Rust f32::asinh

Average Accuracy: 36.2% → 99.5%

Time: 11.1s

Precision: binary32

Cost: 26052

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x}{\frac{1 + \mathsf{hypot}\left(1, x\right)}{x}}\right), x\right)\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))

↓

(FPCore (x)
 :precision binary32
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) -1.5)
   (copysign (log (+ (fabs x) (hypot 1.0 x))) x)
   (copysign (log1p (+ x (/ x (/ (+ 1.0 (hypot 1.0 x)) x)))) x)))

C

float code(float x) {
	return copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
}

↓

float code(float x) {
	float tmp;
	if (copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x) <= -1.5f) {
		tmp = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	} else {
		tmp = copysignf(log1pf((x + (x / ((1.0f + hypotf(1.0f, x)) / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	return copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
end

↓

function code(x)
	tmp = Float32(0.0)
	if (copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x) <= Float32(-1.5))
		tmp = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x);
	else
		tmp = copysign(log1p(Float32(x + Float32(x / Float32(Float32(Float32(1.0) + hypot(Float32(1.0), x)) / x)))), x);
	end
	return tmp
end

Target

Original	36.2%
Target	99.5%
Herbie	99.5%

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

Derivation

1. Split input into 2 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < -1.5

   1. Initial program 49.8%

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

   2. Simplified 99.3%

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

      [Start] 49.8	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      +-commutative [=>] 49.8	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]
      hypot-1-def [=>] 99.3	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   if -1.5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 31.7%

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

   2. Applied egg-rr 97.8%

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

      [Start] 31.7	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      log1p-expm1-u [=>] 31.7	\[ \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]
      expm1-udef [=>] 31.7	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]
      add-exp-log [<=] 31.7	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1\right), x\right) \]
      associate--l+ [=>] 81.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right| + \left(\sqrt{x \cdot x + 1} - 1\right)}\right), x\right) \]
      add-sqr-sqrt [=>] 48.9	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \left(\sqrt{x \cdot x + 1} - 1\right)\right), x\right) \]
      fabs-sqr [=>] 48.9	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \left(\sqrt{x \cdot x + 1} - 1\right)\right), x\right) \]
      add-sqr-sqrt [<=] 81.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x} + \left(\sqrt{x \cdot x + 1} - 1\right)\right), x\right) \]
      +-commutative [=>] 81.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\sqrt{\color{blue}{1 + x \cdot x}} - 1\right)\right), x\right) \]
      sqr-abs [<=] 81.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\sqrt{1 + \color{blue}{\left|x\right| \cdot \left|x\right|}} - 1\right)\right), x\right) \]
      hypot-1-def [=>] 97.8	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\color{blue}{\mathsf{hypot}\left(1, \left|x\right|\right)} - 1\right)\right), x\right) \]
      add-sqr-sqrt [=>] 65.8	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) - 1\right)\right), x\right) \]
      fabs-sqr [=>] 65.8	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) - 1\right)\right), x\right) \]
      add-sqr-sqrt [<=] 97.8	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, \color{blue}{x}\right) - 1\right)\right), x\right) \]

   3. Applied egg-rr 83.1%

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

      [Start] 97.8	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right), x\right) \]
      flip-- [=>] 81.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right) - 1 \cdot 1}{\mathsf{hypot}\left(1, x\right) + 1}}\right), x\right) \]
      div-inv [=>] 81.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right) - 1 \cdot 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}}\right), x\right) \]
      metadata-eval [=>] 81.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right) - \color{blue}{1}\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      hypot-udef [=>] 81.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right) - 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      hypot-udef [=>] 81.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} - 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      add-sqr-sqrt [<=] 81.5	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\color{blue}{\left(1 \cdot 1 + x \cdot x\right)} - 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      metadata-eval [=>] 81.5	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\left(\color{blue}{1} + x \cdot x\right) - 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      +-commutative [=>] 81.5	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\color{blue}{\left(x \cdot x + 1\right)} - 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      associate--l+ [=>] 83.1	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\left(x \cdot x + \left(1 - 1\right)\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      metadata-eval [=>] 83.1	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(x \cdot x + \color{blue}{0}\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      +-commutative [=>] 83.1	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(x \cdot x + 0\right) \cdot \frac{1}{\color{blue}{1 + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]

   4. Simplified 83.2%

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

      [Start] 83.1	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(x \cdot x + 0\right) \cdot \frac{1}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      associate-*r/ [=>] 83.2	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\frac{\left(x \cdot x + 0\right) \cdot 1}{1 + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]
      +-rgt-identity [=>] 83.2	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{\color{blue}{\left(x \cdot x\right)} \cdot 1}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      *-rgt-identity [=>] 83.2	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{\color{blue}{x \cdot x}}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   5. Applied egg-rr 97.6%

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

      [Start] 83.2	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      expm1-log1p-u [=>] 83.1	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot x}{1 + \mathsf{hypot}\left(1, x\right)}\right)\right)}\right), x\right) \]
      expm1-udef [=>] 81.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot x}{1 + \mathsf{hypot}\left(1, x\right)}\right)} - 1\right)}\right), x\right) \]
      associate-/l* [=>] 97.6	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\frac{1 + \mathsf{hypot}\left(1, x\right)}{x}}}\right)} - 1\right)\right), x\right) \]

   6. Simplified 99.6%

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

      [Start] 97.6	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(e^{\mathsf{log1p}\left(\frac{x}{\frac{1 + \mathsf{hypot}\left(1, x\right)}{x}}\right)} - 1\right)\right), x\right) \]
      expm1-def [=>] 99.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\frac{1 + \mathsf{hypot}\left(1, x\right)}{x}}\right)\right)}\right), x\right) \]
      expm1-log1p [=>] 99.6	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\frac{x}{\frac{1 + \mathsf{hypot}\left(1, x\right)}{x}}}\right), x\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x}{\frac{1 + \mathsf{hypot}\left(1, x\right)}{x}}\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.2%
Cost	10020

\[\begin{array}{l} \mathbf{if}\;x \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{1}{\frac{1}{\mathsf{hypot}\left(1, x\right) + -1}}\right), x\right)\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	10020

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

Alternative 3
Accuracy	98.8%
Cost	9896

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.009999999776482582:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 4
Accuracy	99.4%
Cost	9896

\[\begin{array}{l} \mathbf{if}\;x \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.009999999776482582:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 5
Accuracy	98.2%
Cost	9892

\[\begin{array}{l} \mathbf{if}\;x \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]

Alternative 6
Accuracy	98.3%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 7
Accuracy	98.1%
Cost	6760

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 8
Accuracy	83.9%
Cost	6664

\[\begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 9
Accuracy	97.3%
Cost	6664

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

Alternative 10
Accuracy	97.5%
Cost	6664

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

Alternative 11
Accuracy	97.4%
Cost	6664

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

Alternative 12
Accuracy	68.8%
Cost	6564

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

Alternative 13
Accuracy	62.4%
Cost	6532

\[\begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 14
Accuracy	54.1%
Cost	3264

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

Reproduce

herbie shell --seed 2023126 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :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))