Rust f32::asinh

Average Error: 20.6 → 12.7

Time: 44.8s

Precision: binary32

Cost: 13512

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -19999999961012896000:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\frac{x \cdot x - \left(-1 - 3 \cdot \left(1 + x \cdot x\right)\right)}{4}}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary32
 (if (<= x -19999999961012896000.0)
   (copysign (log (- (fabs x) x)) x)
   (if (<= x 500.0)
     (copysign
      (log
       (+
        (fabs x)
        (sqrt (/ (- (* x x) (- -1.0 (* 3.0 (+ 1.0 (* x x))))) 4.0))))
      x)
     (copysign (log (/ 0.5 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 <= -19999999961012896000.0f) {
		tmp = copysignf(logf((fabsf(x) - x)), x);
	} else if (x <= 500.0f) {
		tmp = copysignf(logf((fabsf(x) + sqrtf((((x * x) - (-1.0f - (3.0f * (1.0f + (x * x))))) / 4.0f)))), x);
	} else {
		tmp = copysignf(logf((0.5f / 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(-19999999961012896000.0))
		tmp = copysign(log(Float32(abs(x) - x)), x);
	elseif (x <= Float32(500.0))
		tmp = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(Float32(x * x) - Float32(Float32(-1.0) - Float32(Float32(3.0) * Float32(Float32(1.0) + Float32(x * x))))) / Float32(4.0))))), x);
	else
		tmp = copysign(log(Float32(Float32(0.5) / 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)
	tmp = single(0.0);
	if (x <= single(-19999999961012896000.0))
		tmp = sign(x) * abs(log((abs(x) - x)));
	elseif (x <= single(500.0))
		tmp = sign(x) * abs(log((abs(x) + sqrt((((x * x) - (single(-1.0) - (single(3.0) * (single(1.0) + (x * x))))) / single(4.0))))));
	else
		tmp = sign(x) * abs(log((single(0.5) / x)));
	end
	tmp_2 = tmp;
end

Target

Original	20.6
Target	0.1
Herbie	12.7

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

Derivation

1. Split input into 3 regimes

2. if x < -2e19

   1. Initial program 32.0

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

   2. Taylor expanded in x around -inf 0.4

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

   3. Simplified 0.4

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

      [Start] 0.4	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + -1 \cdot x\right), x\right) \]
      rational_best-simplify-1 [=>] 0.4	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{x \cdot -1}\right), x\right) \]
      rational_best-simplify-10 [=>] 0.4	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(-x\right)}\right), x\right) \]

   4. Applied egg-rr 0.4

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

   5. Simplified 0.4

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

      [Start] 0.4	\[ \mathsf{copysign}\left(\log \left(\left|x\right| - x\right) + 0, x\right) \]
      rational_best-simplify-3 [<=] 0.4	\[ \mathsf{copysign}\left(\color{blue}{0 + \log \left(\left|x\right| - x\right)}, x\right) \]
      rational_best-simplify-6 [=>] 0.4	\[ \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| - x\right)}, x\right) \]

   if -2e19 < x < 500

   1. Initial program 19.5

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

   2. Applied egg-rr 19.7

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

   3. Simplified 19.7

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

      [Start] 19.7	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\frac{\left(x \cdot x + 1\right) \cdot 3}{4} - \frac{-1 - x \cdot x}{4}}\right), x\right) \]
      rational_best-simplify-66 [=>] 19.7	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{\frac{\left(x \cdot x + 1\right) \cdot 3 - \left(-1 - x \cdot x\right)}{4}}}\right), x\right) \]
      rational_best-simplify-51 [=>] 19.7	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\frac{\color{blue}{x \cdot x - \left(-1 - \left(x \cdot x + 1\right) \cdot 3\right)}}{4}}\right), x\right) \]
      rational_best-simplify-1 [=>] 19.7	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\frac{x \cdot x - \left(-1 - \color{blue}{3 \cdot \left(x \cdot x + 1\right)}\right)}{4}}\right), x\right) \]
      rational_best-simplify-3 [=>] 19.7	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\frac{x \cdot x - \left(-1 - 3 \cdot \color{blue}{\left(1 + x \cdot x\right)}\right)}{4}}\right), x\right) \]

   if 500 < x

   1. Initial program 17.2

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

   2. Taylor expanded in x around inf 0.3

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

   3. Simplified 0.3

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

      [Start] 0.3	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(0.5 \cdot \frac{1}{x} + x\right)\right), x\right) \]
      rational_best-simplify-3 [=>] 0.3	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(x + 0.5 \cdot \frac{1}{x}\right)}\right), x\right) \]
      rational_best-simplify-55 [=>] 0.3	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \color{blue}{1 \cdot \frac{0.5}{x}}\right)\right), x\right) \]
      rational_best-simplify-1 [=>] 0.3	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \color{blue}{\frac{0.5}{x} \cdot 1}\right)\right), x\right) \]
      rational_best-simplify-7 [=>] 0.3	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \color{blue}{\frac{0.5}{x}}\right)\right), x\right) \]

   4. Taylor expanded in x around 0 0.0

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.5}{x}\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 12.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19999999961012896000:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\frac{x \cdot x - \left(-1 - 3 \cdot \left(1 + x \cdot x\right)\right)}{4}}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	12.6
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -19999999961012896000:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + 1\right) + \left(\sqrt{1 + x \cdot x} + -1\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 2
Error	12.6
Cost	13192

\[\begin{array}{l} \mathbf{if}\;x \leq -19999999961012896000:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 3
Error	13.3
Cost	9924

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

Alternative 4
Error	13.4
Cost	9864

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

Alternative 5
Error	17.7
Cost	9796

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

Alternative 6
Error	19.0
Cost	9732

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

Alternative 7
Error	19.0
Cost	6628

\[\begin{array}{l} \mathbf{if}\;x \leq -9.999999350456404 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 8
Error	20.8
Cost	6596

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

Alternative 9
Error	23.4
Cost	6528

\[\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right) \]

Alternative 10
Error	27.7
Cost	6464

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

Reproduce

herbie shell --seed 2023099 
(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))