Rust f32::asinh

Percentage Accurate: 38.0% → 99.5%

Time: 2.3s

Alternatives: 4

Speedup: 1.6×

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: 38.0% 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

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

FPCore

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

C

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

Julia

function code(x)
	return copysign(asinh(abs(x)), x)
end

MATLAB

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

Derivation

1. Initial program 38.0%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
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-fabs.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. lift-sqrt.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. sqr-abs-rev N/A

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

   8. asinh-def-rev N/A

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

   9. lower-asinh.f32 N/A

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

   10. lift-fabs.f32 99.5

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

3. Applied rewrites 99.5%

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

Alternative 2: 45.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(1.0))
		tmp = copysign(log(Float32(abs(x) - Float32(-1.0))), x);
	else
		tmp = copysign(log(Float32(abs(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(log((abs(x) - single(-1.0))));
	else
		tmp = sign(x) * abs(log((abs(x) + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 38.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. add-flip N/A

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

      3. metadata-eval N/A

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

      4. lower--.f32 N/A

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

      5. lift-fabs.f32 31.7

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

   4. Applied rewrites 31.7%

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

   if 1 < x

   1. Initial program 38.0%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. add-flip N/A

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

      5. metadata-eval N/A

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

      6. lower--.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lift-fabs.f32 29.8

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

   4. Applied rewrites 29.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lift-fabs.f32 29.8

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

   7. Applied rewrites 29.8%

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

Alternative 3: 31.7% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (copysign (log (- (fabs x) -1.0)) x))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 38.0%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. add-flip N/A

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

   3. metadata-eval N/A

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

   4. lower--.f32 N/A

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

   5. lift-fabs.f32 31.7

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

4. Applied rewrites 31.7%

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

Alternative 4: 13.7% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (copysign (log x) x))

C

float code(float x) {
	return copysignf(logf(x), x);
}

Julia

function code(x)
	return copysign(log(x), x)
end

MATLAB

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

Derivation

1. Initial program 38.0%

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

2. Taylor expanded in x around inf

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

   1. log-pow-rev N/A

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

   2. inv-pow N/A

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

   3. pow-pow N/A

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

   4. lower-special-* N/A

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

   5. metadata-eval N/A

      \[\leadsto \mathsf{copysign}\left(\log \left({x}^{1}\right), x\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{copysign}\left(\log \left({x}^{\left(\frac{2}{2}\right)}\right), x\right) \]

   7. lower-special-/ N/A

      \[\leadsto \mathsf{copysign}\left(\log \left({x}^{\left(\frac{2}{2}\right)}\right), x\right) \]

   8. sqrt-pow1 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\sqrt{{x}^{2}}\right), x\right) \]

   9. pow2 N/A

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

   10. rem-sqrt-square-rev N/A

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

   11. log-fabs N/A

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

   12. lower-log.f32 13.7

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

4. Applied rewrites 13.7%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log x}, x\right) \]
5. Add Preprocessing

Developer Target 1: 99.7% 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 2025132 
(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))