Rust f32::asinh

Percentage Accurate: 38.0% → 99.4%

Time: 10.3s

Precision: binary32

Cost: 35944

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.0010000000474974513:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

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 -0.10000000149011612)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.0010000000474974513)
       (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) 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 <= -0.10000000149011612f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.0010000000474974513f) {
		tmp = copysignf((x + (-0.16666666666666666f * powf(x, 3.0f))), 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(-0.10000000149011612))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.0010000000474974513))
		tmp = copysign(Float32(x + Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0)))), 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(-0.10000000149011612))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (t_0 <= single(0.0010000000474974513))
		tmp = sign(x) * abs((x + (single(-0.16666666666666666) * (x ^ single(3.0)))));
	else
		tmp = sign(x) * abs(log((x + hypot(single(1.0), x))));
	end
	tmp_2 = tmp;
end

Target

Original	38.0%
Target	99.6%
Herbie	99.4%

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

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) < -0.100000001

   1. Initial program 49.8%

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

   2. Applied egg-rr 12.3%

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

      [Start] 49.8	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      flip-+ [=>] 6.1	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]
      frac-2neg [=>] 6.1	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-\left(\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}\right)}{-\left(\left|x\right| - \sqrt{x \cdot x + 1}\right)}\right)}, x\right) \]
      log-div [=>] 6.1	\[ \mathsf{copysign}\left(\color{blue}{\log \left(-\left(\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}\right)\right) - \log \left(-\left(\left|x\right| - \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]
      sqr-abs [=>] 6.1	\[ \mathsf{copysign}\left(\log \left(-\left(\color{blue}{x \cdot x} - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}\right)\right) - \log \left(-\left(\left|x\right| - \sqrt{x \cdot x + 1}\right)\right), x\right) \]
      add-sqr-sqrt [<=] 6.2	\[ \mathsf{copysign}\left(\log \left(-\left(x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}\right)\right) - \log \left(-\left(\left|x\right| - \sqrt{x \cdot x + 1}\right)\right), x\right) \]
      fma-def [=>] 6.2	\[ \mathsf{copysign}\left(\log \left(-\left(x \cdot x - \color{blue}{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \log \left(-\left(\left|x\right| - \sqrt{x \cdot x + 1}\right)\right), x\right) \]
      add-sqr-sqrt [=>] -0.0	\[ \mathsf{copysign}\left(\log \left(-\left(x \cdot x - \mathsf{fma}\left(x, x, 1\right)\right)\right) - \log \left(-\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \sqrt{x \cdot x + 1}\right)\right), x\right) \]
      fabs-sqr [=>] -0.0	\[ \mathsf{copysign}\left(\log \left(-\left(x \cdot x - \mathsf{fma}\left(x, x, 1\right)\right)\right) - \log \left(-\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \sqrt{x \cdot x + 1}\right)\right), x\right) \]
      add-sqr-sqrt [<=] 12.3	\[ \mathsf{copysign}\left(\log \left(-\left(x \cdot x - \mathsf{fma}\left(x, x, 1\right)\right)\right) - \log \left(-\left(\color{blue}{x} - \sqrt{x \cdot x + 1}\right)\right), x\right) \]
      +-commutative [=>] 12.3	\[ \mathsf{copysign}\left(\log \left(-\left(x \cdot x - \mathsf{fma}\left(x, x, 1\right)\right)\right) - \log \left(-\left(x - \sqrt{\color{blue}{1 + x \cdot x}}\right)\right), x\right) \]

   3. Simplified 99.9%

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

      [Start] 12.3	\[ \mathsf{copysign}\left(\log \left(-\left(x \cdot x - \mathsf{fma}\left(x, x, 1\right)\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]
      neg-mul-1 [=>] 12.3	\[ \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot \left(x \cdot x - \mathsf{fma}\left(x, x, 1\right)\right)\right)} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]
      fma-udef [=>] 12.3	\[ \mathsf{copysign}\left(\log \left(-1 \cdot \left(x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]
      associate--r+ [=>] 46.3	\[ \mathsf{copysign}\left(\log \left(-1 \cdot \color{blue}{\left(\left(x \cdot x - x \cdot x\right) - 1\right)}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]
      +-inverses [=>] 99.9	\[ \mathsf{copysign}\left(\log \left(-1 \cdot \left(\color{blue}{0} - 1\right)\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]
      metadata-eval [=>] 99.9	\[ \mathsf{copysign}\left(\log \left(-1 \cdot \color{blue}{-1}\right) - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]
      metadata-eval [=>] 99.9	\[ \mathsf{copysign}\left(\log \color{blue}{1} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]
      metadata-eval [=>] 99.9	\[ \mathsf{copysign}\left(\color{blue}{0} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right), x\right) \]
      neg-sub0 [<=] 99.9	\[ \mathsf{copysign}\left(\color{blue}{-\log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]
      neg-sub0 [=>] 99.9	\[ \mathsf{copysign}\left(-\log \color{blue}{\left(0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]
      associate--r- [=>] 99.9	\[ \mathsf{copysign}\left(-\log \color{blue}{\left(\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]
      neg-sub0 [<=] 99.9	\[ \mathsf{copysign}\left(-\log \left(\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      +-commutative [<=] 99.9	\[ \mathsf{copysign}\left(-\log \color{blue}{\left(\mathsf{hypot}\left(1, x\right) + \left(-x\right)\right)}, x\right) \]
      sub-neg [<=] 99.9	\[ \mathsf{copysign}\left(-\log \color{blue}{\left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   if -0.100000001 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.00100000005

   1. Initial program 18.5%

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

   2. Applied egg-rr 18.7%

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

      [Start] 18.5	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      *-un-lft-identity [=>] 18.5	\[ \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]
      *-commutative [=>] 18.5	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]
      log-prod [=>] 18.5	\[ \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]
      add-sqr-sqrt [=>] 9.3	\[ \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]
      fabs-sqr [=>] 9.3	\[ \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]
      add-sqr-sqrt [<=] 18.6	\[ \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]
      +-commutative [=>] 18.6	\[ \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]
      hypot-1-def [=>] 18.7	\[ \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]
      metadata-eval [=>] 18.7	\[ \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Simplified 18.7%

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

      [Start] 18.7	\[ \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0, x\right) \]
      +-rgt-identity [=>] 18.7	\[ \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

   4. Taylor expanded in x around 0 100.0%

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

   if 0.00100000005 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 53.1%

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

   2. Applied egg-rr 99.9%

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

      [Start] 53.1	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      *-un-lft-identity [=>] 53.1	\[ \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]
      *-commutative [=>] 53.1	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]
      log-prod [=>] 53.1	\[ \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]
      add-sqr-sqrt [=>] 53.1	\[ \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]
      fabs-sqr [=>] 53.1	\[ \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]
      add-sqr-sqrt [<=] 53.1	\[ \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]
      +-commutative [=>] 53.1	\[ \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]
      hypot-1-def [=>] 99.9	\[ \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]
      metadata-eval [=>] 99.9	\[ \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Simplified 99.9%

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

      [Start] 99.9	\[ \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0, x\right) \]
      +-rgt-identity [=>] 99.9	\[ \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.0010000000474974513:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.3%
Cost	9896

\[\begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.0010000000474974513:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 2
Accuracy	98.0%
Cost	6792

\[\begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \]

Alternative 3
Accuracy	97.6%
Cost	6760

\[\begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 4
Accuracy	83.8%
Cost	6664

\[\begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\frac{x \cdot \left(x + 2\right)}{x + 2}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 5
Accuracy	96.9%
Cost	6664

\[\begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\frac{x \cdot \left(x + 2\right)}{x + 2}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 6
Accuracy	68.9%
Cost	6564

\[\begin{array}{l} \mathbf{if}\;x \leq -100:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 7
Accuracy	62.8%
Cost	6532

\[\begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 8
Accuracy	54.9%
Cost	3264

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

Reproduce

herbie shell --seed 2023163 
(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))