Rust f32::asinh

Percentage Accurate: 38.5% → 99.6%

Time: 4.4s

Alternatives: 3

Speedup: 2.2×

Specification

Math

\[\begin{array}{l} \\ \sinh^{-1} x \end{array} \]

FPCore

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

C

float code(float x) {
	return asinhf(x);
}

Julia

function code(x)
	return asinh(x)
end

MATLAB

function tmp = code(x)
	tmp = asinh(x);
end

Initial Program: 38.5% 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: 99.6% accurate, 1.1× 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 36.7%

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

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

   4. lift-+.f32 N/A

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

   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. sqr-abs-rev N/A

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

   7. lift-fabs.f32 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. lift-fabs.f32 N/A

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

   9. asinh-def-rev N/A

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

   10. lower-asinh.f32 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 62.4% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x 1.0) (copysign x x) (copysign (log1p x) x)))

C

float code(float x) {
	float tmp;
	if (x <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(log1pf(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 33.5%

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

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

      4. lift-+.f32 N/A

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

      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. sqr-abs-rev N/A

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

      7. lift-fabs.f32 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. lift-fabs.f32 N/A

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

      9. asinh-def-rev N/A

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

      10. lower-asinh.f32 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-fabs.f32 N/A

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

      2. neg-fabs N/A

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

      3. lift-neg.f32 N/A

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

      4. rem-sqrt-square N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f32 N/A

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

      7. lower-sqrt.f32 N/A

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

      8. lower-sqrt.f32 69.2

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

   6. Applied rewrites 69.2%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot {\left(\sqrt{-1}\right)}^{2}}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 64.7%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 64.7%

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

   if 1 < x

   1. Initial program 50.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 11.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(\frac{0.5}{\left|x\right| - -1} \cdot x, x, \mathsf{log1p}\left(\left|x\right|\right)\right)}, x\right) \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 11.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign}\left(\frac{1}{2} \cdot \color{blue}{\frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 11.1%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 44.7%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(x\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 54.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 36.7%

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

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

   4. lift-+.f32 N/A

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

   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. sqr-abs-rev N/A

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

   7. lift-fabs.f32 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. lift-fabs.f32 N/A

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

   9. asinh-def-rev N/A

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

   10. lower-asinh.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-fabs.f32 N/A

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

   2. neg-fabs N/A

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

   3. lift-neg.f32 N/A

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

   4. rem-sqrt-square N/A

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

   5. sqrt-prod N/A

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

   6. lower-*.f32 N/A

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

   7. lower-sqrt.f32 N/A

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

   8. lower-sqrt.f32 56.0

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

6. Applied rewrites 56.0%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot {\left(\sqrt{-1}\right)}^{2}}, x\right) \]
8. Step-by-step derivation

9. Applied rewrites 54.4%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 54.4%

    \[\leadsto \mathsf{copysign}\left(x, x\right) \]
13. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.6× 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 2025019 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :alt
  (! :herbie-platform default (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))