Rust f32::asinh

Average Accuracy: 35.9% → 98.7%

Time: 11.5s

Precision: binary32

Cost: 9892

Math

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

↓

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

FPCore

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

↓

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

Target

Original	35.9%
Target	99.5%
Herbie	98.7%

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

Derivation

1. Split input into 3 regimes

2. if x < -0.5

   1. Initial program 50.9%

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

   2. Applied egg-rr 99.3%

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

      [Start] 50.9	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      flip-+ [=>] 8.4	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]
      div-inv [=>] 8.4	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]
      add-sqr-sqrt [<=] 8.1	\[ \mathsf{copysign}\left(\log \left(\left(\left|x\right| \cdot \left|x\right| - \color{blue}{\left(x \cdot x + 1\right)}\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]
      associate--r+ [=>] 8.2	\[ \mathsf{copysign}\left(\log \left(\color{blue}{\left(\left(\left|x\right| \cdot \left|x\right| - x \cdot x\right) - 1\right)} \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]
      sqr-abs [<=] 8.2	\[ \mathsf{copysign}\left(\log \left(\left(\left(\left|x\right| \cdot \left|x\right| - \color{blue}{\left|x\right| \cdot \left|x\right|}\right) - 1\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]
      distribute-lft-out-- [=>] 8.2	\[ \mathsf{copysign}\left(\log \left(\left(\color{blue}{\left|x\right| \cdot \left(\left|x\right| - \left|x\right|\right)} - 1\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]
      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 [<=] 8.2	\[ \mathsf{copysign}\left(\log \left(\left(\color{blue}{x} \cdot \left(\left|x\right| - \left|x\right|\right) - 1\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]
      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 [<=] 0.1	\[ \mathsf{copysign}\left(\log \left(\left(x \cdot \left(\color{blue}{x} - \left|x\right|\right) - 1\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]
      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 [<=] 8.2	\[ \mathsf{copysign}\left(\log \left(\left(x \cdot \left(x - \color{blue}{x}\right) - 1\right) \cdot \frac{1}{\left|x\right| - \sqrt{x \cdot x + 1}}\right), x\right) \]

   3. Simplified 99.3%

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

      [Start] 99.3	\[ \mathsf{copysign}\left(\log \left(\left(x \cdot \left(x - x\right) - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      +-inverses [=>] 99.3	\[ \mathsf{copysign}\left(\log \left(\left(x \cdot \color{blue}{0} - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      mul0-rgt [=>] 99.3	\[ \mathsf{copysign}\left(\log \left(\left(\color{blue}{0} - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      metadata-eval [=>] 99.3	\[ \mathsf{copysign}\left(\log \left(\color{blue}{-1} \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      associate-*r/ [=>] 99.3	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-1 \cdot 1}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
      metadata-eval [=>] 99.3	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      metadata-eval [<=] 99.3	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\frac{1}{-1}}}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      associate-/r* [<=] 99.3	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}, x\right) \]
      neg-mul-1 [<=] 99.3	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]
      neg-sub0 [=>] 99.3	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]
      associate--r- [=>] 99.3	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right), x\right) \]
      neg-sub0 [<=] 99.3	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      mul-1-neg [<=] 99.3	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{-1 \cdot x} + \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      +-commutative [<=] 99.3	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + -1 \cdot x}}\right), x\right) \]
      mul-1-neg [=>] 99.3	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) + \color{blue}{\left(-x\right)}}\right), x\right) \]
      sub-neg [<=] 99.3	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right), x\right) \]

   if -0.5 < x < 0.5

   1. Initial program 21.1%

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

   2. Applied egg-rr 97.3%

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

      [Start] 21.1	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      log1p-expm1-u [=>] 21.1	\[ \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]
      expm1-udef [=>] 21.1	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]
      add-exp-log [<=] 21.1	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1\right), x\right) \]
      associate--l+ [=>] 97.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right| + \left(\sqrt{x \cdot x + 1} - 1\right)}\right), x\right) \]
      add-sqr-sqrt [=>] 48.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \left(\sqrt{x \cdot x + 1} - 1\right)\right), x\right) \]
      fabs-sqr [=>] 48.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \left(\sqrt{x \cdot x + 1} - 1\right)\right), x\right) \]
      add-sqr-sqrt [<=] 97.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x} + \left(\sqrt{x \cdot x + 1} - 1\right)\right), x\right) \]
      +-commutative [=>] 97.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\sqrt{\color{blue}{1 + x \cdot x}} - 1\right)\right), x\right) \]
      sqr-abs [<=] 97.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\sqrt{1 + \color{blue}{\left|x\right| \cdot \left|x\right|}} - 1\right)\right), x\right) \]
      hypot-1-def [=>] 97.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\color{blue}{\mathsf{hypot}\left(1, \left|x\right|\right)} - 1\right)\right), x\right) \]
      add-sqr-sqrt [=>] 49.0	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) - 1\right)\right), x\right) \]
      fabs-sqr [=>] 49.0	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) - 1\right)\right), x\right) \]
      add-sqr-sqrt [<=] 97.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, \color{blue}{x}\right) - 1\right)\right), x\right) \]

   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] 97.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right), x\right) \]
      flip-- [=>] 97.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right) - 1 \cdot 1}{\mathsf{hypot}\left(1, x\right) + 1}}\right), x\right) \]
      div-inv [=>] 97.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \color{blue}{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right) - 1 \cdot 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}}\right), x\right) \]
      hypot-udef [=>] 97.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right) - 1 \cdot 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      hypot-udef [=>] 97.3	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} - 1 \cdot 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      add-sqr-sqrt [<=] 97.6	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\color{blue}{\left(1 \cdot 1 + x \cdot x\right)} - 1 \cdot 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      metadata-eval [=>] 97.6	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\left(\color{blue}{1} + x \cdot x\right) - 1 \cdot 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      +-commutative [=>] 97.6	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\color{blue}{\left(x \cdot x + 1\right)} - 1 \cdot 1\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      metadata-eval [=>] 97.6	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\left(x \cdot x + 1\right) - \color{blue}{1}\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right) + 1}\right), x\right) \]
      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 99.6%

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

   6. Simplified 99.6%

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

      [Start] 99.6	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + 0.5 \cdot {x}^{2}}\right), x\right) \]
      unpow2 [=>] 99.6	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + 0.5 \cdot \color{blue}{\left(x \cdot x\right)}}\right), x\right) \]

   if 0.5 < x

   1. Initial program 49.4%

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

   2. Applied egg-rr 99.0%

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

      [Start] 49.4	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      log1p-expm1-u [=>] 49.4	\[ \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]
      expm1-udef [=>] 49.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]
      add-exp-log [<=] 49.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1\right), x\right) \]
      associate--l+ [=>] 49.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right| + \left(\sqrt{x \cdot x + 1} - 1\right)}\right), x\right) \]
      add-sqr-sqrt [=>] 49.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \left(\sqrt{x \cdot x + 1} - 1\right)\right), x\right) \]
      fabs-sqr [=>] 49.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \left(\sqrt{x \cdot x + 1} - 1\right)\right), x\right) \]
      add-sqr-sqrt [<=] 49.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{x} + \left(\sqrt{x \cdot x + 1} - 1\right)\right), x\right) \]
      +-commutative [=>] 49.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\sqrt{\color{blue}{1 + x \cdot x}} - 1\right)\right), x\right) \]
      sqr-abs [<=] 49.4	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\sqrt{1 + \color{blue}{\left|x\right| \cdot \left|x\right|}} - 1\right)\right), x\right) \]
      hypot-1-def [=>] 99.0	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\color{blue}{\mathsf{hypot}\left(1, \left|x\right|\right)} - 1\right)\right), x\right) \]
      add-sqr-sqrt [=>] 99.0	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) - 1\right)\right), x\right) \]
      fabs-sqr [=>] 99.0	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) - 1\right)\right), x\right) \]
      add-sqr-sqrt [<=] 99.0	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, \color{blue}{x}\right) - 1\right)\right), x\right) \]

   3. Taylor expanded in x around inf 96.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)}, x\right) \]

   4. Simplified 96.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log 2 + \log x}, x\right) \]
      Proof

      [Start] 96.6	\[ \mathsf{copysign}\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right), x\right) \]
      mul-1-neg [=>] 96.6	\[ \mathsf{copysign}\left(\log 2 + \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)}, x\right) \]
      log-rec [=>] 96.6	\[ \mathsf{copysign}\left(\log 2 + \left(-\color{blue}{\left(-\log x\right)}\right), x\right) \]
      remove-double-neg [=>] 96.6	\[ \mathsf{copysign}\left(\log 2 + \color{blue}{\log x}, x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

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

Alternatives

Alternative 1
Accuracy	98.2%
Cost	22916

\[\begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -0.0006000000284984708:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right), x\right)\\ \end{array} \]

Alternative 2
Accuracy	97.9%
Cost	9864

\[\begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log 2 + \log x, x\right)\\ \end{array} \]

Alternative 3
Accuracy	98.2%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \frac{x \cdot x}{2 + \left(x \cdot x\right) \cdot 0.5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \]

Alternative 4
Accuracy	98.2%
Cost	6792

\[\begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \]

Alternative 5
Accuracy	97.9%
Cost	6760

\[\begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 6
Accuracy	83.7%
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 7
Accuracy	97.2%
Cost	6664

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

Alternative 8
Accuracy	68.8%
Cost	6564

\[\begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 9
Accuracy	62.3%
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 10
Accuracy	54.1%
Cost	3264

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

Reproduce

herbie shell --seed 2023133 
(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))