add-exp-log 4.9%

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

   8. Applied egg-rr 4.9%

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

   9. Taylor expanded in x around -inf 98.7%

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

   if -1.25 < x < 1e-3

   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. Step-by-step derivation

      1. add-cbrt-cube 7.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 7.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 7.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 2.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 2.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 7.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   4. Applied egg-rr 7.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   5. Taylor expanded in x around 0 99.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(-0.5 \cdot {x}^{3} + 3 \cdot x\right)}, x\right) \]
   6. Step-by-step derivation

      1. distribute-rgt-in 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(-0.5 \cdot {x}^{3}\right) \cdot 0.3333333333333333 + \left(3 \cdot x\right) \cdot 0.3333333333333333}, x\right) \]

      2. *-commutative 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left({x}^{3} \cdot -0.5\right)} \cdot 0.3333333333333333 + \left(3 \cdot x\right) \cdot 0.3333333333333333, x\right) \]

      3. associate-*l* 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{3} \cdot \left(-0.5 \cdot 0.3333333333333333\right)} + \left(3 \cdot x\right) \cdot 0.3333333333333333, x\right) \]

      4. metadata-eval 99.0%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot \color{blue}{-0.16666666666666666} + \left(3 \cdot x\right) \cdot 0.3333333333333333, x\right) \]

      5. *-commutative 99.0%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot -0.16666666666666666 + \color{blue}{0.3333333333333333 \cdot \left(3 \cdot x\right)}, x\right) \]

      6. associate-*r* 99.5%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot -0.16666666666666666 + \color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot x}, x\right) \]

      7. metadata-eval 99.5%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot -0.16666666666666666 + \color{blue}{1} \cdot x, x\right) \]

      8. *-un-lft-identity 99.5%

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

   7. Applied egg-rr 99.5%

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

   if 1e-3 < x

   1. Initial program 54.8%

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

      1. +-commutative 54.8%

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

      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. add0 100.0%

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

      4. +-commutative 100.0%

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

      5. hypot-1-def 54.8%

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

      6. +-commutative 54.8%

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

      7. +-commutative 54.8%

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

      8. add-sqr-sqrt 54.8%

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

      9. fabs-sqr 54.8%

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

      10. add-sqr-sqrt 54.8%

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

      11. +-commutative 54.8%

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

      12. hypot-1-def 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. add0 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\left(1 + \log \left(\frac{-0.5}{x}\right)\right) + -1, x\right)\\ \mathbf{elif}\;x \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\left(1 + \log \left(\frac{-0.5}{x}\right)\right) + -1, x\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\left(1 + \log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\right) + -1, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (+ (+ 1.0 (log (/ -0.5 x))) -1.0) x)
   (if (<= x 0.96)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (+ (+ 1.0 (log (+ (* x 2.0) (* 0.5 (/ 1.0 x))))) -1.0) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = copysign(((1.0 + log((-0.5 / x))) + -1.0), x);
	} else if (x <= 0.96) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(((1.0 + log(((x * 2.0) + (0.5 * (1.0 / x))))) + -1.0), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.copySign(((1.0 + Math.log((-0.5 / x))) + -1.0), x);
	} else if (x <= 0.96) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(((1.0 + Math.log(((x * 2.0) + (0.5 * (1.0 / x))))) + -1.0), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.copysign(((1.0 + math.log((-0.5 / x))) + -1.0), x)
	elif x <= 0.96:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(((1.0 + math.log(((x * 2.0) + (0.5 * (1.0 / x))))) + -1.0), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = copysign(Float64(Float64(1.0 + log(Float64(-0.5 / x))) + -1.0), x);
	elseif (x <= 0.96)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(Float64(Float64(1.0 + log(Float64(Float64(x * 2.0) + Float64(0.5 * Float64(1.0 / x))))) + -1.0), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = sign(x) * abs(((1.0 + log((-0.5 / x))) + -1.0));
	elseif (x <= 0.96)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(((1.0 + log(((x * 2.0) + (0.5 * (1.0 / x))))) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[With[{TMP1 = Abs[N[(N[(1.0 + N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 0.96], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[(N[(1.0 + N[Log[N[(N[(x * 2.0), $MachinePrecision] + N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $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 50.7%

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

      1. +-commutative 50.7%

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

      2. hypot-1-def 100.0%

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

      3. add0 100.0%

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

      4. +-commutative 100.0%

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

      5. hypot-1-def 50.7%

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

      6. +-commutative 50.7%

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

      7. +-commutative 50.7%

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

      8. add-sqr-sqrt 0.0%

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

      9. fabs-sqr 0.0%

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

      10. add-sqr-sqrt 4.9%

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

      11. +-commutative 4.9%

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

      12. hypot-1-def 4.9%

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

   4. Applied egg-rr 4.9%

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

      1. add0 4.9%

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

   6. Simplified 4.9%

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

      1. expm1-log1p-u 0.0%

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

      2. log1p-define 0.0%

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

      3. expm1-undefine 0.0%

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

      4. add-exp-log 4.9%

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

   8. Applied egg-rr 4.9%

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

   9. Taylor expanded in x around -inf 98.7%

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

   if -1.25 < x < 0.95999999999999996

   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. Step-by-step derivation

      1. add-cbrt-cube 7.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 7.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 7.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 2.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 2.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 7.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   4. Applied egg-rr 7.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   5. Taylor expanded in x around 0 99.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(-0.5 \cdot {x}^{3} + 3 \cdot x\right)}, x\right) \]
   6. Step-by-step derivation

      1. distribute-rgt-in 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(-0.5 \cdot {x}^{3}\right) \cdot 0.3333333333333333 + \left(3 \cdot x\right) \cdot 0.3333333333333333}, x\right) \]

      2. *-commutative 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left({x}^{3} \cdot -0.5\right)} \cdot 0.3333333333333333 + \left(3 \cdot x\right) \cdot 0.3333333333333333, x\right) \]

      3. associate-*l* 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{3} \cdot \left(-0.5 \cdot 0.3333333333333333\right)} + \left(3 \cdot x\right) \cdot 0.3333333333333333, x\right) \]

      4. metadata-eval 99.0%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot \color{blue}{-0.16666666666666666} + \left(3 \cdot x\right) \cdot 0.3333333333333333, x\right) \]

      5. *-commutative 99.0%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot -0.16666666666666666 + \color{blue}{0.3333333333333333 \cdot \left(3 \cdot x\right)}, x\right) \]

      6. associate-*r* 99.5%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot -0.16666666666666666 + \color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot x}, x\right) \]

      7. metadata-eval 99.5%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot -0.16666666666666666 + \color{blue}{1} \cdot x, x\right) \]

      8. *-un-lft-identity 99.5%

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

   7. Applied egg-rr 99.5%

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

   if 0.95999999999999996 < x

   1. Initial program 54.8%

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

      1. +-commutative 54.8%

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

      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. add0 100.0%

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

      4. +-commutative 100.0%

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

      5. hypot-1-def 54.8%

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

      6. +-commutative 54.8%

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

      7. +-commutative 54.8%

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

      8. add-sqr-sqrt 54.8%

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

      9. fabs-sqr 54.8%

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

      10. add-sqr-sqrt 54.8%

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

      11. +-commutative 54.8%

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

      12. hypot-1-def 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. add0 100.0%

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

   6. Simplified 100.0%

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

      1. expm1-log1p-u 98.2%

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

      2. log1p-define 98.2%

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

      3. expm1-undefine 98.2%

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

      4. add-exp-log 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around inf 99.2%

      \[\leadsto \mathsf{copysign}\left(\left(1 + \log \color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)}\right) - 1, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\left(1 + \log \left(\frac{-0.5}{x}\right)\right) + -1, x\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\left(1 + \log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\right) + -1, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{e + x \cdot e}{e}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.0)
   (copysign (- (log (/ -1.0 x))) x)
   (if (<= x 1.55)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (/ (+ E (* x E)) E)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = copysign(-log((-1.0 / x)), x);
	} else if (x <= 1.55) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log(((((double) M_E) + (x * ((double) M_E))) / ((double) M_E))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = Math.copySign(-Math.log((-1.0 / x)), x);
	} else if (x <= 1.55) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(Math.log(((Math.E + (x * Math.E)) / Math.E)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.0:
		tmp = math.copysign(-math.log((-1.0 / x)), x)
	elif x <= 1.55:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(math.log(((math.e + (x * math.e)) / math.e)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.0)
		tmp = copysign(Float64(-log(Float64(-1.0 / x))), x);
	elseif (x <= 1.55)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(log(Float64(Float64(exp(1) + Float64(x * exp(1))) / exp(1))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.0)
		tmp = sign(x) * abs(-log((-1.0 / x)));
	elseif (x <= 1.55)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(log(((2.71828182845904523536 + (x * 2.71828182845904523536)) / 2.71828182845904523536)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.0], N[With[{TMP1 = Abs[(-N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.55], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(E + N[(x * E), $MachinePrecision]), $MachinePrecision] / E), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 50.7%

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

   3. Taylor expanded in x around -inf 31.4%

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

      1. mul-1-neg 31.4%

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

   5. Simplified 31.4%

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

   if -2 < x < 1.55000000000000004

   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. Step-by-step derivation

      1. add-cbrt-cube 7.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 7.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 7.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 2.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 2.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 7.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   4. Applied egg-rr 7.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   5. Taylor expanded in x around 0 99.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(-0.5 \cdot {x}^{3} + 3 \cdot x\right)}, x\right) \]
   6. Step-by-step derivation

      1. distribute-rgt-in 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(-0.5 \cdot {x}^{3}\right) \cdot 0.3333333333333333 + \left(3 \cdot x\right) \cdot 0.3333333333333333}, x\right) \]

      2. *-commutative 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left({x}^{3} \cdot -0.5\right)} \cdot 0.3333333333333333 + \left(3 \cdot x\right) \cdot 0.3333333333333333, x\right) \]

      3. associate-*l* 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{3} \cdot \left(-0.5 \cdot 0.3333333333333333\right)} + \left(3 \cdot x\right) \cdot 0.3333333333333333, x\right) \]

      4. metadata-eval 99.0%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot \color{blue}{-0.16666666666666666} + \left(3 \cdot x\right) \cdot 0.3333333333333333, x\right) \]

      5. *-commutative 99.0%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot -0.16666666666666666 + \color{blue}{0.3333333333333333 \cdot \left(3 \cdot x\right)}, x\right) \]

      6. associate-*r* 99.5%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot -0.16666666666666666 + \color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot x}, x\right) \]

      7. metadata-eval 99.5%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot -0.16666666666666666 + \color{blue}{1} \cdot x, x\right) \]

      8. *-un-lft-identity 99.5%

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

   7. Applied egg-rr 99.5%

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

   if 1.55000000000000004 < x

   1. Initial program 54.8%

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

      1. add-exp-log 54.8%

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

      2. *-un-lft-identity 54.8%

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

      3. exp-prod 54.8%

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

      4. expm1-log1p-u 53.9%

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

      5. expm1-undefine 53.9%

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

      6. pow-sub 53.9%

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

   4. Applied egg-rr 98.2%

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

      1. exp-1-e 98.2%

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

      2. log1p-undefine 98.2%

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

      3. rem-exp-log 100.0%

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

      4. unpow1 100.0%

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

      5. exp-1-e 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 31.1%

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

      1. log-E 31.1%

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

      2. *-rgt-identity 31.1%

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

   9. Simplified 31.1%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{e + x \cdot e}{e}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\left(1 + \log \left(x \cdot 2\right)\right) + -1, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.0)
   (copysign (- (log (/ -1.0 x))) x)
   (if (<= x 1.25)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (+ (+ 1.0 (log (* x 2.0))) -1.0) x))))

C

double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = copysign(-log((-1.0 / x)), x);
	} else if (x <= 1.25) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(((1.0 + log((x * 2.0))) + -1.0), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = Math.copySign(-Math.log((-1.0 / x)), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(((1.0 + Math.log((x * 2.0))) + -1.0), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.0:
		tmp = math.copysign(-math.log((-1.0 / x)), x)
	elif x <= 1.25:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(((1.0 + math.log((x * 2.0))) + -1.0), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.0)
		tmp = copysign(Float64(-log(Float64(-1.0 / x))), x);
	elseif (x <= 1.25)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(Float64(Float64(1.0 + log(Float64(x * 2.0))) + -1.0), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.0)
		tmp = sign(x) * abs(-log((-1.0 / x)));
	elseif (x <= 1.25)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(((1.0 + log((x * 2.0))) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.0], N[With[{TMP1 = Abs[(-N[Log[N[(-1.0 / 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[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[(N[(1.0 + N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 50.7%

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

   3. Taylor expanded in x around -inf 31.4%

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

      1. mul-1-neg 31.4%

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

   5. Simplified 31.4%

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

   if -2 < x < 1.25

   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. Step-by-step derivation

      1. add-cbrt-cube 7.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 7.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 7.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 2.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 2.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 7.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   4. Applied egg-rr 7.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   5. Taylor expanded in x around 0 99.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(-0.5 \cdot {x}^{3} + 3 \cdot x\right)}, x\right) \]
   6. Step-by-step derivation

      1. distribute-rgt-in 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(-0.5 \cdot {x}^{3}\right) \cdot 0.3333333333333333 + \left(3 \cdot x\right) \cdot 0.3333333333333333}, x\right) \]

      2. *-commutative 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left({x}^{3} \cdot -0.5\right)} \cdot 0.3333333333333333 + \left(3 \cdot x\right) \cdot 0.3333333333333333, x\right) \]

      3. associate-*l* 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{3} \cdot \left(-0.5 \cdot 0.3333333333333333\right)} + \left(3 \cdot x\right) \cdot 0.3333333333333333, x\right) \]

      4. metadata-eval 99.0%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot \color{blue}{-0.16666666666666666} + \left(3 \cdot x\right) \cdot 0.3333333333333333, x\right) \]

      5. *-commutative 99.0%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot -0.16666666666666666 + \color{blue}{0.3333333333333333 \cdot \left(3 \cdot x\right)}, x\right) \]

      6. associate-*r* 99.5%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot -0.16666666666666666 + \color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot x}, x\right) \]

      7. metadata-eval 99.5%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot -0.16666666666666666 + \color{blue}{1} \cdot x, x\right) \]

      8. *-un-lft-identity 99.5%

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

   7. Applied egg-rr 99.5%

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

   if 1.25 < x

   1. Initial program 54.8%

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

      1. +-commutative 54.8%

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

      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. add0 100.0%

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

      4. +-commutative 100.0%

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

      5. hypot-1-def 54.8%

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

      6. +-commutative 54.8%

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

      7. +-commutative 54.8%

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

      8. add-sqr-sqrt 54.8%

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

      9. fabs-sqr 54.8%

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

      10. add-sqr-sqrt 54.8%

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

      11. +-commutative 54.8%

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

      12. hypot-1-def 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. add0 100.0%

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

   6. Simplified 100.0%

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

      1. expm1-log1p-u 98.2%

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

      2. log1p-define 98.2%

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

      3. expm1-undefine 98.2%

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

      4. add-exp-log 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around inf 98.7%

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

       1. *-commutative 98.7%

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

   11. Simplified 98.7%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\left(1 + \log \left(x \cdot 2\right)\right) + -1, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\left(1 + \log \left(\frac{-0.5}{x}\right)\right) + -1, x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\left(1 + \log \left(x \cdot 2\right)\right) + -1, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (+ (+ 1.0 (log (/ -0.5 x))) -1.0) x)
   (if (<= x 1.25)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (+ (+ 1.0 (log (* x 2.0))) -1.0) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = copysign(((1.0 + log((-0.5 / x))) + -1.0), x);
	} else if (x <= 1.25) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(((1.0 + log((x * 2.0))) + -1.0), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.copySign(((1.0 + Math.log((-0.5 / x))) + -1.0), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(((1.0 + Math.log((x * 2.0))) + -1.0), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.copysign(((1.0 + math.log((-0.5 / x))) + -1.0), x)
	elif x <= 1.25:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(((1.0 + math.log((x * 2.0))) + -1.0), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = copysign(Float64(Float64(1.0 + log(Float64(-0.5 / x))) + -1.0), x);
	elseif (x <= 1.25)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(Float64(Float64(1.0 + log(Float64(x * 2.0))) + -1.0), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = sign(x) * abs(((1.0 + log((-0.5 / x))) + -1.0));
	elseif (x <= 1.25)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(((1.0 + log((x * 2.0))) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[With[{TMP1 = Abs[N[(N[(1.0 + N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $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[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[(N[(1.0 + N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $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 50.7%

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

      1. +-commutative 50.7%

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

      2. hypot-1-def 100.0%

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

      3. add0 100.0%

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

      4. +-commutative 100.0%

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

      5. hypot-1-def 50.7%

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

      6. +-commutative 50.7%

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

      7. +-commutative 50.7%

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

      8. add-sqr-sqrt 0.0%

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

      9. fabs-sqr 0.0%

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

      10. add-sqr-sqrt 4.9%

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

      11. +-commutative 4.9%

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

      12. hypot-1-def 4.9%

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

   4. Applied egg-rr 4.9%

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

      1. add0 4.9%

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

   6. Simplified 4.9%

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

      1. expm1-log1p-u 0.0%

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

      2. log1p-define 0.0%

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

      3. expm1-undefine 0.0%

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

      4. add-exp-log 4.9%

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

   8. Applied egg-rr 4.9%

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

   9. Taylor expanded in x around -inf 98.7%

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

   if -1.25 < x < 1.25

   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. Step-by-step derivation

      1. add-cbrt-cube 7.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 7.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 7.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 2.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 2.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 7.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   4. Applied egg-rr 7.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   5. Taylor expanded in x around 0 99.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(-0.5 \cdot {x}^{3} + 3 \cdot x\right)}, x\right) \]
   6. Step-by-step derivation

      1. distribute-rgt-in 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(-0.5 \cdot {x}^{3}\right) \cdot 0.3333333333333333 + \left(3 \cdot x\right) \cdot 0.3333333333333333}, x\right) \]

      2. *-commutative 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left({x}^{3} \cdot -0.5\right)} \cdot 0.3333333333333333 + \left(3 \cdot x\right) \cdot 0.3333333333333333, x\right) \]

      3. associate-*l* 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{3} \cdot \left(-0.5 \cdot 0.3333333333333333\right)} + \left(3 \cdot x\right) \cdot 0.3333333333333333, x\right) \]

      4. metadata-eval 99.0%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot \color{blue}{-0.16666666666666666} + \left(3 \cdot x\right) \cdot 0.3333333333333333, x\right) \]

      5. *-commutative 99.0%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot -0.16666666666666666 + \color{blue}{0.3333333333333333 \cdot \left(3 \cdot x\right)}, x\right) \]

      6. associate-*r* 99.5%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot -0.16666666666666666 + \color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot x}, x\right) \]

      7. metadata-eval 99.5%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot -0.16666666666666666 + \color{blue}{1} \cdot x, x\right) \]

      8. *-un-lft-identity 99.5%

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

   7. Applied egg-rr 99.5%

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

   if 1.25 < x

   1. Initial program 54.8%

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

      1. +-commutative 54.8%

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

      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. add0 100.0%

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

      4. +-commutative 100.0%

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

      5. hypot-1-def 54.8%

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

      6. +-commutative 54.8%

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

      7. +-commutative 54.8%

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

      8. add-sqr-sqrt 54.8%

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

      9. fabs-sqr 54.8%

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

      10. add-sqr-sqrt 54.8%

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

      11. +-commutative 54.8%

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

      12. hypot-1-def 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. add0 100.0%

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

   6. Simplified 100.0%

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

      1. expm1-log1p-u 98.2%

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

      2. log1p-define 98.2%

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

      3. expm1-undefine 98.2%

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

      4. add-exp-log 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around inf 98.7%

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

       1. *-commutative 98.7%

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

   11. Simplified 98.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\left(1 + \log \left(\frac{-0.5}{x}\right)\right) + -1, x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\left(1 + \log \left(x \cdot 2\right)\right) + -1, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.0)
   (copysign (- (log (/ -1.0 x))) x)
   (if (<= x 2.0)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log x) x))))

C

double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = copysign(-log((-1.0 / x)), x);
	} else if (x <= 2.0) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = Math.copySign(-Math.log((-1.0 / x)), x);
	} else if (x <= 2.0) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(Math.log(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.0:
		tmp = math.copysign(-math.log((-1.0 / x)), x)
	elif x <= 2.0:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(math.log(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.0)
		tmp = copysign(Float64(-log(Float64(-1.0 / x))), x);
	elseif (x <= 2.0)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.0)
		tmp = sign(x) * abs(-log((-1.0 / x)));
	elseif (x <= 2.0)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.0], N[With[{TMP1 = Abs[(-N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 2.0], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 50.7%

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

   3. Taylor expanded in x around -inf 31.4%

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

      1. mul-1-neg 31.4%

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

   5. Simplified 31.4%

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

   if -2 < x < 2

   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. Step-by-step derivation

      1. add-cbrt-cube 7.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 7.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 7.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 2.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 2.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 7.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   4. Applied egg-rr 7.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   5. Taylor expanded in x around 0 99.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(-0.5 \cdot {x}^{3} + 3 \cdot x\right)}, x\right) \]
   6. Step-by-step derivation

      1. distribute-rgt-in 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(-0.5 \cdot {x}^{3}\right) \cdot 0.3333333333333333 + \left(3 \cdot x\right) \cdot 0.3333333333333333}, x\right) \]

      2. *-commutative 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left({x}^{3} \cdot -0.5\right)} \cdot 0.3333333333333333 + \left(3 \cdot x\right) \cdot 0.3333333333333333, x\right) \]

      3. associate-*l* 99.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{3} \cdot \left(-0.5 \cdot 0.3333333333333333\right)} + \left(3 \cdot x\right) \cdot 0.3333333333333333, x\right) \]

      4. metadata-eval 99.0%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot \color{blue}{-0.16666666666666666} + \left(3 \cdot x\right) \cdot 0.3333333333333333, x\right) \]

      5. *-commutative 99.0%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot -0.16666666666666666 + \color{blue}{0.3333333333333333 \cdot \left(3 \cdot x\right)}, x\right) \]

      6. associate-*r* 99.5%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot -0.16666666666666666 + \color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot x}, x\right) \]

      7. metadata-eval 99.5%

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot -0.16666666666666666 + \color{blue}{1} \cdot x, x\right) \]

      8. *-un-lft-identity 99.5%

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

   7. Applied egg-rr 99.5%

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

   if 2 < x

   1. Initial program 54.8%

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

   3. Taylor expanded in x around inf 31.1%

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

      1. mul-1-neg 31.1%

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

      2. log-rec 31.1%

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

      3. remove-double-neg 31.1%

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

   5. Simplified 31.1%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 3.15:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.2)
   (copysign (- (log (/ -1.0 x))) x)
   (if (<= x 3.15) (copysign x x) (copysign (log x) x))))

C

double code(double x) {
	double tmp;
	if (x <= -3.2) {
		tmp = copysign(-log((-1.0 / x)), x);
	} else if (x <= 3.15) {
		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(-Math.log((-1.0 / x)), x);
	} else if (x <= 3.15) {
		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(-math.log((-1.0 / x)), x)
	elif x <= 3.15:
		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(Float64(-log(Float64(-1.0 / x))), x);
	elseif (x <= 3.15)
		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(-log((-1.0 / x)));
	elseif (x <= 3.15)
		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[(-N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 3.15], 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 3 regimes

2. if x < -3.2000000000000002

   1. Initial program 50.7%

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

   3. Taylor expanded in x around -inf 31.4%

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

      1. mul-1-neg 31.4%

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

   5. Simplified 31.4%

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

   if -3.2000000000000002 < x < 3.14999999999999991

   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. Step-by-step derivation

      1. add-cbrt-cube 7.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 7.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 7.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 7.7%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 2.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 2.6%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 7.9%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   4. Applied egg-rr 7.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   5. Taylor expanded in x around 0 98.3%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot x\right)}, x\right) \]
   6. Step-by-step derivation

      1. *-commutative 98.3%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   7. Simplified 98.3%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   8. Taylor expanded in x around 0 98.8%

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

   if 3.14999999999999991 < x

   1. Initial program 54.8%

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

   3. Taylor expanded in x around inf 31.1%

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

      1. mul-1-neg 31.1%

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

      2. log-rec 31.1%

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

      3. remove-double-neg 31.1%

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

   5. Simplified 31.1%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 3.15:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.15:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.15) (copysign x x) (copysign (log x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 3.15) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.15) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.15:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.15)
		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 <= 3.15)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.15], 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 < 3.14999999999999991

   1. Initial program 20.3%

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

      1. add-cbrt-cube 15.5%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

      2. pow1/3 15.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

      3. log-pow 15.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      4. pow3 15.4%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

      5. log-pow 20.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      6. +-commutative 20.2%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

      7. hypot-1-def 34.5%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

      8. add-sqr-sqrt 1.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      9. fabs-sqr 1.9%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

      10. add-sqr-sqrt 7.0%

          \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   4. Applied egg-rr 7.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   5. Taylor expanded in x around 0 71.1%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot x\right)}, x\right) \]
   6. Step-by-step derivation

      1. *-commutative 71.1%

         \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   7. Simplified 71.1%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

   8. Taylor expanded in x around 0 71.5%

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

   if 3.14999999999999991 < x

   1. Initial program 54.8%

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

   3. Taylor expanded in x around inf 31.1%

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

      1. mul-1-neg 31.1%

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

      2. log-rec 31.1%

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

      3. remove-double-neg 31.1%

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

   5. Simplified 31.1%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.15:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

   1. add-cbrt-cube 23.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\sqrt[3]{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]

   2. pow1/3 23.1%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left({\left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}^{0.3333333333333333}\right)}, x\right) \]

   3. log-pow 23.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right) \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

   4. pow3 23.1%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}^{3}\right)}, x\right) \]

   5. log-pow 30.3%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

   6. +-commutative 30.3%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

   7. hypot-1-def 53.6%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)\right), x\right) \]

   8. add-sqr-sqrt 30.5%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   9. fabs-sqr 30.5%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

   10. add-sqr-sqrt 34.1%

       \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]

4. Applied egg-rr 34.1%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

5. Taylor expanded in x around 0 52.0%

   \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(3 \cdot x\right)}, x\right) \]
6. Step-by-step derivation

   1. *-commutative 52.0%

      \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

7. Simplified 52.0%

   \[\leadsto \mathsf{copysign}\left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)}, x\right) \]

8. Taylor expanded in x around 0 52.2%

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

9. Final simplification 52.2%

   \[\leadsto \mathsf{copysign}\left(x, x\right) \]
10. 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 2024046 
(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))