Rust f64::asinh

Average Error: 45.7 → 0.1

Time: 7.3s

Precision: binary64

Cost: 46084

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) -10.0)
   (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
   (copysign (log1p (- x (/ x (/ (- -1.0 (hypot 1.0 x)) 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 (copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x) <= -10.0) {
		tmp = copysign(log((1.0 / (hypot(1.0, x) - x))), x);
	} else {
		tmp = copysign(log1p((x - (x / ((-1.0 - hypot(1.0, x)) / 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 (Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x) <= -10.0) {
		tmp = Math.copySign(Math.log((1.0 / (Math.hypot(1.0, x) - x))), x);
	} else {
		tmp = Math.copySign(Math.log1p((x - (x / ((-1.0 - Math.hypot(1.0, x)) / 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 math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x) <= -10.0:
		tmp = math.copysign(math.log((1.0 / (math.hypot(1.0, x) - x))), x)
	else:
		tmp = math.copysign(math.log1p((x - (x / ((-1.0 - math.hypot(1.0, x)) / 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 (copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x) <= -10.0)
		tmp = copysign(log(Float64(1.0 / Float64(hypot(1.0, x) - x))), x);
	else
		tmp = copysign(log1p(Float64(x - Float64(x / Float64(Float64(-1.0 - hypot(1.0, x)) / x)))), x);
	end
	return 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[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], -10.0], N[With[{TMP1 = Abs[N[Log[N[(1.0 / N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + N[(x - N[(x / N[(N[(-1.0 - N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Target

Original	45.7
Target	0.0
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < -10

   1. Initial program 32.4

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

   2. Applied egg-rr 32.4

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

   3. Simplified 0.0

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

      [Start] 32.4	\[ \mathsf{copysign}\left(\log \left(\left(\left(x \cdot x\right) \cdot 0 - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      associate-*r/ [=>] 32.4	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left(\left(x \cdot x\right) \cdot 0 - 1\right) \cdot 1}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
      mul0-rgt [=>] 0.0	\[ \mathsf{copysign}\left(\log \left(\frac{\left(\color{blue}{0} - 1\right) \cdot 1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      metadata-eval [=>] 0.0	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{-1} \cdot 1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      metadata-eval [=>] 0.0	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      metadata-eval [<=] 0.0	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\frac{1}{-1}}}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      associate-/r* [<=] 0.0	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}, x\right) \]
      neg-mul-1 [<=] 0.0	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]
      neg-sub0 [=>] 0.0	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]
      associate--r- [=>] 0.0	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]
      neg-sub0 [<=] 0.0	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      mul-1-neg [<=] 0.0	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{-1 \cdot x} + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      +-commutative [<=] 0.0	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + -1 \cdot x}}\right), x\right) \]
      mul-1-neg [=>] 0.0	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) + \color{blue}{\left(-x\right)}}\right), x\right) \]
      sub-neg [<=] 0.0	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right), x\right) \]

   if -10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x)

   1. Initial program 50.0

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

   2. Applied egg-rr 39.4

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

   3. Simplified 0.4

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

      [Start] 39.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, x\right) + \left(x - 1\right)\right), x\right) \]
      +-commutative [=>] 39.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(x - 1\right) + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      associate-+l- [=>] 0.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x - \left(1 - \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]

   4. Applied egg-rr 10.6

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

   5. Simplified 0.1

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

      [Start] 10.6	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x - \frac{-\left(0 - x \cdot x\right)}{\left(-\mathsf{hypot}\left(1, x\right)\right) + -1}\right), x\right) \]
      sub0-neg [=>] 10.6	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x - \frac{-\color{blue}{\left(-x \cdot x\right)}}{\left(-\mathsf{hypot}\left(1, x\right)\right) + -1}\right), x\right) \]
      remove-double-neg [=>] 10.6	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x - \frac{\color{blue}{x \cdot x}}{\left(-\mathsf{hypot}\left(1, x\right)\right) + -1}\right), x\right) \]
      associate-/l* [=>] 0.1	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x - \color{blue}{\frac{x}{\frac{\left(-\mathsf{hypot}\left(1, x\right)\right) + -1}{x}}}\right), x\right) \]
      +-commutative [=>] 0.1	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x - \frac{x}{\frac{\color{blue}{-1 + \left(-\mathsf{hypot}\left(1, x\right)\right)}}{x}}\right), x\right) \]
      sub-neg [<=] 0.1	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x - \frac{x}{\frac{\color{blue}{-1 - \mathsf{hypot}\left(1, x\right)}}{x}}\right), x\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.2
Cost	19784

\[\begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.00096:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 2
Error	0.0
Cost	19784

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00094:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.00096:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 3
Error	0.3
Cost	19780

\[\begin{array}{l} \mathbf{if}\;x \leq -0.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]

Alternative 4
Error	0.3
Cost	13832

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x - \frac{x}{x \cdot -0.5 + \frac{-2}{x}}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \]

Alternative 5
Error	0.8
Cost	13828

\[\begin{array}{l} \mathbf{if}\;x \leq -0.3:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x - \frac{x}{-1 + \frac{-0.5 - x}{x \cdot x}}\right), x\right)\\ \end{array} \]

Alternative 6
Error	0.3
Cost	13576

\[\begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \]

Alternative 7
Error	0.4
Cost	13512

\[\begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 8
Error	11.3
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 9
Error	0.5
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 10
Error	22.3
Cost	13124

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

Alternative 11
Error	26.1
Cost	13060

\[\begin{array}{l} \mathbf{if}\;x \leq 1.58:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 12
Error	30.2
Cost	6528

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

Reproduce

herbie shell --seed 2023039 
(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))