Rust f32::asinh

Percentage Accurate: 37.8% → 99.6%

Time: 2.2s

Alternatives: 5

Speedup: 1.7×

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: 37.8% 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, 1.7× speedup

Math

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

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(x), x)
end

MATLAB

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

Derivation

1. Initial program 37.8%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
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. pow2 N/A

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

   8. +-commutative N/A

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

   9. +-commutative N/A

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

   10. pow2 N/A

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

   11. sqr-abs-rev N/A

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

   12. asinh-def-rev N/A

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

   13. lower-asinh.f32 N/A

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

   14. lift-fabs.f32 99.6

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in x around 0

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

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

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

   2. sqrt-unprod N/A

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

   3. rem-square-sqrt 99.6

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

6. Applied rewrites 99.6%

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

Alternative 2: 33.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = Float32(0.0)
	if (copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x) <= Float32(1.0))
		tmp = copysign(log(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 ((sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))))) <= single(1.0))
		tmp = sign(x) * abs(log((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 (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x) < 1

   1. Initial program 37.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 31.2%

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

      1. lift-fabs.f32 N/A

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

      2. lift-+.f32 N/A

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

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

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

      4. sqrt-unprod N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. lower-+.f32 20.5

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

   6. Applied rewrites 20.5%

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

   if 1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 37.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 31.2%

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

   5. Taylor expanded in x around inf

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

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

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

      2. sqrt-unprod N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. rem-square-sqrt N/A

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

      6. sqrt-unprod N/A

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

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

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

      8. fabs-mul N/A

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

      9. mul-1-neg N/A

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

      10. neg-fabs N/A

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

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

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

      12. sqrt-unprod N/A

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

      13. rem-square-sqrt N/A

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

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

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

      15. sqrt-unprod N/A

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

      16. pow2 N/A

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

      17. rem-square-sqrt N/A

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

      18. pow2 N/A

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

      19. pow-div N/A

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

      20. metadata-eval N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(x + {\left(\sqrt{x}\right)}^{0} \cdot x\right), x\right) \]

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. rem-square-sqrt N/A

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

      24. sqrt-unprod N/A

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

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

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

      26. fabs-mul N/A

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

      27. mul-1-neg N/A

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

      28. neg-fabs N/A

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

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

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

      30. sqrt-unprod N/A

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

      31. rem-square-sqrt N/A

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

   7. Applied rewrites 26.7%

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

Alternative 3: 26.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return copysign(log(Float32(x + x)), x)
end

MATLAB

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

Derivation

1. Initial program 37.8%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 31.2%

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

5. Taylor expanded in x around inf

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

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

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

   2. sqrt-unprod N/A

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

   5. rem-square-sqrt N/A

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

   6. sqrt-unprod N/A

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

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

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

   8. fabs-mul N/A

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

   9. mul-1-neg N/A

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

   10. neg-fabs N/A

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

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

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

   12. sqrt-unprod N/A

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

   13. rem-square-sqrt N/A

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

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

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

   15. sqrt-unprod N/A

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

   16. pow2 N/A

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

   17. rem-square-sqrt N/A

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

   18. pow2 N/A

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

   19. pow-div N/A

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

   20. metadata-eval N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(x + {\left(\sqrt{x}\right)}^{0} \cdot x\right), x\right) \]

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. rem-square-sqrt N/A

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

   24. sqrt-unprod N/A

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

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

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

   26. fabs-mul N/A

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

   27. mul-1-neg N/A

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

   28. neg-fabs N/A

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

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

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

   30. sqrt-unprod N/A

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

   31. rem-square-sqrt N/A

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

7. Applied rewrites 26.7%

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

Alternative 4: 14.6% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (copysign (+ (log x) 1.0) x))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 37.8%

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

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. unpow-1 N/A

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

   3. inv-pow N/A

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

   4. pow-neg N/A

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

   5. metadata-eval N/A

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

   6. unpow1 N/A

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

   7. lower-log.f32 13.6

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

4. Applied rewrites 13.6%

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

5. Taylor expanded in x around inf

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

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

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

   2. sqrt-unprod N/A

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

   3. mul-1-neg N/A

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

   4. neg-log N/A

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

   5. inv-pow N/A

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

   6. pow-neg N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. sqrt-pow2 N/A

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

   10. pow-to-exp N/A

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

   11. rem-log-exp N/A

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

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

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

   13. sqrt-unprod N/A

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

   14. pow2 N/A

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

   15. rem-square-sqrt N/A

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

   16. pow2 N/A

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

   17. pow-div N/A

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

   18. metadata-eval N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\sqrt{x}\right) \cdot 2 + {\left(\sqrt{x}\right)}^{0}, x\right) \]

   19. metadata-eval N/A

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

   20. lower-+.f32 N/A

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

7. Applied rewrites 14.6%

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

Alternative 5: 13.6% 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 37.8%

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

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. unpow-1 N/A

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

   3. inv-pow N/A

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

   4. pow-neg N/A

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

   5. metadata-eval N/A

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

   6. unpow1 N/A

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

   7. lower-log.f32 13.6

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

4. Applied rewrites 13.6%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log x}, x\right) \]
5. 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 2025143 
(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))