Rust f64::asinh

Percentage Accurate: 29.5% → 99.9%

Time: 2.4s

Alternatives: 8

Speedup: 1.8×

Specification

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]

Initial Program: 29.5% 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: 99.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\sinh^{-1} x, x\right) \end{array} \]

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.5%

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

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{copysign}\left(\sinh^{-1} x, x\right)} \]
4. Add Preprocessing

Alternative 2: 82.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -3.2) {
		tmp = copysign(log(fabs(x)), x);
	} else if (x <= 1.25) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log((2.0 * x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.2) {
		tmp = Math.copySign(Math.log(Math.abs(x)), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log((2.0 * x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.2:
		tmp = math.copysign(math.log(math.fabs(x)), x)
	elif x <= 1.25:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log((2.0 * x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.2)
		tmp = copysign(log(abs(x)), x);
	elseif (x <= 1.25)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float64(2.0 * x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.2)
		tmp = sign(x) * abs(log(abs(x)));
	elseif (x <= 1.25)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((2.0 * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.2], N[With[{TMP1 = Abs[N[Log[N[Abs[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[N[(2.0 * x), $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 29.5%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 26.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 18.2%

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

   if -3.2000000000000002 < x < 1.25

   1. Initial program 29.5%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 52.9%

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

   if 1.25 < x

   1. Initial program 29.5%

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

   3. Applied rewrites 18.0%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 25.6%

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

Alternative 3: 65.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right|\right), x\right)\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(1 + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.2)
   (copysign (log (fabs x)) x)
   (if (<= x 1.6) (copysign x x) (copysign (log (+ 1.0 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -3.2) {
		tmp = copysign(log(fabs(x)), x);
	} else if (x <= 1.6) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log((1.0 + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.2) {
		tmp = Math.copySign(Math.log(Math.abs(x)), x);
	} else if (x <= 1.6) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log((1.0 + x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.2:
		tmp = math.copysign(math.log(math.fabs(x)), x)
	elif x <= 1.6:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log((1.0 + x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.2)
		tmp = copysign(log(abs(x)), x);
	elseif (x <= 1.6)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float64(1.0 + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.2)
		tmp = sign(x) * abs(log(abs(x)));
	elseif (x <= 1.6)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((1.0 + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.2], N[With[{TMP1 = Abs[N[Log[N[Abs[x], $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], 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[N[(1.0 + x), $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 29.5%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 26.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 18.2%

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

   if -3.2000000000000002 < x < 1.6000000000000001

   1. Initial program 29.5%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 52.9%

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

   if 1.6000000000000001 < x

   1. Initial program 29.5%

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

   3. Applied rewrites 18.0%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 11.5%

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

Alternative 4: 65.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right|\right), x\right)\\ \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (fabs x)) x)))
   (if (<= x -3.2) t_0 (if (<= x 3.2) (copysign x x) t_0))))

C

double code(double x) {
	double t_0 = copysign(log(fabs(x)), x);
	double tmp;
	if (x <= -3.2) {
		tmp = t_0;
	} else if (x <= 3.2) {
		tmp = copysign(x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log(Math.abs(x)), x);
	double tmp;
	if (x <= -3.2) {
		tmp = t_0;
	} else if (x <= 3.2) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log(math.fabs(x)), x)
	tmp = 0
	if x <= -3.2:
		tmp = t_0
	elif x <= 3.2:
		tmp = math.copysign(x, x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = copysign(log(abs(x)), x)
	tmp = 0.0
	if (x <= -3.2)
		tmp = t_0;
	elseif (x <= 3.2)
		tmp = copysign(x, x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log(abs(x)));
	tmp = 0.0;
	if (x <= -3.2)
		tmp = t_0;
	elseif (x <= 3.2)
		tmp = sign(x) * abs(x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[Abs[x], $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[x, -3.2], t$95$0, If[LessEqual[x, 3.2], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2000000000000002 or 3.2000000000000002 < x

   1. Initial program 29.5%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 26.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 18.2%

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

   if -3.2000000000000002 < x < 3.2000000000000002

   1. Initial program 29.5%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 52.9%

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

Alternative 5: 57.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(2, x\right)\\ \mathbf{elif}\;x \leq 6:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(2 \cdot 3, 2 \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.0)
   (copysign 2.0 x)
   (if (<= x 6.0) (copysign x x) (copysign (* 2.0 3.0) (* 2.0 3.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = copysign(2.0, x);
	} else if (x <= 6.0) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign((2.0 * 3.0), (2.0 * 3.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = Math.copySign(2.0, x);
	} else if (x <= 6.0) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign((2.0 * 3.0), (2.0 * 3.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.0:
		tmp = math.copysign(2.0, x)
	elif x <= 6.0:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign((2.0 * 3.0), (2.0 * 3.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.0)
		tmp = copysign(2.0, x);
	elseif (x <= 6.0)
		tmp = copysign(x, x);
	else
		tmp = copysign(Float64(2.0 * 3.0), Float64(2.0 * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.0)
		tmp = sign(x) * abs(2.0);
	elseif (x <= 6.0)
		tmp = sign(x) * abs(x);
	else
		tmp = sign((2.0 * 3.0)) * abs((2.0 * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.0], N[With[{TMP1 = Abs[2.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 6.0], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[(2.0 * 3.0), $MachinePrecision]], TMP2 = Sign[N[(2.0 * 3.0), $MachinePrecision]]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 29.5%

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

   3. Applied rewrites 99.9%

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

   5. Applied rewrites 9.9%

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

   if -2 < x < 6

   1. Initial program 29.5%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 52.9%

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

   if 6 < x

   1. Initial program 29.5%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 52.9%

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

   8. Applied rewrites 10.0%

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

   10. Applied rewrites 5.9%

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

   12. Applied rewrites 6.0%

       \[\leadsto \mathsf{copysign}\left(2 \cdot \color{blue}{3}, 2 \cdot 2\right) \]

   14. Applied rewrites 6.0%

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

Alternative 6: 57.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(2, x\right)\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(2 \cdot 2, 2 \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.0)
   (copysign 2.0 x)
   (if (<= x 4.0) (copysign x x) (copysign (* 2.0 2.0) (* 2.0 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = copysign(2.0, x);
	} else if (x <= 4.0) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign((2.0 * 2.0), (2.0 * 2.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = Math.copySign(2.0, x);
	} else if (x <= 4.0) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign((2.0 * 2.0), (2.0 * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.0:
		tmp = math.copysign(2.0, x)
	elif x <= 4.0:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign((2.0 * 2.0), (2.0 * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.0)
		tmp = copysign(2.0, x);
	elseif (x <= 4.0)
		tmp = copysign(x, x);
	else
		tmp = copysign(Float64(2.0 * 2.0), Float64(2.0 * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.0)
		tmp = sign(x) * abs(2.0);
	elseif (x <= 4.0)
		tmp = sign(x) * abs(x);
	else
		tmp = sign((2.0 * 2.0)) * abs((2.0 * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.0], N[With[{TMP1 = Abs[2.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 4.0], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[(2.0 * 2.0), $MachinePrecision]], TMP2 = Sign[N[(2.0 * 2.0), $MachinePrecision]]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 29.5%

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

   3. Applied rewrites 99.9%

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

   5. Applied rewrites 9.9%

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

   if -2 < x < 4

   1. Initial program 29.5%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 52.9%

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

   if 4 < x

   1. Initial program 29.5%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 52.9%

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

   8. Applied rewrites 10.0%

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

   10. Applied rewrites 5.9%

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

Alternative 7: 57.3% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.0)
   (copysign 2.0 x)
   (if (<= x 2.0) (copysign x x) (copysign 2.0 x))))

C

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

Java

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

Python

def code(x):
	tmp = 0
	if x <= -2.0:
		tmp = math.copysign(2.0, x)
	elif x <= 2.0:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(2.0, x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.0)
		tmp = copysign(2.0, x);
	elseif (x <= 2.0)
		tmp = copysign(x, x);
	else
		tmp = copysign(2.0, x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.0)
		tmp = sign(x) * abs(2.0);
	elseif (x <= 2.0)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -2 or 2 < x

   1. Initial program 29.5%

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

   3. Applied rewrites 99.9%

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

   5. Applied rewrites 9.9%

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

   if -2 < x < 2

   1. Initial program 29.5%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 52.9%

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

Alternative 8: 9.9% accurate, 5.4× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (copysign 2.0 x))

C

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

Java

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

Python

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

Julia

function code(x)
	return copysign(2.0, x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.5%

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

3. Applied rewrites 99.9%

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

5. Applied rewrites 9.9%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{2}, x\right) \]
6. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.4× 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 2025159 
(FPCore (x)
  :name "Rust f64::asinh"
  :precision binary64

  :alt
  (! :herbie-platform c (let* ((ax (fabs x)) (ix (/ 1 ax))) (copysign (log1p (+ ax (/ ax (+ (hypot 1 ix) ix)))) x)))

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