Rust f64::asinh

Average Error: 45.2 → 29.2

Time: 2.4s

Precision: binary64

Cost: 26376

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 5000000:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -4e+154)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 5000000.0)
     (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)
     (copysign (log (/ 0.5 x)) x))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -4e+154) {
		tmp = copysign(log((-0.5 / x)), x);
	} else if (x <= 5000000.0) {
		tmp = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	} else {
		tmp = copysign(log((0.5 / x)), x);
	}
	return tmp;
}

Java

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

↓

public static double code(double x) {
	double tmp;
	if (x <= -4e+154) {
		tmp = Math.copySign(Math.log((-0.5 / x)), x);
	} else if (x <= 5000000.0) {
		tmp = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	} else {
		tmp = Math.copySign(Math.log((0.5 / x)), x);
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -4e+154:
		tmp = math.copysign(math.log((-0.5 / x)), x)
	elif x <= 5000000.0:
		tmp = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	else:
		tmp = math.copysign(math.log((0.5 / x)), x)
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -4e+154)
		tmp = copysign(log(Float64(-0.5 / x)), x);
	elseif (x <= 5000000.0)
		tmp = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x);
	else
		tmp = copysign(log(Float64(0.5 / x)), x);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4e+154)
		tmp = sign(x) * abs(log((-0.5 / x)));
	elseif (x <= 5000000.0)
		tmp = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
	else
		tmp = sign(x) * abs(log((0.5 / x)));
	end
	tmp_2 = tmp;
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]

↓

code[x_] := If[LessEqual[x, -4e+154], N[With[{TMP1 = Abs[N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 5000000.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Target

Original	45.2
Target	0.1
Herbie	29.2

\[\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, \frac{1}{\left|x\right|}\right) + \frac{1}{\left|x\right|}}\right), x\right) \]

Derivation

1. Split input into 3 regimes

2. if x < -4.00000000000000015e154

   1. Initial program 64.0

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

   2. Taylor expanded in x around -inf 0.1

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

   3. Simplified 0.1

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

      [Start] 0.1	\[ \mathsf{copysign}\left(\log \left(\left(\left|x\right| + -1 \cdot x\right) - 0.5 \cdot \frac{1}{x}\right), x\right) \]
      rational.json-simplify-41 [=>] 0.1	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\left(-0.5 \cdot \frac{1}{x}\right) + \left(\left|x\right| + -1 \cdot x\right)\right)}, x\right) \]
      rational.json-simplify-1 [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\left(-0.5 \cdot \frac{1}{x}\right) + \color{blue}{\left(-1 \cdot x + \left|x\right|\right)}\right), x\right) \]
      rational.json-simplify-33 [=>] 0.1	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left(-0.5 \cdot \frac{1}{x}\right) + -1 \cdot x\right) + \left|x\right|\right)}, x\right) \]
      rational.json-simplify-1 [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\color{blue}{\left(-1 \cdot x + \left(-0.5 \cdot \frac{1}{x}\right)\right)} + \left|x\right|\right), x\right) \]
      rational.json-simplify-2 [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\left(\color{blue}{x \cdot -1} + \left(-0.5 \cdot \frac{1}{x}\right)\right) + \left|x\right|\right), x\right) \]
      rational.json-simplify-9 [<=] 0.1	\[ \mathsf{copysign}\left(\log \left(\left(\color{blue}{\left(-x\right)} + \left(-0.5 \cdot \frac{1}{x}\right)\right) + \left|x\right|\right), x\right) \]
      rational.json-simplify-41 [<=] 0.1	\[ \mathsf{copysign}\left(\log \left(\color{blue}{\left(\left(-0.5 \cdot \frac{1}{x}\right) - x\right)} + \left|x\right|\right), x\right) \]
      rational.json-simplify-9 [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\left(\color{blue}{\left(0.5 \cdot \frac{1}{x}\right) \cdot -1} - x\right) + \left|x\right|\right), x\right) \]
      rational.json-simplify-2 [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\left(\color{blue}{-1 \cdot \left(0.5 \cdot \frac{1}{x}\right)} - x\right) + \left|x\right|\right), x\right) \]
      rational.json-simplify-2 [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\left(-1 \cdot \color{blue}{\left(\frac{1}{x} \cdot 0.5\right)} - x\right) + \left|x\right|\right), x\right) \]
      rational.json-simplify-31 [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\left(\color{blue}{\frac{1}{x} \cdot \left(-1 \cdot 0.5\right)} - x\right) + \left|x\right|\right), x\right) \]
      metadata-eval [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\left(\frac{1}{x} \cdot \color{blue}{-0.5} - x\right) + \left|x\right|\right), x\right) \]

   4. Taylor expanded in x around 0 0.0

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

   if -4.00000000000000015e154 < x < 5e6

   1. Initial program 46.3

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

   if 5e6 < x

   1. Initial program 33.0

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

   2. Taylor expanded in x around inf 0.2

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

   3. Taylor expanded in x around 0 0.0

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

4. Final simplification 29.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 5000000:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	30.3
Cost	19972

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0.5 \cdot \frac{1}{x} + \left(\left|x\right| + x\right)\right), x\right)\\ \end{array} \]

Alternative 2
Error	30.3
Cost	19972

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{1}{x} \cdot -0.5 - x\right) + \left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0.5 \cdot \frac{1}{x} + \left(\left|x\right| + x\right)\right), x\right)\\ \end{array} \]

Alternative 3
Error	41.4
Cost	13188

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]

Alternative 4
Error	30.4
Cost	13188

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 5
Error	52.2
Cost	13124

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]

Alternative 6
Error	58.1
Cost	12928

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

Reproduce

herbie shell --seed 2023053 
(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))