Rust f64::asinh

Percentage Accurate: 30.1% → 99.3%

Time: 5.8s

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: 30.1% 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: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (log (- (fabs x) x)) x)
   (if (<= x 1.25)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log1p (+ -1.0 (* x 2.0))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (x <= 1.25) {
		tmp = copysign((x + (pow(x, 3.0) * -0.16666666666666666)), x);
	} else {
		tmp = copysign(log1p((-1.0 + (x * 2.0))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.copySign(Math.log((Math.abs(x) - x)), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign((x + (Math.pow(x, 3.0) * -0.16666666666666666)), x);
	} else {
		tmp = Math.copySign(Math.log1p((-1.0 + (x * 2.0))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.copysign(math.log((math.fabs(x) - x)), x)
	elif x <= 1.25:
		tmp = math.copysign((x + (math.pow(x, 3.0) * -0.16666666666666666)), x)
	else:
		tmp = math.copysign(math.log1p((-1.0 + (x * 2.0))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (x <= 1.25)
		tmp = copysign(Float64(x + Float64((x ^ 3.0) * -0.16666666666666666)), x);
	else
		tmp = copysign(log1p(Float64(-1.0 + Float64(x * 2.0))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + N[(-1.0 + N[(x * 2.0), $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 50.9%

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

   2. Taylor expanded in x around -inf 98.9%

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

   if -1.25 < x < 1.25

   1. Initial program 7.2%

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

      1. log1p-expm1-u 7.2%

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

      2. expm1-udef 7.2%

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

      3. add-exp-log 7.2%

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

      4. +-commutative 7.2%

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

      5. metadata-eval 7.2%

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

      6. hypot-udef 7.2%

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

      7. associate--l+ 98.8%

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

      8. add-sqr-sqrt 52.7%

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

      9. fabs-sqr 52.7%

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

      10. add-sqr-sqrt 98.8%

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

   3. Applied egg-rr 98.8%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   if 1.25 < x

   1. Initial program 55.7%

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

      1. log1p-expm1-u 55.7%

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

      2. expm1-udef 55.7%

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

      3. add-exp-log 55.7%

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

      4. +-commutative 55.7%

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

      5. metadata-eval 55.7%

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

      6. hypot-udef 100.0%

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

      7. associate--l+ 100.0%

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

      8. add-sqr-sqrt 100.0%

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

      9. fabs-sqr 100.0%

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

      10. add-sqr-sqrt 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{2 \cdot x - 1}\right), x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + x \cdot 2\right), x\right)\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(-1 - x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) -5.0)
   (copysign (log1p (+ (hypot 1.0 x) (- -1.0 x))) x)
   (copysign (log1p (+ x (+ (hypot 1.0 x) -1.0))) x)))

C

double code(double x) {
	double tmp;
	if (copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x) <= -5.0) {
		tmp = copysign(log1p((hypot(1.0, x) + (-1.0 - x))), x);
	} else {
		tmp = copysign(log1p((x + (hypot(1.0, x) + -1.0))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x) <= -5.0) {
		tmp = Math.copySign(Math.log1p((Math.hypot(1.0, x) + (-1.0 - x))), x);
	} else {
		tmp = Math.copySign(Math.log1p((x + (Math.hypot(1.0, x) + -1.0))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x) <= -5.0:
		tmp = math.copysign(math.log1p((math.hypot(1.0, x) + (-1.0 - x))), x)
	else:
		tmp = math.copysign(math.log1p((x + (math.hypot(1.0, x) + -1.0))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x) <= -5.0)
		tmp = copysign(log1p(Float64(hypot(1.0, x) + Float64(-1.0 - x))), x);
	else
		tmp = copysign(log1p(Float64(x + Float64(hypot(1.0, x) + -1.0))), x);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 50.9%

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

      1. log1p-expm1-u 50.9%

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

      2. expm1-udef 50.9%

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

      3. add-exp-log 50.9%

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

      4. +-commutative 50.9%

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

      5. metadata-eval 50.9%

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

      6. hypot-udef 100.0%

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

      7. associate--l+ 100.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. fabs-sqr 0.0%

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

      10. add-sqr-sqrt 5.1%

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

   3. Applied egg-rr 5.1%

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

      1. +-commutative 5.1%

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

      2. associate-+l- 5.1%

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

      3. metadata-eval 5.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) - \left(\color{blue}{\left(1 + 0\right)} - x\right)\right), x\right) \]

      4. associate-+r- 5.1%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) - \color{blue}{\left(1 + \left(0 - x\right)\right)}\right), x\right) \]

      5. neg-sub0 5.1%

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

      6. mul-1-neg 5.1%

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

      7. add-sqr-sqrt 7.1%

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

      8. fabs-sqr 7.1%

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

      9. add-sqr-sqrt 5.1%

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

      10. mul-1-neg 5.1%

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

      11. neg-fabs 5.1%

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

      12. add-sqr-sqrt 0.0%

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

      13. fabs-sqr 0.0%

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

      14. add-sqr-sqrt 100.0%

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

      15. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

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

   1. Initial program 22.8%

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

      1. log1p-expm1-u 22.8%

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

      2. expm1-udef 22.8%

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

      3. add-exp-log 22.8%

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

      4. +-commutative 22.8%

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

      5. metadata-eval 22.8%

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

      6. hypot-udef 37.1%

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

      7. associate--l+ 99.2%

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

      8. add-sqr-sqrt 67.9%

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

      9. fabs-sqr 67.9%

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

      10. add-sqr-sqrt 99.2%

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

   3. Applied egg-rr 99.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -14500:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -14500.0)
   (copysign (log (- (fabs x) x)) x)
   (copysign (log1p (+ x (+ (hypot 1.0 x) -1.0))) x)))

C

double code(double x) {
	double tmp;
	if (x <= -14500.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else {
		tmp = copysign(log1p((x + (hypot(1.0, x) + -1.0))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -14500.0) {
		tmp = Math.copySign(Math.log((Math.abs(x) - x)), x);
	} else {
		tmp = Math.copySign(Math.log1p((x + (Math.hypot(1.0, x) + -1.0))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -14500.0:
		tmp = math.copysign(math.log((math.fabs(x) - x)), x)
	else:
		tmp = math.copysign(math.log1p((x + (math.hypot(1.0, x) + -1.0))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -14500.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	else
		tmp = copysign(log1p(Float64(x + Float64(hypot(1.0, x) + -1.0))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -14500.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + N[(x + N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -14500

   1. Initial program 49.5%

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

   2. Taylor expanded in x around -inf 100.0%

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

   if -14500 < x

   1. Initial program 23.7%

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

      1. log1p-expm1-u 23.7%

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

      2. expm1-udef 23.7%

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

      3. add-exp-log 23.7%

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

      4. +-commutative 23.7%

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

      5. metadata-eval 23.7%

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

      6. hypot-udef 37.8%

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

      7. associate--l+ 99.2%

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

      8. add-sqr-sqrt 67.2%

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

      9. fabs-sqr 67.2%

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

      10. add-sqr-sqrt 98.9%

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

   3. Applied egg-rr 98.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -14500:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]

Alternative 4: 82.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.6e-8)
   (copysign (log1p (fabs x)) x)
   (copysign (log (+ x (hypot 1.0 x))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.6e-8) {
		tmp = copysign(log1p(fabs(x)), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.6e-8) {
		tmp = Math.copySign(Math.log1p(Math.abs(x)), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.6e-8:
		tmp = math.copysign(math.log1p(math.fabs(x)), x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.6e-8)
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.6e-8], N[With[{TMP1 = Abs[N[Log[1 + N[Abs[x], $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001e-8

   1. Initial program 23.2%

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

   2. Taylor expanded in x around 0 16.0%

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

   4. Simplified 73.9%

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

   if 1.6000000000000001e-8 < x

   1. Initial program 55.8%

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

      1. log1p-expm1-u 55.8%

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

      2. expm1-udef 55.8%

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

      3. add-exp-log 55.8%

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

      4. +-commutative 55.8%

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

      5. metadata-eval 55.8%

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

      6. hypot-udef 99.3%

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

      7. associate--l+ 99.4%

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

      8. add-sqr-sqrt 99.4%

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

      9. fabs-sqr 99.4%

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

      10. add-sqr-sqrt 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. expm1-log1p-u 97.7%

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

      2. expm1-udef 97.8%

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

      3. log1p-udef 97.8%

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

      4. associate-+r- 97.8%

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

      5. associate-+r- 97.7%

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

      6. +-commutative 97.7%

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

      7. associate-+l+ 97.7%

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

      8. associate-+r- 97.8%

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

      9. add-exp-log 97.8%

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

      10. +-commutative 97.8%

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

      11. log1p-udef 97.8%

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

      12. expm1-udef 97.8%

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

      13. expm1-log1p-u 97.8%

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

   5. Applied egg-rr 97.8%

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

      1. expm1-def 97.7%

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

      2. expm1-log1p 99.3%

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

   7. Simplified 99.3%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 5: 81.8% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (copysign (log1p (fabs x)) x)
   (copysign (log1p (+ -1.0 (* x 2.0))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = copysign(log1p(fabs(x)), x);
	} else {
		tmp = copysign(log1p((-1.0 + (x * 2.0))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = Math.copySign(Math.log1p(Math.abs(x)), x);
	} else {
		tmp = Math.copySign(Math.log1p((-1.0 + (x * 2.0))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = math.copysign(math.log1p(math.fabs(x)), x)
	else:
		tmp = math.copysign(math.log1p((-1.0 + (x * 2.0))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log1p(Float64(-1.0 + Float64(x * 2.0))), x);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 23.4%

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

   2. Taylor expanded in x around 0 16.2%

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

   4. Simplified 73.9%

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

   if 1 < x

   1. Initial program 55.7%

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

      1. log1p-expm1-u 55.7%

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

      2. expm1-udef 55.7%

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

      3. add-exp-log 55.7%

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

      4. +-commutative 55.7%

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

      5. metadata-eval 55.7%

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

      6. hypot-udef 100.0%

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

      7. associate--l+ 100.0%

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

      8. add-sqr-sqrt 100.0%

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

      9. fabs-sqr 100.0%

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

      10. add-sqr-sqrt 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{2 \cdot x - 1}\right), x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.9%

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

Alternative 6: 82.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.96:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.96)
   (copysign (- (log (/ -1.0 x))) x)
   (if (<= x 1.25)
     (copysign (+ x (* (pow x 3.0) -0.16666666666666666)) x)
     (copysign (log1p (+ -1.0 (* x 2.0))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.96) {
		tmp = copysign(-log((-1.0 / x)), x);
	} else if (x <= 1.25) {
		tmp = copysign((x + (pow(x, 3.0) * -0.16666666666666666)), x);
	} else {
		tmp = copysign(log1p((-1.0 + (x * 2.0))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.96) {
		tmp = Math.copySign(-Math.log((-1.0 / x)), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign((x + (Math.pow(x, 3.0) * -0.16666666666666666)), x);
	} else {
		tmp = Math.copySign(Math.log1p((-1.0 + (x * 2.0))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.96:
		tmp = math.copysign(-math.log((-1.0 / x)), x)
	elif x <= 1.25:
		tmp = math.copysign((x + (math.pow(x, 3.0) * -0.16666666666666666)), x)
	else:
		tmp = math.copysign(math.log1p((-1.0 + (x * 2.0))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.96)
		tmp = copysign(Float64(-log(Float64(-1.0 / x))), x);
	elseif (x <= 1.25)
		tmp = copysign(Float64(x + Float64((x ^ 3.0) * -0.16666666666666666)), x);
	else
		tmp = copysign(log1p(Float64(-1.0 + Float64(x * 2.0))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.96], 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[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + N[(-1.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.96

   1. Initial program 50.9%

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

   2. Taylor expanded in x around -inf 99.7%

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

   3. Taylor expanded in x around -inf 31.6%

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

      1. mul-1-neg 31.6%

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

   5. Simplified 31.6%

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

   if -1.96 < x < 1.25

   1. Initial program 7.2%

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

      1. log1p-expm1-u 7.2%

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

      2. expm1-udef 7.2%

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

      3. add-exp-log 7.2%

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

      4. +-commutative 7.2%

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

      5. metadata-eval 7.2%

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

      6. hypot-udef 7.2%

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

      7. associate--l+ 98.8%

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

      8. add-sqr-sqrt 52.7%

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

      9. fabs-sqr 52.7%

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

      10. add-sqr-sqrt 98.8%

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

   3. Applied egg-rr 98.8%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   if 1.25 < x

   1. Initial program 55.7%

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

      1. log1p-expm1-u 55.7%

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

      2. expm1-udef 55.7%

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

      3. add-exp-log 55.7%

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

      4. +-commutative 55.7%

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

      5. metadata-eval 55.7%

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

      6. hypot-udef 100.0%

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

      7. associate--l+ 100.0%

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

      8. add-sqr-sqrt 100.0%

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

      9. fabs-sqr 100.0%

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

      10. add-sqr-sqrt 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{2 \cdot x - 1}\right), x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.96:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + {x}^{3} \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + x \cdot 2\right), x\right)\\ \end{array} \]

Alternative 7: 82.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.2)
   (copysign (- (log (/ -1.0 x))) x)
   (if (<= x 1.25) (copysign x x) (copysign (log1p (+ -1.0 (* x 2.0))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -3.2) {
		tmp = copysign(-log((-1.0 / x)), x);
	} else if (x <= 1.25) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log1p((-1.0 + (x * 2.0))), 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 <= 1.25) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log1p((-1.0 + (x * 2.0))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.2:
		tmp = math.copysign(-math.log((-1.0 / x)), x)
	elif x <= 1.25:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log1p((-1.0 + (x * 2.0))), 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 <= 1.25)
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(Float64(-1.0 + Float64(x * 2.0))), x);
	end
	return 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, 1.25], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + N[(-1.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $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 50.9%

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

   2. Taylor expanded in x around -inf 99.7%

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

   3. Taylor expanded in x around -inf 31.6%

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

      1. mul-1-neg 31.6%

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

   5. Simplified 31.6%

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

   if -3.2000000000000002 < x < 1.25

   1. Initial program 7.2%

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

      1. log1p-expm1-u 7.2%

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

      2. expm1-udef 7.2%

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

      3. add-exp-log 7.2%

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

      4. +-commutative 7.2%

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

      5. metadata-eval 7.2%

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

      6. hypot-udef 7.2%

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

      7. associate--l+ 98.8%

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

      8. add-sqr-sqrt 52.7%

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

      9. fabs-sqr 52.7%

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

      10. add-sqr-sqrt 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. expm1-log1p-u 98.8%

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

      2. expm1-udef 7.2%

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

      3. log1p-udef 7.2%

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

      4. associate-+r- 7.2%

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

      5. associate-+r- 7.2%

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

      6. +-commutative 7.2%

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

      7. associate-+l+ 7.2%

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

      8. associate-+r- 7.2%

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

      9. add-exp-log 7.2%

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

      10. +-commutative 7.2%

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

      11. log1p-udef 7.2%

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

      12. expm1-udef 7.2%

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

      13. expm1-log1p-u 7.2%

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

   5. Applied egg-rr 7.2%

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

      1. expm1-def 7.2%

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

      2. expm1-log1p 7.2%

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

   7. Simplified 7.2%

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

   8. Taylor expanded in x around 0 100.0%

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

   if 1.25 < x

   1. Initial program 55.7%

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

      1. log1p-expm1-u 55.7%

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

      2. expm1-udef 55.7%

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

      3. add-exp-log 55.7%

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

      4. +-commutative 55.7%

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

      5. metadata-eval 55.7%

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

      6. hypot-udef 100.0%

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

      7. associate--l+ 100.0%

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

      8. add-sqr-sqrt 100.0%

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

      9. fabs-sqr 100.0%

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

      10. add-sqr-sqrt 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 100.0%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{2 \cdot x - 1}\right), x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + x \cdot 2\right), x\right)\\ \end{array} \]

Alternative 8: 82.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.2)
   (copysign (- (log (/ -1.0 x))) x)
   (if (<= x 1.25) (copysign x x) (copysign (log1p (* x 2.0)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -3.2) {
		tmp = copysign(-log((-1.0 / x)), x);
	} else if (x <= 1.25) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log1p((x * 2.0)), 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 <= 1.25) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log1p((x * 2.0)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.2:
		tmp = math.copysign(-math.log((-1.0 / x)), x)
	elif x <= 1.25:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log1p((x * 2.0)), 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 <= 1.25)
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(Float64(x * 2.0)), x);
	end
	return 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, 1.25], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + N[(x * 2.0), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2000000000000002

   1. Initial program 50.9%

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

   2. Taylor expanded in x around -inf 99.7%

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

   3. Taylor expanded in x around -inf 31.6%

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

      1. mul-1-neg 31.6%

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

   5. Simplified 31.6%

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

   if -3.2000000000000002 < x < 1.25

   1. Initial program 7.2%

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

      1. log1p-expm1-u 7.2%

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

      2. expm1-udef 7.2%

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

      3. add-exp-log 7.2%

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

      4. +-commutative 7.2%

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

      5. metadata-eval 7.2%

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

      6. hypot-udef 7.2%

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

      7. associate--l+ 98.8%

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

      8. add-sqr-sqrt 52.7%

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

      9. fabs-sqr 52.7%

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

      10. add-sqr-sqrt 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. expm1-log1p-u 98.8%

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

      2. expm1-udef 7.2%

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

      3. log1p-udef 7.2%

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

      4. associate-+r- 7.2%

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

      5. associate-+r- 7.2%

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

      6. +-commutative 7.2%

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

      7. associate-+l+ 7.2%

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

      8. associate-+r- 7.2%

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

      9. add-exp-log 7.2%

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

      10. +-commutative 7.2%

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

      11. log1p-udef 7.2%

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

      12. expm1-udef 7.2%

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

      13. expm1-log1p-u 7.2%

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

   5. Applied egg-rr 7.2%

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

      1. expm1-def 7.2%

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

      2. expm1-log1p 7.2%

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

   7. Simplified 7.2%

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

   8. Taylor expanded in x around 0 100.0%

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

   if 1.25 < x

   1. Initial program 55.7%

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

      1. log1p-expm1-u 55.7%

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

      2. expm1-udef 55.7%

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

      3. add-exp-log 55.7%

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

      4. +-commutative 55.7%

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

      5. metadata-eval 55.7%

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

      6. hypot-udef 100.0%

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

      7. associate--l+ 100.0%

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

      8. add-sqr-sqrt 100.0%

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

      9. fabs-sqr 100.0%

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

      10. add-sqr-sqrt 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 99.2%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{2 \cdot x}\right), x\right) \]
   5. Step-by-step derivation

      1. *-commutative 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x \cdot 2\right), x\right)\\ \end{array} \]

Alternative 9: 75.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.25) (copysign x x) (copysign (log1p (* x 2.0)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log1p((x * 2.0)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log1p((x * 2.0)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.25:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log1p((x * 2.0)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.25)
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(Float64(x * 2.0)), x);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 23.4%

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

      1. log1p-expm1-u 23.4%

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

      2. expm1-udef 23.4%

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

      3. add-exp-log 23.4%

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

      4. +-commutative 23.4%

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

      5. metadata-eval 23.4%

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

      6. hypot-udef 41.6%

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

      7. associate--l+ 99.2%

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

      8. add-sqr-sqrt 33.2%

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

      9. fabs-sqr 33.2%

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

      10. add-sqr-sqrt 64.1%

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

   3. Applied egg-rr 64.1%

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

      1. expm1-log1p-u 63.3%

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

      2. expm1-udef 5.7%

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

      3. log1p-udef 5.7%

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

      4. associate-+r- 4.6%

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

      5. associate-+r- 4.5%

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

      6. +-commutative 4.5%

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

      7. associate-+l+ 4.5%

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

      8. associate-+r- 4.6%

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

      9. add-exp-log 8.3%

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

      10. +-commutative 8.3%

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

      11. log1p-udef 8.3%

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

      12. expm1-udef 8.3%

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

      13. expm1-log1p-u 4.6%

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

   5. Applied egg-rr 4.6%

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

      1. expm1-def 4.5%

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

      2. expm1-log1p 6.5%

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

   7. Simplified 6.5%

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

   8. Taylor expanded in x around 0 64.9%

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

   if 1.25 < x

   1. Initial program 55.7%

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

      1. log1p-expm1-u 55.7%

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

      2. expm1-udef 55.7%

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

      3. add-exp-log 55.7%

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

      4. +-commutative 55.7%

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

      5. metadata-eval 55.7%

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

      6. hypot-udef 100.0%

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

      7. associate--l+ 100.0%

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

      8. add-sqr-sqrt 100.0%

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

      9. fabs-sqr 100.0%

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

      10. add-sqr-sqrt 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 99.2%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{2 \cdot x}\right), x\right) \]
   5. Step-by-step derivation

      1. *-commutative 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x \cdot 2\right), x\right)\\ \end{array} \]

Alternative 10: 59.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.6) (copysign x x) (copysign (log1p x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log1p(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log1p(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.6:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log1p(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.6)
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.6], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   1. Initial program 23.4%

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

      1. log1p-expm1-u 23.4%

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

      2. expm1-udef 23.4%

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

      3. add-exp-log 23.4%

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

      4. +-commutative 23.4%

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

      5. metadata-eval 23.4%

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

      6. hypot-udef 41.6%

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

      7. associate--l+ 99.2%

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

      8. add-sqr-sqrt 33.2%

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

      9. fabs-sqr 33.2%

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

      10. add-sqr-sqrt 64.1%

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

   3. Applied egg-rr 64.1%

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

      1. expm1-log1p-u 63.3%

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

      2. expm1-udef 5.7%

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

      3. log1p-udef 5.7%

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

      4. associate-+r- 4.6%

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

      5. associate-+r- 4.5%

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

      6. +-commutative 4.5%

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

      7. associate-+l+ 4.5%

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

      8. associate-+r- 4.6%

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

      9. add-exp-log 8.3%

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

      10. +-commutative 8.3%

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

      11. log1p-udef 8.3%

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

      12. expm1-udef 8.3%

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

      13. expm1-log1p-u 4.6%

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

   5. Applied egg-rr 4.6%

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

      1. expm1-def 4.5%

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

      2. expm1-log1p 6.5%

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

   7. Simplified 6.5%

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

   8. Taylor expanded in x around 0 64.9%

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

   if 1.6000000000000001 < x

   1. Initial program 55.7%

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

      1. log1p-expm1-u 55.7%

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

      2. expm1-udef 55.7%

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

      3. add-exp-log 55.7%

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

      4. +-commutative 55.7%

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

      5. metadata-eval 55.7%

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

      6. hypot-udef 100.0%

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

      7. associate--l+ 100.0%

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

      8. add-sqr-sqrt 100.0%

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

      9. fabs-sqr 100.0%

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

      10. add-sqr-sqrt 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 31.4%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 11: 52.4% 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 30.8%

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

   1. log1p-expm1-u 30.8%

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

   2. expm1-udef 30.8%

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

   3. add-exp-log 30.8%

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

   4. +-commutative 30.8%

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

   5. metadata-eval 30.8%

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

   6. hypot-udef 55.0%

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

   7. associate--l+ 99.4%

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

   8. add-sqr-sqrt 48.6%

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

   9. fabs-sqr 48.6%

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

   10. add-sqr-sqrt 72.3%

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

3. Applied egg-rr 72.3%

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

   1. expm1-log1p-u 71.4%

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

   2. expm1-udef 27.0%

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

   3. log1p-udef 27.0%

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

   4. associate-+r- 26.2%

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

   5. associate-+r- 26.1%

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

   6. +-commutative 26.1%

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

   7. associate-+l+ 26.1%

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

   8. associate-+r- 26.2%

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

   9. add-exp-log 29.1%

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

   10. +-commutative 29.1%

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

   11. log1p-udef 29.1%

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

   12. expm1-udef 29.1%

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

   13. expm1-log1p-u 26.2%

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

5. Applied egg-rr 26.2%

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

   1. expm1-def 26.2%

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

   2. expm1-log1p 28.0%

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

7. Simplified 28.0%

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

8. Taylor expanded in x around 0 51.2%

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

9. Final simplification 51.2%

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

Developer target: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2023305 
(FPCore (x)
  :name "Rust f64::asinh"
  :precision binary64

  :herbie-target
  (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)

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