Rust f32::asinh

Percentage Accurate: 37.1% → 99.2%

Time: 8.7s

Precision: binary32

Cost: 59016

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \left|x\right| + 1\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(-0.041666666666666664, \left(\frac{3}{t_1} + \frac{3}{{t_1}^{2}}\right) \cdot {x}^{4}, \mathsf{fma}\left(0.5, \frac{x \cdot x}{t_1}, \mathsf{log1p}\left(\left|x\right|\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + \left(x + \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \]

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))
        (t_1 (+ (fabs x) 1.0)))
   (if (<= t_0 -1.0)
     (copysign (log (+ (fabs x) (hypot 1.0 x))) x)
     (if (<= t_0 0.004999999888241291)
       (copysign
        (fma
         -0.041666666666666664
         (* (+ (/ 3.0 t_1) (/ 3.0 (pow t_1 2.0))) (pow x 4.0))
         (fma 0.5 (/ (* x x) t_1) (log1p (fabs x))))
        x)
       (copysign (log1p (+ -1.0 (+ 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 t_1 = fabsf(x) + 1.0f;
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = copysignf(logf((fabsf(x) + hypotf(1.0f, x))), x);
	} else if (t_0 <= 0.004999999888241291f) {
		tmp = copysignf(fmaf(-0.041666666666666664f, (((3.0f / t_1) + (3.0f / powf(t_1, 2.0f))) * powf(x, 4.0f)), fmaf(0.5f, ((x * x) / t_1), log1pf(fabsf(x)))), x);
	} else {
		tmp = copysignf(log1pf((-1.0f + (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)
	t_1 = Float32(abs(x) + Float32(1.0))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = copysign(log(Float32(abs(x) + hypot(Float32(1.0), x))), x);
	elseif (t_0 <= Float32(0.004999999888241291))
		tmp = copysign(fma(Float32(-0.041666666666666664), Float32(Float32(Float32(Float32(3.0) / t_1) + Float32(Float32(3.0) / (t_1 ^ Float32(2.0)))) * (x ^ Float32(4.0))), fma(Float32(0.5), Float32(Float32(x * x) / t_1), log1p(abs(x)))), x);
	else
		tmp = copysign(log1p(Float32(Float32(-1.0) + Float32(x + hypot(Float32(1.0), x)))), x);
	end
	return tmp
end

Target

Original	37.1%
Target	99.5%
Herbie	99.2%

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

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) < -1

   1. Initial program 63.5%

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

   2. Simplified 99.9%

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

      [Start] 63.5%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      +-commutative [=>] 63.5%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]
      hypot-1-def [=>] 99.9%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.00499999989

   1. Initial program 17.4%

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

   2. Simplified 17.5%

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

      [Start] 17.4%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      +-commutative [=>] 17.4%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]
      hypot-1-def [=>] 17.5%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   3. Taylor expanded in x around 0 18.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.041666666666666664 \cdot \left(\left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right) \cdot {x}^{4}\right) + \left(0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \log \left(1 + \left|x\right|\right)\right)}, x\right) \]

   4. Simplified 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(-0.041666666666666664, \left(\frac{3}{1 + \left|x\right|} + \frac{3}{{\left(1 + \left|x\right|\right)}^{2}}\right) \cdot {x}^{4}, \mathsf{fma}\left(0.5, \frac{x \cdot x}{1 + \left|x\right|}, \mathsf{log1p}\left(\left|x\right|\right)\right)\right)}, x\right) \]
      Step-by-step derivation

      [Start] 18.5%	\[ \mathsf{copysign}\left(-0.041666666666666664 \cdot \left(\left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right) \cdot {x}^{4}\right) + \left(0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \log \left(1 + \left|x\right|\right)\right), x\right) \]
      fma-def [=>] 18.5%	\[ \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(-0.041666666666666664, \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right) \cdot {x}^{4}, 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \log \left(1 + \left|x\right|\right)\right)}, x\right) \]
      associate-*r/ [=>] 18.5%	\[ \mathsf{copysign}\left(\mathsf{fma}\left(-0.041666666666666664, \left(\color{blue}{\frac{3 \cdot 1}{1 + \left|x\right|}} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right) \cdot {x}^{4}, 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \log \left(1 + \left|x\right|\right)\right), x\right) \]
      metadata-eval [=>] 18.5%	\[ \mathsf{copysign}\left(\mathsf{fma}\left(-0.041666666666666664, \left(\frac{\color{blue}{3}}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right) \cdot {x}^{4}, 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \log \left(1 + \left|x\right|\right)\right), x\right) \]
      associate-*r/ [=>] 18.5%	\[ \mathsf{copysign}\left(\mathsf{fma}\left(-0.041666666666666664, \left(\frac{3}{1 + \left|x\right|} + \color{blue}{\frac{3 \cdot 1}{{\left(1 + \left|x\right|\right)}^{2}}}\right) \cdot {x}^{4}, 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \log \left(1 + \left|x\right|\right)\right), x\right) \]
      metadata-eval [=>] 18.5%	\[ \mathsf{copysign}\left(\mathsf{fma}\left(-0.041666666666666664, \left(\frac{3}{1 + \left|x\right|} + \frac{\color{blue}{3}}{{\left(1 + \left|x\right|\right)}^{2}}\right) \cdot {x}^{4}, 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \log \left(1 + \left|x\right|\right)\right), x\right) \]
      fma-def [=>] 18.5%	\[ \mathsf{copysign}\left(\mathsf{fma}\left(-0.041666666666666664, \left(\frac{3}{1 + \left|x\right|} + \frac{3}{{\left(1 + \left|x\right|\right)}^{2}}\right) \cdot {x}^{4}, \color{blue}{\mathsf{fma}\left(0.5, \frac{{x}^{2}}{1 + \left|x\right|}, \log \left(1 + \left|x\right|\right)\right)}\right), x\right) \]
      unpow2 [=>] 18.5%	\[ \mathsf{copysign}\left(\mathsf{fma}\left(-0.041666666666666664, \left(\frac{3}{1 + \left|x\right|} + \frac{3}{{\left(1 + \left|x\right|\right)}^{2}}\right) \cdot {x}^{4}, \mathsf{fma}\left(0.5, \frac{\color{blue}{x \cdot x}}{1 + \left|x\right|}, \log \left(1 + \left|x\right|\right)\right)\right), x\right) \]
      log1p-def [=>] 99.9%	\[ \mathsf{copysign}\left(\mathsf{fma}\left(-0.041666666666666664, \left(\frac{3}{1 + \left|x\right|} + \frac{3}{{\left(1 + \left|x\right|\right)}^{2}}\right) \cdot {x}^{4}, \mathsf{fma}\left(0.5, \frac{x \cdot x}{1 + \left|x\right|}, \color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}\right)\right), x\right) \]

   if 0.00499999989 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 45.6%

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

   2. Simplified 96.9%

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

      [Start] 45.6%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      +-commutative [=>] 45.6%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]
      hypot-1-def [=>] 96.9%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   3. Applied egg-rr 96.9%

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

      [Start] 96.9%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      log1p-expm1-u [=>] 96.9%	\[ \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)\right)}, x\right) \]
      expm1-udef [=>] 96.9%	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)} - 1}\right), x\right) \]
      add-exp-log [<=] 96.9%	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)} - 1\right), x\right) \]
      add-sqr-sqrt [=>] 97.0%	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right) - 1\right), x\right) \]
      fabs-sqr [=>] 97.0%	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right) - 1\right), x\right) \]
      add-sqr-sqrt [<=] 96.9%	\[ \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right) - 1\right), x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(-0.041666666666666664, \left(\frac{3}{\left|x\right| + 1} + \frac{3}{{\left(\left|x\right| + 1\right)}^{2}}\right) \cdot {x}^{4}, \mathsf{fma}\left(0.5, \frac{x \cdot x}{\left|x\right| + 1}, \mathsf{log1p}\left(\left|x\right|\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + \left(x + \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	59016

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \left|x\right| + 1\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(-0.041666666666666664, \left(\frac{3}{t_1} + \frac{3}{{t_1}^{2}}\right) \cdot {x}^{4}, \mathsf{fma}\left(0.5, \frac{x \cdot x}{t_1}, \mathsf{log1p}\left(\left|x\right|\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + \left(x + \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \]

Alternative 2
Accuracy	99.1%
Cost	46152

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left({x}^{4}, \frac{-0.125}{x + 1} + \frac{-0.125}{{\left(x + 1\right)}^{2}}, \mathsf{fma}\left(0.5, \frac{x \cdot x}{x + 1}, \mathsf{log1p}\left(x\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + \left(x + \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	42504

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{x \cdot x}{\left|x\right| + 1}, \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + \left(x + \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \]

Alternative 4
Accuracy	99.4%
Cost	36104

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.009999999776482582:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{x \cdot x}{x + 1}, \mathsf{log1p}\left(x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + \left(x + \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \]

Alternative 5
Accuracy	99.4%
Cost	10056

\[\begin{array}{l} \mathbf{if}\;x \leq -0.019999999552965164:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(0.5, \frac{x \cdot x}{x + 1}, \mathsf{log1p}\left(x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + \left(x + \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \]

Alternative 6
Accuracy	98.1%
Cost	9960

\[\begin{array}{l} \mathbf{if}\;x \leq -0.009999999776482582:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 3.9999998989515007 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + \left(x + \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \]

Alternative 7
Accuracy	96.8%
Cost	9896

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 3.9999998989515007 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 8
Accuracy	98.1%
Cost	9896

\[\begin{array}{l} \mathbf{if}\;x \leq -0.009999999776482582:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 3.9999998989515007 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 9
Accuracy	95.8%
Cost	9800

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]

Alternative 10
Accuracy	95.5%
Cost	9800

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]

Alternative 11
Accuracy	95.7%
Cost	6792

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]

Alternative 12
Accuracy	81.8%
Cost	6664

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 13
Accuracy	95.4%
Cost	6664

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 14
Accuracy	68.8%
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 15
Accuracy	5.5%
Cost	6464

\[\mathsf{copysign}\left(\log 0, x\right) \]

Alternative 16
Accuracy	57.6%
Cost	6464

\[\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right) \]

Reproduce

herbie shell --seed 2023277 
(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))