Rust f32::asinh

Average Error: 20.5 → 12.6

Time: 12.7s

Precision: binary32

Cost: 13192

Math

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

↓

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

FPCore

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

↓

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

Target

Original	20.5
Target	0.2
Herbie	12.6

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

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 0.5

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

   3. Simplified 0.5

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

      [Start] 0.5	\[ \mathsf{copysign}\left(\log \left(\left(\left|x\right| + -1 \cdot x\right) - 0.5 \cdot \frac{1}{x}\right), x\right) \]
      rational.json-simplify-2 [=>] 0.5	\[ \mathsf{copysign}\left(\log \left(\left(\left|x\right| + \color{blue}{x \cdot -1}\right) - 0.5 \cdot \frac{1}{x}\right), x\right) \]
      rational.json-simplify-8 [<=] 0.5	\[ \mathsf{copysign}\left(\log \left(\left(\left|x\right| + \color{blue}{\left(-x\right)}\right) - 0.5 \cdot \frac{1}{x}\right), x\right) \]

   4. Taylor expanded in x around 0 0.0

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

   if -2e19 < x < 1

   1. Initial program 19.9

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

   if 1 < x

   1. Initial program 15.9

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

   2. Applied egg-rr 15.9

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

   3. Simplified 15.9

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

      [Start] 15.9	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\left(x \cdot x\right) \cdot \left(1 + \frac{1}{x \cdot x}\right)}\right), x\right) \]
      rational.json-simplify-2 [=>] 15.9	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{\left(1 + \frac{1}{x \cdot x}\right) \cdot \left(x \cdot x\right)}}\right), x\right) \]
      rational.json-simplify-43 [=>] 15.9	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{x \cdot \left(x \cdot \left(1 + \frac{1}{x \cdot x}\right)\right)}}\right), x\right) \]

   4. Applied egg-rr 15.9

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

   5. Simplified 15.9

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot \left(x \cdot \left(1 + \color{blue}{\frac{2}{x} \cdot \frac{0.5}{x}}\right)\right)}\right), x\right) \]
      Proof

      [Start] 15.9	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot \left(x \cdot \left(1 + \frac{2}{x} \cdot \frac{1}{x + x}\right)\right)}\right), x\right) \]
      metadata-eval [<=] 15.9	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot \left(x \cdot \left(1 + \frac{2}{x} \cdot \frac{\color{blue}{0.5 + 0.5}}{x + x}\right)\right)}\right), x\right) \]
      rational.json-simplify-35 [<=] 15.9	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot \left(x \cdot \left(1 + \frac{2}{x} \cdot \color{blue}{\frac{0.5}{x}}\right)\right)}\right), x\right) \]

   6. Taylor expanded in x around inf 0.6

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

   7. Simplified 0.6

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

      [Start] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(0.5 \cdot \frac{1}{x} + x\right)\right), x\right) \]
      rational.json-simplify-1 [=>] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(x + 0.5 \cdot \frac{1}{x}\right)}\right), x\right) \]
      metadata-eval [<=] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + 0.5 \cdot \frac{\color{blue}{\frac{-1}{-1}}}{x}\right)\right), x\right) \]
      rational.json-simplify-44 [<=] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + 0.5 \cdot \color{blue}{\frac{\frac{-1}{x}}{-1}}\right)\right), x\right) \]
      rational.json-simplify-10 [<=] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + 0.5 \cdot \color{blue}{\left(-\frac{-1}{x}\right)}\right)\right), x\right) \]
      rational.json-simplify-8 [=>] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + 0.5 \cdot \color{blue}{\left(\frac{-1}{x} \cdot -1\right)}\right)\right), x\right) \]
      rational.json-simplify-43 [=>] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \color{blue}{\frac{-1}{x} \cdot \left(-1 \cdot 0.5\right)}\right)\right), x\right) \]
      metadata-eval [=>] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{-1}{x} \cdot \color{blue}{-0.5}\right)\right), x\right) \]
      metadata-eval [<=] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{-1}{x} \cdot \color{blue}{\frac{-1}{2}}\right)\right), x\right) \]
      rational.json-simplify-49 [<=] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \color{blue}{\frac{-1 \cdot \frac{-1}{x}}{2}}\right)\right), x\right) \]
      rational.json-simplify-2 [<=] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{\color{blue}{\frac{-1}{x} \cdot -1}}{2}\right)\right), x\right) \]
      rational.json-simplify-8 [<=] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{\color{blue}{-\frac{-1}{x}}}{2}\right)\right), x\right) \]
      rational.json-simplify-10 [=>] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{\color{blue}{\frac{\frac{-1}{x}}{-1}}}{2}\right)\right), x\right) \]
      rational.json-simplify-46 [<=] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{\color{blue}{\frac{-1}{x \cdot -1}}}{2}\right)\right), x\right) \]
      rational.json-simplify-8 [<=] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{\frac{-1}{\color{blue}{-x}}}{2}\right)\right), x\right) \]
      rational.json-simplify-46 [<=] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \color{blue}{\frac{-1}{\left(-x\right) \cdot 2}}\right)\right), x\right) \]
      rational.json-simplify-2 [<=] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{-1}{\color{blue}{2 \cdot \left(-x\right)}}\right)\right), x\right) \]
      rational.json-simplify-46 [=>] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \color{blue}{\frac{\frac{-1}{2}}{-x}}\right)\right), x\right) \]
      metadata-eval [=>] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{\color{blue}{-0.5}}{-x}\right)\right), x\right) \]
      rational.json-simplify-12 [=>] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{-0.5}{\color{blue}{0 - x}}\right)\right), x\right) \]
      rational.json-simplify-50 [=>] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \color{blue}{\frac{--0.5}{x - 0}}\right)\right), x\right) \]
      metadata-eval [=>] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{\color{blue}{0.5}}{x - 0}\right)\right), x\right) \]
      rational.json-simplify-5 [=>] 0.6	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{\color{blue}{x}}\right)\right), x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 12.6

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

Alternatives

Alternative 1
Error	13.3
Cost	13192

\[\begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot \frac{1}{x}}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]

Alternative 2
Error	13.3
Cost	9992

\[\begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - -1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]

Alternative 3
Error	13.3
Cost	9864

\[\begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - -1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 4
Error	18.9
Cost	6596

\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999955487895 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]

Alternative 5
Error	14.7
Cost	6596

\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999955487895 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 6
Error	23.3
Cost	6564

\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999955487895 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]

Alternative 7
Error	27.7
Cost	6464

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

Reproduce

herbie shell --seed 2023067 
(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))