Rust f64::asinh

Average Error: 45.3 → 0.3

Time: 3.8s

Precision: binary64

Cost: 72136

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x \cdot -2 + \frac{-0.5}{x}}\right), x\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\left(1 + \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right) + -1, x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -10.0)
     (copysign (log (/ 1.0 (+ (* x -2.0) (/ -0.5 x)))) x)
     (if (<= t_0 5e-7)
       (copysign (+ x (* -0.16666666666666666 (* x (* x x)))) x)
       (copysign (+ (+ 1.0 (log (+ x (hypot 1.0 x)))) -1.0) x)))))

C

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

↓

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = copysign(log((1.0 / ((x * -2.0) + (-0.5 / x)))), x);
	} else if (t_0 <= 5e-7) {
		tmp = copysign((x + (-0.16666666666666666 * (x * (x * x)))), x);
	} else {
		tmp = copysign(((1.0 + log((x + hypot(1.0, x)))) + -1.0), 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 t_0 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = Math.copySign(Math.log((1.0 / ((x * -2.0) + (-0.5 / x)))), x);
	} else if (t_0 <= 5e-7) {
		tmp = Math.copySign((x + (-0.16666666666666666 * (x * (x * x)))), x);
	} else {
		tmp = Math.copySign(((1.0 + Math.log((x + Math.hypot(1.0, x)))) + -1.0), 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):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -10.0:
		tmp = math.copysign(math.log((1.0 / ((x * -2.0) + (-0.5 / x)))), x)
	elif t_0 <= 5e-7:
		tmp = math.copysign((x + (-0.16666666666666666 * (x * (x * x)))), x)
	else:
		tmp = math.copysign(((1.0 + math.log((x + math.hypot(1.0, x)))) + -1.0), 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)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = copysign(log(Float64(1.0 / Float64(Float64(x * -2.0) + Float64(-0.5 / x)))), x);
	elseif (t_0 <= 5e-7)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * Float64(x * Float64(x * x)))), x);
	else
		tmp = copysign(Float64(Float64(1.0 + log(Float64(x + hypot(1.0, x)))) + -1.0), 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)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = sign(x) * abs(log((1.0 / ((x * -2.0) + (-0.5 / x)))));
	elseif (t_0 <= 5e-7)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x * (x * x)))));
	else
		tmp = sign(x) * abs(((1.0 + log((x + hypot(1.0, x)))) + -1.0));
	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_] := Block[{t$95$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]}, If[LessEqual[t$95$0, -10.0], N[With[{TMP1 = Abs[N[Log[N[(1.0 / N[(N[(x * -2.0), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 5e-7], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[(N[(1.0 + N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Target

Original	45.3
Target	0.0
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 31.6

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

   2. Applied egg-rr 31.6

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

   3. Simplified 0.1

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

      [Start] 31.6	\[ \mathsf{copysign}\left(\log \left(\left(\left(x \cdot x\right) \cdot 0 - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      associate-*r/ [=>] 31.6	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left(\left(x \cdot x\right) \cdot 0 - 1\right) \cdot 1}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
      mul0-rgt [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\frac{\left(\color{blue}{0} - 1\right) \cdot 1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      metadata-eval [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{-1} \cdot 1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      metadata-eval [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      metadata-eval [<=] 0.1	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\frac{1}{-1}}}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      associate-/r* [<=] 0.1	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}, x\right) \]
      neg-mul-1 [<=] 0.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]
      neg-sub0 [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]
      associate--r- [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]
      neg-sub0 [<=] 0.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      mul-1-neg [<=] 0.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{-1 \cdot x} + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      +-commutative [<=] 0.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + -1 \cdot x}}\right), x\right) \]
      mul-1-neg [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) + \color{blue}{\left(-x\right)}}\right), x\right) \]
      sub-neg [<=] 0.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right), x\right) \]

   4. Taylor expanded in x around -inf 0.1

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{-2 \cdot x - 0.5 \cdot \frac{1}{x}}}\right), x\right) \]

   5. Simplified 0.1

      \[\leadsto \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{x \cdot -2 - \frac{0.5}{x}}}\right), x\right) \]
      Proof

      [Start] 0.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{-2 \cdot x - 0.5 \cdot \frac{1}{x}}\right), x\right) \]
      *-commutative [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{x \cdot -2} - 0.5 \cdot \frac{1}{x}}\right), x\right) \]
      associate-*r/ [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{x \cdot -2 - \color{blue}{\frac{0.5 \cdot 1}{x}}}\right), x\right) \]
      metadata-eval [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{x \cdot -2 - \frac{\color{blue}{0.5}}{x}}\right), x\right) \]

   if -10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < 4.99999999999999977e-7

   1. Initial program 58.9

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

   2. Applied egg-rr 59.0

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

   3. Simplified 59.0

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

      [Start] 59.0	\[ \mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) + x\right) + 0, x\right) \]
      +-rgt-identity [=>] 59.0	\[ \mathsf{copysign}\left(\color{blue}{\log \left(\mathsf{hypot}\left(1, x\right) + x\right)}, x\right) \]
      +-commutative [=>] 59.0	\[ \mathsf{copysign}\left(\log \color{blue}{\left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 0.4

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

   5. Applied egg-rr 0.4

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

   if 4.99999999999999977e-7 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x)

   1. Initial program 31.6

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

   2. Applied egg-rr 0.3

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

   3. Simplified 0.3

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

      [Start] 0.3	\[ \mathsf{copysign}\left(\log \left(\mathsf{hypot}\left(1, x\right) + x\right) + 0, x\right) \]
      +-rgt-identity [=>] 0.3	\[ \mathsf{copysign}\left(\color{blue}{\log \left(\mathsf{hypot}\left(1, x\right) + x\right)}, x\right) \]
      +-commutative [=>] 0.3	\[ \mathsf{copysign}\left(\log \color{blue}{\left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

   4. Applied egg-rr 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x \cdot -2 + \frac{-0.5}{x}}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\left(1 + \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right) + -1, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	19784

\[\begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x \cdot -2 + \frac{-0.5}{x}}\right), x\right)\\ \mathbf{elif}\;x \leq 0.00106:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	13576

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \]

Alternative 3
Error	0.3
Cost	13576

\[\begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x \cdot -2 + \frac{-0.5}{x}}\right), x\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \]

Alternative 4
Error	11.2
Cost	13320

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

Alternative 5
Error	0.4
Cost	13320

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

Alternative 6
Error	22.5
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 7
Error	26.3
Cost	13060

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

Alternative 8
Error	30.4
Cost	6528

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

Reproduce

herbie shell --seed 2023083 
(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))