Rust f64::asinh

Average Error: 45.4 → 0.5

Time: 2.7s

Precision: binary64

Cost: 71880

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;t_0 \leq 10^{-11}:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

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 (* x -2.0))) x)
     (if (<= t_0 1e-11)
       (copysign x x)
       (copysign (log (+ x (hypot 1.0 x))) 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((x * -2.0)), x);
	} else if (t_0 <= 1e-11) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log((x + hypot(1.0, 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 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((x * -2.0)), x);
	} else if (t_0 <= 1e-11) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, 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):
	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((x * -2.0)), x)
	elif t_0 <= 1e-11:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, 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)
	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(Float64(-log(Float64(x * -2.0))), x);
	elseif (t_0 <= 1e-11)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, 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)
	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((x * -2.0)));
	elseif (t_0 <= 1e-11)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, 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_] := 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[(x * -2.0), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 1e-11], N[With[{TMP1 = Abs[x], 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]]]]

Target

Original	45.4
Target	0.0
Herbie	0.5

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

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 31.5

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

   2. Simplified 0.0

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

      [Start] 31.5	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      +-commutative [=>] 31.5	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]
      hypot-1-def [=>] 0.0	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   3. Taylor expanded in x around -inf 0.0

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

   4. Simplified 0.1

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

      [Start] 0.0	\[ \mathsf{copysign}\left(\log \left(\left(\left|x\right| + -1 \cdot x\right) - 0.5 \cdot \frac{1}{x}\right), x\right) \]
      rem-square-sqrt [<=] 64.0	\[ \mathsf{copysign}\left(\log \left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + -1 \cdot x\right) - 0.5 \cdot \frac{1}{x}\right), x\right) \]
      fabs-sqr [=>] 64.0	\[ \mathsf{copysign}\left(\log \left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + -1 \cdot x\right) - 0.5 \cdot \frac{1}{x}\right), x\right) \]
      rem-square-sqrt [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\left(\color{blue}{x} + -1 \cdot x\right) - 0.5 \cdot \frac{1}{x}\right), x\right) \]
      mul-1-neg [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\left(x + \color{blue}{\left(-x\right)}\right) - 0.5 \cdot \frac{1}{x}\right), x\right) \]
      sub-neg [<=] 0.1	\[ \mathsf{copysign}\left(\log \left(\color{blue}{\left(x - x\right)} - 0.5 \cdot \frac{1}{x}\right), x\right) \]
      +-inverses [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\color{blue}{0} - 0.5 \cdot \frac{1}{x}\right), x\right) \]
      neg-sub0 [<=] 0.1	\[ \mathsf{copysign}\left(\log \color{blue}{\left(-0.5 \cdot \frac{1}{x}\right)}, x\right) \]
      distribute-lft-neg-in [=>] 0.1	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\left(-0.5\right) \cdot \frac{1}{x}\right)}, x\right) \]
      associate-*r/ [=>] 0.1	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left(-0.5\right) \cdot 1}{x}\right)}, x\right) \]
      metadata-eval [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{-0.5} \cdot 1}{x}\right), x\right) \]
      metadata-eval [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{-0.5}}{x}\right), x\right) \]

   5. Applied egg-rr 0.1

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(x \cdot -2\right)}, x\right) \]

   6. Simplified 0.1

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

      [Start] 0.1	\[ \mathsf{copysign}\left(0 - \log \left(x \cdot -2\right), x\right) \]
      sub0-neg [=>] 0.1	\[ \mathsf{copysign}\left(\color{blue}{-\log \left(x \cdot -2\right)}, x\right) \]

   if -10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < 9.99999999999999939e-12

   1. Initial program 59.0

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

   2. Simplified 59.0

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

      [Start] 59.0	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      +-commutative [=>] 59.0	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]
      hypot-1-def [=>] 59.0	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   3. Taylor expanded in x around 0 58.7

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

   4. Simplified 0.6

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{x}{1 + x} \cdot x, \mathsf{log1p}\left(x\right)\right)}, x\right) \]
      Proof

      [Start] 58.7	\[ \mathsf{copysign}\left(0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \log \left(1 + \left|x\right|\right), x\right) \]
      fma-def [=>] 58.7	\[ \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \left|x\right|}, \log \left(1 + \left|x\right|\right)\right)}, x\right) \]
      unpow2 [=>] 58.7	\[ \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{\color{blue}{x \cdot x}}{1 + \left|x\right|}, \log \left(1 + \left|x\right|\right)\right), x\right) \]
      associate-/l* [=>] 58.7	\[ \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \color{blue}{\frac{x}{\frac{1 + \left|x\right|}{x}}}, \log \left(1 + \left|x\right|\right)\right), x\right) \]
      associate-/r/ [=>] 58.7	\[ \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \color{blue}{\frac{x}{1 + \left|x\right|} \cdot x}, \log \left(1 + \left|x\right|\right)\right), x\right) \]
      rem-square-sqrt [<=] 61.7	\[ \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{x}{1 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|} \cdot x, \log \left(1 + \left|x\right|\right)\right), x\right) \]
      fabs-sqr [=>] 61.7	\[ \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{x}{1 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot x, \log \left(1 + \left|x\right|\right)\right), x\right) \]
      rem-square-sqrt [=>] 58.8	\[ \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{x}{1 + \color{blue}{x}} \cdot x, \log \left(1 + \left|x\right|\right)\right), x\right) \]
      log1p-def [=>] 0.7	\[ \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{x}{1 + x} \cdot x, \color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}\right), x\right) \]
      rem-square-sqrt [<=] 32.8	\[ \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{x}{1 + x} \cdot x, \mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right), x\right) \]
      fabs-sqr [=>] 32.8	\[ \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{x}{1 + x} \cdot x, \mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right), x\right) \]
      rem-square-sqrt [=>] 0.6	\[ \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{x}{1 + x} \cdot x, \mathsf{log1p}\left(\color{blue}{x}\right)\right), x\right) \]

   5. Taylor expanded in x around 0 0.7

      \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \color{blue}{{x}^{2}}, \mathsf{log1p}\left(x\right)\right), x\right) \]

   6. Simplified 0.7

      \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \color{blue}{x \cdot x}, \mathsf{log1p}\left(x\right)\right), x\right) \]
      Proof

      [Start] 0.7	\[ \mathsf{copysign}\left(\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{log1p}\left(x\right)\right), x\right) \]
      unpow2 [=>] 0.7	\[ \mathsf{copysign}\left(\mathsf{fma}\left(0.5, \color{blue}{x \cdot x}, \mathsf{log1p}\left(x\right)\right), x\right) \]

   7. Taylor expanded in x around 0 0.6

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

   if 9.99999999999999939e-12 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x)

   1. Initial program 32.1

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

   2. Simplified 0.5

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

      [Start] 32.1	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      +-commutative [=>] 32.1	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]
      hypot-1-def [=>] 0.5	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   3. Applied egg-rr 0.5

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

   4. Simplified 0.5

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

      [Start] 0.5	\[ \mathsf{copysign}\left(0 + \log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      +-lft-identity [=>] 0.5	\[ \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 10^{-11}:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	13576

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \]

Alternative 2
Error	11.1
Cost	13320

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

Alternative 3
Error	0.5
Cost	13320

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

Alternative 4
Error	0.5
Cost	13320

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

Alternative 5
Error	22.4
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 6
Error	27.4
Cost	12928

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

Alternative 7
Error	30.4
Cost	6528

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

Reproduce

herbie shell --seed 2023045 
(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))