Rust f32::asinh

Percentage Accurate: 38.1% → 99.5%

Time: 3.5s

Alternatives: 2

Speedup: 2.1×

Specification

Math

\[\sinh^{-1} x \]

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: 38.1% accurate, 1.0× speedup

Math

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

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, 1.1× speedup

Math

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

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 38.1%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
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. 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 99.5%

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

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: 54.8% accurate, 2.1× speedup

Math

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

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 38.1%

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

2. Taylor expanded in x around -inf

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

   1. lower-*.f32 29.5%

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

4. Applied rewrites 29.5%

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

   1. lift-*.f32 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f32 29.5%

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

6. Applied rewrites 29.5%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. lower--.f32 N/A

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

   5. rem-square-sqrt N/A

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

   6. sqrt-unprod N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. sqr-neg-rev N/A

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

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

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

   11. lift-fabs.f32 N/A

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

   12. lift-fabs.f32 N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| - \left(\mathsf{neg}\left(\mathsf{Rewrite<=}\left(*-rgt-identity, \left(\left|x\right| \cdot 1\right)\right)\right)\right)\right), x\right) \]

   13. lift-fabs.f32 N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| - \mathsf{Rewrite<=}\left(distribute-rgt-neg-out, \left(\left|x\right| \cdot \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right) \]

   14. lift-fabs.f32 N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| - \left|x\right| \cdot \mathsf{Rewrite=>}\left(metadata-eval, -1\right)\right), x\right) \]

   15. lift-fabs.f32 N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| - \mathsf{Rewrite=>}\left(lift-fabs.f32, \left(\left|x\right|\right)\right) \cdot -1\right), x\right) \]

   16. lift-fabs.f32 N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| - \mathsf{Rewrite=>}\left(rem-sqrt-square-rev, \left(\sqrt{x \cdot x}\right)\right) \cdot -1\right), x\right) \]

   17. lift-fabs.f32 N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| - \sqrt{\mathsf{Rewrite=>}\left(sqr-neg-rev, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)\right)} \cdot -1\right), x\right) \]

   18. lift-fabs.f32 N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| - \sqrt{\mathsf{Rewrite<=}\left(lift-neg.f32, \left(-x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \cdot -1\right), x\right) \]

   19. lift-fabs.f32 N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| - \sqrt{\left(-x\right) \cdot \mathsf{Rewrite<=}\left(lift-neg.f32, \left(-x\right)\right)} \cdot -1\right), x\right) \]

   20. lift-fabs.f32 N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| - \mathsf{Rewrite<=}\left(sqrt-unprod, \left(\sqrt{-x} \cdot \sqrt{-x}\right)\right) \cdot -1\right), x\right) \]

   21. lift-fabs.f32 N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| - \mathsf{Rewrite=>}\left(rem-square-sqrt, \left(-x\right)\right) \cdot -1\right), x\right) \]

   22. lift-fabs.f32 N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| - \mathsf{Rewrite=>}\left(lift-neg.f32, \left(\mathsf{neg}\left(x\right)\right)\right) \cdot -1\right), x\right) \]

   23. lift-fabs.f32 N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| - \mathsf{Rewrite=>}\left(distribute-lft-neg-out, \left(\mathsf{neg}\left(x \cdot -1\right)\right)\right)\right), x\right) \]

   24. lift-fabs.f32 N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| - \mathsf{Rewrite<=}\left(distribute-rgt-neg-out, \left(x \cdot \left(\mathsf{neg}\left(-1\right)\right)\right)\right)\right), x\right) \]

   25. lift-fabs.f32 N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| - x \cdot \mathsf{Rewrite=>}\left(metadata-eval, 1\right)\right), x\right) \]

   26. lift-fabs.f32 N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| - \mathsf{Rewrite=>}\left(*-rgt-identity, x\right)\right), x\right) \]

8. Applied rewrites 29.5%

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

9. Taylor expanded in x around 0

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

    1. lower-*.f32 54.8%

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

11. Applied rewrites 54.8%

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

Reproduce

herbie shell --seed 2025258 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32
  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))