Rust f32::asinh

Average Accuracy: 38.0% → 61.0%

Time: 8.6s

Precision: binary32

Cost: 39528

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq 0.2199999988079071:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 0.2199999988079071)
     (copysign (log (+ (fabs x) (hypot 1.0 x))) x)
     (if (<= t_0 0.20000000298023224)
       (copysign
        (+
         (* -0.16666666666666666 (pow x 3.0))
         (+ (* 0.075 (pow x 5.0)) (+ x (* -0.044642857142857144 (pow x 7.0)))))
        x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

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

↓

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= 0.2199999988079071f) {
		tmp = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = copysignf(((-0.16666666666666666f * powf(x, 3.0f)) + ((0.075f * powf(x, 5.0f)) + (x + (-0.044642857142857144f * powf(x, 7.0f))))), x);
	} else {
		tmp = copysignf(logf((x + hypotf(1.0f, 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)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.2199999988079071))
		tmp = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x);
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = copysign(Float32(Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0))) + Float32(Float32(Float32(0.075) * (x ^ Float32(5.0))) + Float32(x + Float32(Float32(-0.044642857142857144) * (x ^ Float32(7.0)))))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
	tmp = single(0.0);
	if (t_0 <= single(0.2199999988079071))
		tmp = sign(x) * abs(log((abs(x) + hypot(single(1.0), x))));
	elseif (t_0 <= single(0.20000000298023224))
		tmp = sign(x) * abs(((single(-0.16666666666666666) * (x ^ single(3.0))) + ((single(0.075) * (x ^ single(5.0))) + (x + (single(-0.044642857142857144) * (x ^ single(7.0)))))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Target

Original	38.0%
Target	99.5%
Herbie	61.0%

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

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.219999999

   1. Initial program 33.4%

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

   2. Simplified 51.1%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
      Step-by-step derivation

      [Start] 33.4	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      +-commutative [=>] 33.4	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]
      hypot-1-def [=>] 51.1	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   if 0.219999999 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.200000003

   1. Initial program 39.1%

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

   2. Simplified 64.3%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
      Step-by-step derivation

      [Start] 39.1	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      +-commutative [=>] 39.1	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]
      hypot-1-def [=>] 64.3	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   3. Applied egg-rr 41.4%

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

      [Start] 64.3	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      *-un-lft-identity [=>] 64.3	\[ \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]
      log-prod [=>] 64.3	\[ \mathsf{copysign}\left(\color{blue}{\log 1 + \log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]
      metadata-eval [=>] 64.3	\[ \mathsf{copysign}\left(\color{blue}{0} + \log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      *-un-lft-identity [=>] 64.3	\[ \mathsf{copysign}\left(0 + \log \color{blue}{\left(1 \cdot \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]
      *-un-lft-identity [<=] 64.3	\[ \mathsf{copysign}\left(0 + \log \color{blue}{\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]
      add-sqr-sqrt [=>] 33.1	\[ \mathsf{copysign}\left(0 + \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      fabs-sqr [=>] 33.1	\[ \mathsf{copysign}\left(0 + \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      add-sqr-sqrt [<=] 41.4	\[ \mathsf{copysign}\left(0 + \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]

   4. Simplified 41.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]
      Step-by-step derivation

      [Start] 41.4	\[ \mathsf{copysign}\left(0 + \log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      +-lft-identity [=>] 41.4	\[ \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

   5. Taylor expanded in x around 0 48.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right)}, x\right) \]

   if 0.200000003 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 53.8%

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

   2. Simplified 98.6%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
      Step-by-step derivation

      [Start] 53.8	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      +-commutative [=>] 53.8	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]
      hypot-1-def [=>] 98.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   3. Applied egg-rr 98.6%

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

      [Start] 98.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      *-un-lft-identity [=>] 98.6	\[ \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]
      log-prod [=>] 98.6	\[ \mathsf{copysign}\left(\color{blue}{\log 1 + \log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]
      metadata-eval [=>] 98.6	\[ \mathsf{copysign}\left(\color{blue}{0} + \log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      *-un-lft-identity [=>] 98.6	\[ \mathsf{copysign}\left(0 + \log \color{blue}{\left(1 \cdot \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]
      *-un-lft-identity [<=] 98.6	\[ \mathsf{copysign}\left(0 + \log \color{blue}{\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]
      add-sqr-sqrt [=>] 98.5	\[ \mathsf{copysign}\left(0 + \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      fabs-sqr [=>] 98.5	\[ \mathsf{copysign}\left(0 + \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      add-sqr-sqrt [<=] 98.6	\[ \mathsf{copysign}\left(0 + \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]

   4. Simplified 98.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]
      Step-by-step derivation

      [Start] 98.6	\[ \mathsf{copysign}\left(0 + \log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      +-lft-identity [=>] 98.6	\[ \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.2199999988079071:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	61.0%
Cost	36168

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 2
Accuracy	60.9%
Cost	10120

\[\begin{array}{l} \mathbf{if}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(-1 + \left(1 + \log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 3
Accuracy	60.9%
Cost	10020

\[\begin{array}{l} \mathbf{if}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(-1 + \left(1 + \log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 4
Accuracy	51.7%
Cost	9896

\[\begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 5
Accuracy	61.0%
Cost	9896

\[\begin{array}{l} \mathbf{if}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 6
Accuracy	51.3%
Cost	6792

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

Alternative 7
Accuracy	37.7%
Cost	6664

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

Alternative 8
Accuracy	51.2%
Cost	6664

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

Alternative 9
Accuracy	51.3%
Cost	6664

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

Alternative 10
Accuracy	51.1%
Cost	6664

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

Alternative 11
Accuracy	24.7%
Cost	6564

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

Alternative 12
Accuracy	62.5%
Cost	6532

\[\begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 13
Accuracy	54.4%
Cost	3264

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

Reproduce

herbie shell --seed 2023159 
(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))