Rust f32::asinh

Average Error: 20.4 → 0.1

Time: 9.4s

Precision: binary32

Cost: 26180

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -4.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, \frac{x}{1 + \mathsf{hypot}\left(1, x\right)}, x\right)\right), x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary32
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) -4.5)
   (copysign (log (/ -0.5 x)) x)
   (copysign (log1p (fma 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) <= -4.5f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else {
		tmp = copysignf(log1pf(fmaf(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(-4.5))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	else
		tmp = copysign(log1p(fma(x, Float32(x / Float32(Float32(1.0) + hypot(Float32(1.0), x))), x)), x);
	end
	return tmp
end

Target

Original	20.4
Target	0.1
Herbie	0.1

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

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) < -4.5

   1. Initial program 16.7

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

   2. Simplified 0.2

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

      [Start] 16.7	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      +-commutative [=>] 16.7	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]
      hypot-1-def [=>] 0.2	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   3. Taylor expanded in x around -inf 0.2

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

   4. Simplified 0.1

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

      [Start] 0.2	\[ \mathsf{copysign}\left(\log \left(\left(\left|x\right| + -1 \cdot x\right) - 0.5 \cdot \frac{1}{x}\right), x\right) \]
      rem-square-sqrt [<=] 32.0	\[ \mathsf{copysign}\left(\log \left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + -1 \cdot x\right) - 0.5 \cdot \frac{1}{x}\right), x\right) \]
      fabs-sqr [=>] 32.0	\[ \mathsf{copysign}\left(\log \left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + -1 \cdot x\right) - 0.5 \cdot \frac{1}{x}\right), x\right) \]
      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	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{-0.5}}{x}\right), x\right) \]

   if -4.5 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 21.5

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

   2. Applied egg-rr 0.7

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

   3. Applied egg-rr 5.3

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

   4. Simplified 5.3

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

      [Start] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(x \cdot x + 0\right) \cdot \frac{1}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      associate-*r/ [=>] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\frac{\left(x \cdot x + 0\right) \cdot 1}{1 + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]
      +-rgt-identity [=>] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{\color{blue}{\left(x \cdot x\right)} \cdot 1}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      *-rgt-identity [=>] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{\color{blue}{x \cdot x}}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   5. Applied egg-rr 16.3

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

   6. Simplified 0.1

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

      [Start] 16.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\frac{x}{\frac{1 + \mathsf{hypot}\left(1, x\right)}{x}} + \left(x + 1\right)\right) - 1\right), x\right) \]
      associate--l+ [=>] 16.1	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\frac{x}{\frac{1 + \mathsf{hypot}\left(1, x\right)}{x}} + \left(\left(x + 1\right) - 1\right)}\right), x\right) \]
      +-commutative [=>] 16.1	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(\left(x + 1\right) - 1\right) + \frac{x}{\frac{1 + \mathsf{hypot}\left(1, x\right)}{x}}}\right), x\right) \]
      associate--l+ [=>] 0.1	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(x + \left(1 - 1\right)\right)} + \frac{x}{\frac{1 + \mathsf{hypot}\left(1, x\right)}{x}}\right), x\right) \]
      metadata-eval [=>] 0.1	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + \color{blue}{0}\right) + \frac{x}{\frac{1 + \mathsf{hypot}\left(1, x\right)}{x}}\right), x\right) \]
      associate-/l* [<=] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + 0\right) + \color{blue}{\frac{x \cdot x}{1 + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]
      metadata-eval [<=] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + 0\right) + \frac{x \cdot x}{\color{blue}{\left(2 + -1\right)} + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      associate-+r+ [<=] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + 0\right) + \frac{x \cdot x}{\color{blue}{2 + \left(-1 + \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]
      metadata-eval [<=] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + 0\right) + \frac{x \cdot x}{2 + \left(\color{blue}{\left(0 - 1\right)} + \mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]
      associate--r- [<=] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + 0\right) + \frac{x \cdot x}{2 + \color{blue}{\left(0 - \left(1 - \mathsf{hypot}\left(1, x\right)\right)\right)}}\right), x\right) \]
      neg-sub0 [<=] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + 0\right) + \frac{x \cdot x}{2 + \color{blue}{\left(-\left(1 - \mathsf{hypot}\left(1, x\right)\right)\right)}}\right), x\right) \]
      neg-mul-1 [=>] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + 0\right) + \frac{x \cdot x}{2 + \color{blue}{-1 \cdot \left(1 - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]
      sub-neg [=>] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + 0\right) + \frac{x \cdot x}{2 + -1 \cdot \color{blue}{\left(1 + \left(-\mathsf{hypot}\left(1, x\right)\right)\right)}}\right), x\right) \]
      distribute-lft-in [=>] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + 0\right) + \frac{x \cdot x}{2 + \color{blue}{\left(-1 \cdot 1 + -1 \cdot \left(-\mathsf{hypot}\left(1, x\right)\right)\right)}}\right), x\right) \]
      metadata-eval [=>] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + 0\right) + \frac{x \cdot x}{2 + \left(\color{blue}{-1} + -1 \cdot \left(-\mathsf{hypot}\left(1, x\right)\right)\right)}\right), x\right) \]
      associate-+r+ [=>] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + 0\right) + \frac{x \cdot x}{\color{blue}{\left(2 + -1\right) + -1 \cdot \left(-\mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]
      metadata-eval [=>] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + 0\right) + \frac{x \cdot x}{\color{blue}{1} + -1 \cdot \left(-\mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]
      metadata-eval [<=] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + 0\right) + \frac{x \cdot x}{\color{blue}{-1 \cdot -1} + -1 \cdot \left(-\mathsf{hypot}\left(1, x\right)\right)}\right), x\right) \]
      distribute-lft-in [<=] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + 0\right) + \frac{x \cdot x}{\color{blue}{-1 \cdot \left(-1 + \left(-\mathsf{hypot}\left(1, x\right)\right)\right)}}\right), x\right) \]
      metadata-eval [<=] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + 0\right) + \frac{x \cdot x}{\color{blue}{\frac{1}{-1}} \cdot \left(-1 + \left(-\mathsf{hypot}\left(1, x\right)\right)\right)}\right), x\right) \]
      sub-neg [<=] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + 0\right) + \frac{x \cdot x}{\frac{1}{-1} \cdot \color{blue}{\left(-1 - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]
      associate-/r/ [<=] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + 0\right) + \frac{x \cdot x}{\color{blue}{\frac{1}{\frac{-1}{-1 - \mathsf{hypot}\left(1, x\right)}}}}\right), x\right) \]
      associate-/l* [<=] 5.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + 0\right) + \color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{-1 - \mathsf{hypot}\left(1, x\right)}}{1}}\right), x\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -4.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, \frac{x}{1 + \mathsf{hypot}\left(1, x\right)}, x\right)\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	23044

\[\begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -4.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + x \cdot \frac{x}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	10020

\[\begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x \cdot \left(1 + \frac{x}{1 + \mathsf{hypot}\left(1, x\right)}\right)\right), x\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	9896

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 4
Error	0.2
Cost	9896

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 5
Error	0.6
Cost	9892

\[\begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]

Alternative 6
Error	0.5
Cost	6984

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

Alternative 7
Error	0.5
Cost	6792

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

Alternative 8
Error	0.6
Cost	6760

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

Alternative 9
Error	5.2
Cost	6664

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

Alternative 10
Error	0.9
Cost	6664

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

Alternative 11
Error	10.0
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 12
Error	12.1
Cost	6532

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

Alternative 13
Error	14.8
Cost	3264

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

Reproduce

herbie shell --seed 2023066 
(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))