Rust f32::asinh

Percentage Accurate: 37.3% → 99.5%

Time: 2.1s

Alternatives: 3

Speedup: 3.1×

Specification

Math

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

FPCore

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

C

float code(float x) {
	return copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
}

Julia

function code(x)
	return copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
end

Initial Program: 37.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x) {
	return copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
}

Julia

function code(x)
	return copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
end

Alternative 1: 99.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\sinh^{-1} \left(-x\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign (asinh (- x)) x))

C

float code(float x) {
	return copysignf(asinhf(-x), x);
}

Julia

function code(x)
	return copysign(asinh(Float32(-x)), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(asinh(-x));
end

Derivation

1. Initial program 37.3%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Step-by-step derivation

   1. lift-log.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. sum-to-mult N/A

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

   4. log-prod N/A

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

   5. lower-+.f32 N/A

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

3. Applied rewrites 57.2%

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

   1. lift-+.f32 N/A

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

5. Applied rewrites 99.5%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\sinh^{-1} \left(-x\right)}, x\right) \]
6. Add Preprocessing

Alternative 2: 76.3% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 1.0) (copysign (* -1.0 x) x) (copysign (log (+ x x)) x)))

C

float code(float x) {
	float tmp;
	if (x <= 1.0f) {
		tmp = copysignf((-1.0f * x), x);
	} else {
		tmp = copysignf(logf((x + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(1.0))
		tmp = copysign(Float32(Float32(-1.0) * x), x);
	else
		tmp = copysign(log(Float32(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(1.0))
		tmp = sign(x) * abs((single(-1.0) * x));
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 37.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. lift-log.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. sum-to-mult N/A

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

      4. log-prod N/A

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

      5. lower-+.f32 N/A

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

   3. Applied rewrites 57.2%

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

      1. lift-+.f32 N/A

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

   5. Applied rewrites 99.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\sinh^{-1} \left(-x\right)}, x\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-1 \cdot x}, x\right) \]
   7. Step-by-step derivation

      1. lower-*.f32 54.2

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

   8. Applied rewrites 54.2%

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

   if 1 < x

   1. Initial program 37.3%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-fabs.f32 29.0

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

   4. Applied rewrites 29.0%

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

   6. Applied rewrites 26.3%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(-x\right) - x\right), x\right)} \]
   7. Step-by-step derivation

      1. lift--.f32 N/A

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

      2. flip-- N/A

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

      3. lift-neg.f32 N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(-x\right) - x \cdot x}{\left(-x\right) + x}\right), x\right) \]

      4. lift-neg.f32 N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right) - x \cdot x}{\left(-x\right) + x}\right), x\right) \]

      5. sqr-neg-rev N/A

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

      6. sqr-neg-rev N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{x \cdot x - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}{\left(-x\right) + x}\right), x\right) \]

      7. lift-neg.f32 N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{x \cdot x - \left(-x\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}{\left(-x\right) + x}\right), x\right) \]

      8. lift-neg.f32 N/A

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

      9. +-commutative N/A

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

      10. flip-- N/A

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

      11. lift-neg.f32 N/A

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

      12. add-flip-rev N/A

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

      13. lower-+.f32 27.5

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

   8. Applied rewrites 27.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + x\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 54.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(-1 \cdot x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign (* -1.0 x) x))

C

float code(float x) {
	return copysignf((-1.0f * x), x);
}

Julia

function code(x)
	return copysign(Float32(Float32(-1.0) * x), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs((single(-1.0) * x));
end

Derivation

1. Initial program 37.3%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Step-by-step derivation

   1. lift-log.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. sum-to-mult N/A

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

   4. log-prod N/A

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

   5. lower-+.f32 N/A

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

3. Applied rewrites 57.2%

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

   1. lift-+.f32 N/A

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

5. Applied rewrites 99.5%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\sinh^{-1} \left(-x\right)}, x\right) \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign}\left(\color{blue}{-1 \cdot x}, x\right) \]
7. Step-by-step derivation

   1. lower-*.f32 54.2

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

8. Applied rewrites 54.2%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{-1 \cdot x}, x\right) \]
9. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (/ 1.0 (fabs x))))
   (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0)))) x)))

C

float code(float x) {
	float t_0 = 1.0f / fabsf(x);
	return copysignf(log1pf((fabsf(x) + (fabsf(x) / (hypotf(1.0f, t_0) + t_0)))), x);
}

Julia

function code(x)
	t_0 = Float32(Float32(1.0) / abs(x))
	return copysign(log1p(Float32(abs(x) + Float32(abs(x) / Float32(hypot(Float32(1.0), t_0) + t_0)))), x)
end

Reproduce

herbie shell --seed 2025148 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :alt
  (! :herbie-platform c (let* ((ax (fabs x)) (ix (/ 1 ax))) (copysign (log1p (+ ax (/ ax (+ (hypot 1 ix) ix)))) x)))

  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))