Rust f64::asinh

Percentage Accurate: 28.9% → 99.3%

Time: 12.8s

Alternatives: 8

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: 28.9% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) -20.0)
   (copysign (log (+ (fabs x) (hypot 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) <= -20.0) {
		tmp = copysign(log((fabs(x) + hypot(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) <= -20.0) {
		tmp = Math.copySign(Math.log((Math.abs(x) + Math.hypot(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) <= -20.0:
		tmp = math.copysign(math.log((math.fabs(x) + math.hypot(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) <= -20.0)
		tmp = copysign(log(Float64(abs(x) + hypot(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], -20.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], 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) < -20

   1. Initial program 47.7%

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

      1. +-commutative 47.7%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   if -20 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x)

   1. Initial program 21.7%

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

      1. +-commutative 21.7%

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

      2. hypot-1-def 37.8%

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

   3. Simplified 37.8%

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

      1. log1p-expm1-u 37.8%

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

      2. expm1-udef 37.8%

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

      3. add-exp-log 37.8%

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

      4. associate--l+ 99.5%

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

      5. add-sqr-sqrt 70.0%

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

      6. fabs-sqr 70.0%

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

      7. add-sqr-sqrt 99.5%

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

   5. Applied egg-rr 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]

Alternative 2: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 48.5%

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

      1. +-commutative 48.5%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. +-lft-identity 100.0%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. fabs-sqr 0.0%

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

      5. add-sqr-sqrt 4.6%

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

   5. Applied egg-rr 4.6%

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

   7. Simplified 4.6%

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

   8. Taylor expanded in x around -inf 98.4%

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

   if -1.25 < x < 1e-3

   1. Initial program 6.4%

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

      1. +-commutative 6.4%

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

      2. hypot-1-def 6.4%

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

   3. Simplified 6.4%

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

      1. expm1-log1p-u 6.4%

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

      2. expm1-udef 6.4%

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

      3. flip3-- 6.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{{\left(e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}}, x\right) \]

      4. add-sqr-sqrt 3.1%

         \[\leadsto \mathsf{copysign}\left(\frac{{\left(e^{\mathsf{log1p}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}, x\right) \]

      5. fabs-sqr 3.1%

         \[\leadsto \mathsf{copysign}\left(\frac{{\left(e^{\mathsf{log1p}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}, x\right) \]

      6. add-sqr-sqrt 6.4%

         \[\leadsto \mathsf{copysign}\left(\frac{{\left(e^{\mathsf{log1p}\left(\log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}, x\right) \]

      7. metadata-eval 6.4%

         \[\leadsto \mathsf{copysign}\left(\frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - \color{blue}{1}}{e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}, x\right) \]

   5. Applied egg-rr 6.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - 1}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}}, x\right) \]
   6. Step-by-step derivation

      1. *-un-lft-identity 6.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{1 \cdot \frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - 1}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}}, x\right) \]

      2. sub-neg 6.5%

         \[\leadsto \mathsf{copysign}\left(1 \cdot \frac{\color{blue}{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} + \left(-1\right)}}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}, x\right) \]

      3. metadata-eval 6.5%

         \[\leadsto \mathsf{copysign}\left(1 \cdot \frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} + \color{blue}{-1}}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}, x\right) \]

      4. +-commutative 6.5%

         \[\leadsto \mathsf{copysign}\left(1 \cdot \frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} + -1}{\color{blue}{\left(1 + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right) + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}}}, x\right) \]

      5. *-rgt-identity 6.5%

         \[\leadsto \mathsf{copysign}\left(1 \cdot \frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} + -1}{\left(1 + \color{blue}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}}\right) + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}}, x\right) \]

   7. Applied egg-rr 6.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{1 \cdot \frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} + -1}{1 + \left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + {\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{2}\right)}}, x\right) \]
   8. Step-by-step derivation

      1. *-lft-identity 6.5%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} + -1}{1 + \left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + {\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{2}\right)}}, x\right) \]

      2. +-commutative 6.5%

         \[\leadsto \mathsf{copysign}\left(\frac{\color{blue}{-1 + {\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3}}}{1 + \left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + {\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{2}\right)}, x\right) \]

   9. Simplified 6.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{-1 + {\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3}}{1 + \left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + {\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{2}\right)}}, x\right) \]

   10. Taylor expanded in x around 0 100.0%

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

       1. *-commutative 100.0%

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

   12. Simplified 100.0%

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

   if 1e-3 < x

   1. Initial program 51.2%

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

      1. +-commutative 51.2%

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

      2. hypot-1-def 99.6%

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

   3. Simplified 99.6%

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

      1. +-lft-identity 99.6%

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

      2. +-commutative 99.6%

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

      3. add-sqr-sqrt 99.6%

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

      4. fabs-sqr 99.6%

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

      5. add-sqr-sqrt 99.6%

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

   5. Applied egg-rr 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 99.5%

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

Alternative 3: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9000:\\ \;\;\;\;\mathsf{copysign}\left(\log 0.5 + \log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -9000.0)
   (copysign (+ (log 0.5) (log (/ -1.0 x))) x)
   (copysign (log1p (+ x (+ (hypot 1.0 x) -1.0))) x)))

C

double code(double x) {
	double tmp;
	if (x <= -9000.0) {
		tmp = copysign((log(0.5) + log((-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 (x <= -9000.0) {
		tmp = Math.copySign((Math.log(0.5) + Math.log((-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 x <= -9000.0:
		tmp = math.copysign((math.log(0.5) + math.log((-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 (x <= -9000.0)
		tmp = copysign(Float64(log(0.5) + log(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[x, -9000.0], N[With[{TMP1 = Abs[N[(N[Log[0.5], $MachinePrecision] + N[Log[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 x < -9e3

   1. Initial program 47.7%

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

      1. +-commutative 47.7%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. +-lft-identity 100.0%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. fabs-sqr 0.0%

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

      5. add-sqr-sqrt 3.1%

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

   5. Applied egg-rr 3.1%

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

   7. Simplified 3.1%

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

   8. Taylor expanded in x around -inf 99.6%

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

   if -9e3 < x

   1. Initial program 21.7%

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

      1. +-commutative 21.7%

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

      2. hypot-1-def 37.8%

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

   3. Simplified 37.8%

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

      1. log1p-expm1-u 37.8%

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

      2. expm1-udef 37.8%

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

      3. add-exp-log 37.8%

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

      4. associate--l+ 99.5%

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

      5. add-sqr-sqrt 70.0%

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

      6. fabs-sqr 70.0%

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

      7. add-sqr-sqrt 99.5%

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

   5. Applied egg-rr 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9000:\\ \;\;\;\;\mathsf{copysign}\left(\log 0.5 + \log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]

Alternative 4: 81.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.25e-8)
   (copysign (log1p (fabs x)) x)
   (copysign (log (+ x (hypot 1.0 x))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.25e-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 <= 1.25e-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 <= 1.25e-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 <= 1.25e-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, 1.25e-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 < 1.2499999999999999e-8

   1. Initial program 20.5%

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

      1. +-commutative 20.5%

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

      2. hypot-1-def 37.6%

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

   3. Simplified 37.6%

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

   4. Taylor expanded in x around 0 14.8%

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

   6. Simplified 76.8%

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

   if 1.2499999999999999e-8 < x

   1. Initial program 51.2%

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

      1. +-commutative 51.2%

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

      2. hypot-1-def 99.6%

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

   3. Simplified 99.6%

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

      1. +-lft-identity 99.6%

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

      2. +-commutative 99.6%

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

      3. add-sqr-sqrt 99.6%

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

      4. fabs-sqr 99.6%

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

      5. add-sqr-sqrt 99.6%

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

   5. Applied egg-rr 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 5: 65.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.1%

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

   1. +-commutative 28.1%

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

   2. hypot-1-def 53.1%

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

3. Simplified 53.1%

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

4. Taylor expanded in x around 0 18.9%

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

6. Simplified 65.5%

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

7. Final simplification 65.5%

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

Alternative 6: 65.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 48.5%

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

      1. +-commutative 48.5%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around -inf 31.5%

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

   if -1 < x

   1. Initial program 21.3%

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

      1. +-commutative 21.3%

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

      2. hypot-1-def 37.5%

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

   3. Simplified 37.5%

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

   4. Taylor expanded in x around 0 14.7%

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

      1. log1p-def 76.8%

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

      2. add-sqr-sqrt 47.7%

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

      3. fabs-sqr 47.7%

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

      4. add-sqr-sqrt 76.8%

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

      5. *-un-lft-identity 76.8%

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

   6. Applied egg-rr 76.8%

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

      1. *-lft-identity 76.8%

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

   8. Simplified 76.8%

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

4. Final simplification 65.4%

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

Alternative 7: 59.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 22.0%

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

      1. +-commutative 22.0%

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

      2. hypot-1-def 38.8%

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

   3. Simplified 38.8%

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

      1. expm1-log1p-u 38.2%

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

      2. expm1-udef 38.3%

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

      3. flip3-- 38.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{{\left(e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}}, x\right) \]

      4. add-sqr-sqrt 3.9%

         \[\leadsto \mathsf{copysign}\left(\frac{{\left(e^{\mathsf{log1p}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}, x\right) \]

      5. fabs-sqr 3.9%

         \[\leadsto \mathsf{copysign}\left(\frac{{\left(e^{\mathsf{log1p}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}, x\right) \]

      6. add-sqr-sqrt 6.1%

         \[\leadsto \mathsf{copysign}\left(\frac{{\left(e^{\mathsf{log1p}\left(\log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}, x\right) \]

      7. metadata-eval 6.1%

         \[\leadsto \mathsf{copysign}\left(\frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - \color{blue}{1}}{e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}, x\right) \]

   5. Applied egg-rr 6.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - 1}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}}, x\right) \]
   6. Step-by-step derivation

      1. *-un-lft-identity 6.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{1 \cdot \frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - 1}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}}, x\right) \]

      2. sub-neg 6.2%

         \[\leadsto \mathsf{copysign}\left(1 \cdot \frac{\color{blue}{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} + \left(-1\right)}}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}, x\right) \]

      3. metadata-eval 6.2%

         \[\leadsto \mathsf{copysign}\left(1 \cdot \frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} + \color{blue}{-1}}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}, x\right) \]

      4. +-commutative 6.2%

         \[\leadsto \mathsf{copysign}\left(1 \cdot \frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} + -1}{\color{blue}{\left(1 + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right) + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}}}, x\right) \]

      5. *-rgt-identity 6.2%

         \[\leadsto \mathsf{copysign}\left(1 \cdot \frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} + -1}{\left(1 + \color{blue}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}}\right) + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}}, x\right) \]

   7. Applied egg-rr 6.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{1 \cdot \frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} + -1}{1 + \left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + {\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{2}\right)}}, x\right) \]
   8. Step-by-step derivation

      1. *-lft-identity 6.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} + -1}{1 + \left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + {\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{2}\right)}}, x\right) \]

      2. +-commutative 6.2%

         \[\leadsto \mathsf{copysign}\left(\frac{\color{blue}{-1 + {\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3}}}{1 + \left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + {\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{2}\right)}, x\right) \]

   9. Simplified 6.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{-1 + {\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3}}{1 + \left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + {\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{2}\right)}}, x\right) \]

   10. Taylor expanded in x around 0 67.8%

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

   if 1.55000000000000004 < x

   1. Initial program 48.3%

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

      1. +-commutative 48.3%

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

      2. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 31.7%

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

      1. log1p-def 31.7%

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

      2. add-sqr-sqrt 31.7%

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

      3. fabs-sqr 31.7%

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

      4. add-sqr-sqrt 31.7%

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

      5. *-un-lft-identity 31.7%

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

   6. Applied egg-rr 31.7%

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

      1. *-lft-identity 31.7%

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

   8. Simplified 31.7%

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

4. Final simplification 59.4%

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

Alternative 8: 53.0% 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 28.1%

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

   1. +-commutative 28.1%

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

   2. hypot-1-def 53.1%

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

3. Simplified 53.1%

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

   1. expm1-log1p-u 52.2%

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

   2. expm1-udef 52.2%

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

   3. flip3-- 52.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{{\left(e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}}, x\right) \]

   4. add-sqr-sqrt 26.0%

      \[\leadsto \mathsf{copysign}\left(\frac{{\left(e^{\mathsf{log1p}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}, x\right) \]

   5. fabs-sqr 26.0%

      \[\leadsto \mathsf{copysign}\left(\frac{{\left(e^{\mathsf{log1p}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}, x\right) \]

   6. add-sqr-sqrt 27.6%

      \[\leadsto \mathsf{copysign}\left(\frac{{\left(e^{\mathsf{log1p}\left(\log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}, x\right) \]

   7. metadata-eval 27.6%

      \[\leadsto \mathsf{copysign}\left(\frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - \color{blue}{1}}{e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}, x\right) \]

5. Applied egg-rr 27.7%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - 1}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}}, x\right) \]
6. Step-by-step derivation

   1. *-un-lft-identity 27.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{1 \cdot \frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} - 1}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}}, x\right) \]

   2. sub-neg 27.7%

      \[\leadsto \mathsf{copysign}\left(1 \cdot \frac{\color{blue}{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} + \left(-1\right)}}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}, x\right) \]

   3. metadata-eval 27.7%

      \[\leadsto \mathsf{copysign}\left(1 \cdot \frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} + \color{blue}{-1}}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + \left(1 + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right)}, x\right) \]

   4. +-commutative 27.7%

      \[\leadsto \mathsf{copysign}\left(1 \cdot \frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} + -1}{\color{blue}{\left(1 + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot 1\right) + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}}}, x\right) \]

   5. *-rgt-identity 27.7%

      \[\leadsto \mathsf{copysign}\left(1 \cdot \frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} + -1}{\left(1 + \color{blue}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}}\right) + e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} \cdot e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}}, x\right) \]

7. Applied egg-rr 27.7%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{1 \cdot \frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} + -1}{1 + \left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + {\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{2}\right)}}, x\right) \]
8. Step-by-step derivation

   1. *-lft-identity 27.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{{\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3} + -1}{1 + \left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + {\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{2}\right)}}, x\right) \]

   2. +-commutative 27.7%

      \[\leadsto \mathsf{copysign}\left(\frac{\color{blue}{-1 + {\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3}}}{1 + \left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + {\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{2}\right)}, x\right) \]

9. Simplified 27.7%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{-1 + {\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{3}}{1 + \left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} + {\left(e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)}^{2}\right)}}, x\right) \]

10. Taylor expanded in x around 0 53.1%

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

11. Final simplification 53.1%

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

Developer target: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2023326 
(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))