Rust f32::asinh

Average Accuracy: 36.1% → 99.4%

Time: 10.6s

Precision: binary32

Cost: 9896

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary32
 (if (<= x -0.20000000298023224)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 0.10000000149011612)
     (copysign (log1p (+ x (/ (* x x) (+ 2.0 (* (* x x) 0.5))))) 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 tmp;
	if (x <= -0.20000000298023224f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 0.10000000149011612f) {
		tmp = copysignf(log1pf((x + ((x * x) / (2.0f + ((x * x) * 0.5f))))), 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)
	tmp = Float32(0.0)
	if (x <= Float32(-0.20000000298023224))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.10000000149011612))
		tmp = copysign(log1p(Float32(x + Float32(Float32(x * x) / Float32(Float32(2.0) + Float32(Float32(x * x) * Float32(0.5)))))), x);
	else
		tmp = copysign(log(Float32(x + hypot(Float32(1.0), x))), x);
	end
	return tmp
end

Target

Original	36.1%
Target	99.5%
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 x < -0.200000003

   1. Initial program 51.4%

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

   2. Applied egg-rr 99.1%

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

      [Start] 51.4	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      flip-+ [=>] 9.0	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]
      div-inv [=>] 9.0	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]
      add-sqr-sqrt [<=] 8.9	\[ \mathsf{copysign}\left(\log \left(\left(\left|x\right| \cdot \left|x\right| - \color{blue}{\left(x \cdot x + 1\right)}\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]
      associate--r+ [=>] 8.9	\[ \mathsf{copysign}\left(\log \left(\color{blue}{\left(\left(\left|x\right| \cdot \left|x\right| - x \cdot x\right) - 1\right)} \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]
      sqr-abs [<=] 8.9	\[ \mathsf{copysign}\left(\log \left(\left(\left(\left|x\right| \cdot \left|x\right| - \color{blue}{\left|x\right| \cdot \left|x\right|}\right) - 1\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]
      distribute-lft-out-- [=>] 8.9	\[ \mathsf{copysign}\left(\log \left(\left(\color{blue}{\left|x\right| \cdot \left(\left|x\right| - \left|x\right|\right)} - 1\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]
      add-sqr-sqrt [=>] -0.0	\[ \mathsf{copysign}\left(\log \left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left(\left|x\right| - \left|x\right|\right) - 1\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]
      fabs-sqr [=>] -0.0	\[ \mathsf{copysign}\left(\log \left(\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left(\left|x\right| - \left|x\right|\right) - 1\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]
      add-sqr-sqrt [<=] 8.9	\[ \mathsf{copysign}\left(\log \left(\left(\color{blue}{x} \cdot \left(\left|x\right| - \left|x\right|\right) - 1\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]
      add-sqr-sqrt [=>] -0.0	\[ \mathsf{copysign}\left(\log \left(\left(x \cdot \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \left|x\right|\right) - 1\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]
      fabs-sqr [=>] -0.0	\[ \mathsf{copysign}\left(\log \left(\left(x \cdot \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \left|x\right|\right) - 1\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]
      add-sqr-sqrt [<=] 0.5	\[ \mathsf{copysign}\left(\log \left(\left(x \cdot \left(\color{blue}{x} - \left|x\right|\right) - 1\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]
      add-sqr-sqrt [=>] -0.0	\[ \mathsf{copysign}\left(\log \left(\left(x \cdot \left(x - \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) - 1\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]
      fabs-sqr [=>] -0.0	\[ \mathsf{copysign}\left(\log \left(\left(x \cdot \left(x - \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) - 1\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]
      add-sqr-sqrt [<=] 8.9	\[ \mathsf{copysign}\left(\log \left(\left(x \cdot \left(x - \color{blue}{x}\right) - 1\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]

   3. Simplified 99.1%

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

      [Start] 99.1	\[ \mathsf{copysign}\left(\log \left(\left(x \cdot \left(x - x\right) - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      *-commutative [=>] 99.1	\[ \mathsf{copysign}\left(\log \left(\left(\color{blue}{\left(x - x\right) \cdot x} - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      +-inverses [=>] 99.1	\[ \mathsf{copysign}\left(\log \left(\left(\color{blue}{0} \cdot x - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      mul0-lft [=>] 99.1	\[ \mathsf{copysign}\left(\log \left(\left(\color{blue}{0} - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      metadata-eval [=>] 99.1	\[ \mathsf{copysign}\left(\log \left(\color{blue}{-1} \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      associate-*r/ [=>] 99.1	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-1 \cdot 1}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
      metadata-eval [=>] 99.1	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      metadata-eval [<=] 99.1	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\frac{1}{-1}}}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      associate-/r* [<=] 99.1	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}, x\right) \]
      neg-mul-1 [<=] 99.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]
      neg-sub0 [=>] 99.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]
      associate--r- [=>] 99.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]
      neg-sub0 [<=] 99.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      mul-1-neg [<=] 99.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{-1 \cdot x} + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      +-commutative [<=] 99.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + -1 \cdot x}}\right), x\right) \]
      mul-1-neg [=>] 99.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) + \color{blue}{\left(-x\right)}}\right), x\right) \]
      sub-neg [<=] 99.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right), x\right) \]

   4. Applied egg-rr 99.2%

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

      [Start] 99.1	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right) \]
      log-div [=>] 99.2	\[ \mathsf{copysign}\left(\color{blue}{\log 1 - \log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]
      sub-neg [=>] 99.2	\[ \mathsf{copysign}\left(\color{blue}{\log 1 + \left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)}, x\right) \]
      metadata-eval [=>] 99.2	\[ \mathsf{copysign}\left(\color{blue}{0} + \left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right), x\right) \]

   5. Simplified 99.2%

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

      [Start] 99.2	\[ \mathsf{copysign}\left(0 + \left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right), x\right) \]
      +-lft-identity [=>] 99.2	\[ \mathsf{copysign}\left(\color{blue}{-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   if -0.200000003 < x < 0.100000001

   1. Initial program 20.2%

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

   2. Applied egg-rr 97.2%

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

      [Start] 20.2	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      log1p-expm1-u [=>] 20.2	\[ \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]
      expm1-udef [=>] 20.2	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]
      add-exp-log [<=] 20.2	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1\right), x\right) \]
      associate--l+ [=>] 97.2	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right| + \left(\sqrt{x \cdot x + 1} - 1\right)}\right), x\right) \]
      add-sqr-sqrt [=>] 47.8	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \left(\sqrt{x \cdot x + 1} - 1\right)\right), x\right) \]
      fabs-sqr [=>] 47.8	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \left(\sqrt{x \cdot x + 1} - 1\right)\right), x\right) \]
      add-sqr-sqrt [<=] 97.2	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x} + \left(\sqrt{x \cdot x + 1} - 1\right)\right), x\right) \]
      +-commutative [=>] 97.2	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\sqrt{\color{blue}{1 + x \cdot x}} - 1\right)\right), x\right) \]
      sqr-abs [<=] 97.2	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\sqrt{1 + \color{blue}{\left|x\right| \cdot \left|x\right|}} - 1\right)\right), x\right) \]
      hypot-1-def [=>] 97.2	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\color{blue}{\mathsf{hypot}\left(1, \left|x\right|\right)} - 1\right)\right), x\right) \]
      add-sqr-sqrt [=>] 48.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) - 1\right)\right), x\right) \]
      fabs-sqr [=>] 48.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) - 1\right)\right), x\right) \]
      add-sqr-sqrt [<=] 97.2	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, \color{blue}{x}\right) - 1\right)\right), x\right) \]

   3. Applied egg-rr 99.9%

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

      [Start] 97.2	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right), x\right) \]
      flip-- [=>] 97.2	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right) - 1 \cdot 1}{\mathsf{hypot}\left(1, x\right) + 1}}\right), x\right) \]
      div-inv [=>] 97.2	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right) - 1 \cdot 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}}\right), x\right) \]
      hypot-udef [=>] 97.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right) - 1 \cdot 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      hypot-udef [=>] 97.2	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} - 1 \cdot 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      add-sqr-sqrt [<=] 97.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\color{blue}{\left(1 \cdot 1 + x \cdot x\right)} - 1 \cdot 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      metadata-eval [=>] 97.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\left(\color{blue}{1} + x \cdot x\right) - 1 \cdot 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      +-commutative [=>] 97.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\color{blue}{\left(x \cdot x + 1\right)} - 1 \cdot 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      metadata-eval [=>] 97.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\left(x \cdot x + 1\right) - \color{blue}{1}\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      associate--l+ [=>] 99.9	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\left(x \cdot x + \left(1 - 1\right)\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      metadata-eval [=>] 99.9	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(x \cdot x + \color{blue}{0}\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      +-commutative [=>] 99.9	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(x \cdot x + 0\right) \cdot \frac{1}{\color{blue}{1 + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]

   4. Simplified 99.9%

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

      [Start] 99.9	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(x \cdot x + 0\right) \cdot \frac{1}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      associate-*r/ [=>] 99.9	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\frac{\left(x \cdot x + 0\right) \cdot 1}{1 + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]
      +-rgt-identity [=>] 99.9	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{\color{blue}{\left(x \cdot x\right)} \cdot 1}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      *-rgt-identity [=>] 99.9	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{\color{blue}{x \cdot x}}{1 + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   5. Taylor expanded in x around 0 99.9%

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

   6. Simplified 99.9%

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{\color{blue}{2 + 0.5 \cdot \left(x \cdot x\right)}}\right), x\right) \]
      Proof

      [Start] 99.9	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + 0.5 \cdot {x}^{2}}\right), x\right) \]
      unpow2 [=>] 99.9	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + 0.5 \cdot \color{blue}{\left(x \cdot x\right)}}\right), x\right) \]

   if 0.100000001 < x

   1. Initial program 51.6%

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

   2. Applied egg-rr 98.8%

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

      [Start] 51.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      *-un-lft-identity [=>] 51.6	\[ \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]
      log-prod [=>] 51.6	\[ \mathsf{copysign}\left(\color{blue}{\log 1 + \log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}, x\right) \]
      +-commutative [=>] 51.6	\[ \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]
      add-sqr-sqrt [=>] 51.6	\[ \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]
      fabs-sqr [=>] 51.6	\[ \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]
      add-sqr-sqrt [<=] 51.6	\[ \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]
      +-commutative [=>] 51.6	\[ \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]
      sqr-abs [<=] 51.6	\[ \mathsf{copysign}\left(\log \left(x + \sqrt{1 + \color{blue}{\left|x\right| \cdot \left|x\right|}}\right) + \log 1, x\right) \]
      hypot-1-def [=>] 98.8	\[ \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, \left|x\right|\right)}\right) + \log 1, x\right) \]
      add-sqr-sqrt [=>] 98.8	\[ \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right) + \log 1, x\right) \]
      fabs-sqr [=>] 98.8	\[ \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right) + \log 1, x\right) \]
      add-sqr-sqrt [<=] 98.8	\[ \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, \color{blue}{x}\right)\right) + \log 1, x\right) \]
      metadata-eval [=>] 98.8	\[ \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Simplified 98.8%

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

      [Start] 98.8	\[ \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0, x\right) \]
      +-rgt-identity [=>] 98.8	\[ \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.20000000298023224:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.1%
Cost	22916

\[\begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.8700000047683716:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]

Alternative 2
Accuracy	98.2%
Cost	9896

\[\begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x \cdot -2 + \frac{-0.5}{x}}\right), x\right)\\ \mathbf{elif}\;x \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 3
Accuracy	97.7%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x \cdot -2 + \frac{-0.5}{x}}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \]

Alternative 4
Accuracy	97.8%
Cost	6792

\[\begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \]

Alternative 5
Accuracy	98.3%
Cost	6792

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

Alternative 6
Accuracy	97.5%
Cost	6760

\[\begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x \cdot 2 + -1\right), x\right)\\ \end{array} \]

Alternative 7
Accuracy	96.9%
Cost	6728

\[\begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x \cdot 2 + -1\right), x\right)\\ \end{array} \]

Alternative 8
Accuracy	84.2%
Cost	6664

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

Alternative 9
Accuracy	96.9%
Cost	6664

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

Alternative 10
Accuracy	68.9%
Cost	6564

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

Alternative 11
Accuracy	63.2%
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 12
Accuracy	54.8%
Cost	3264

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

Reproduce

herbie shell --seed 2023131 
(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))