Rust f32::asinh

Percentage Accurate: 37.2% → 99.6%

Time: 2.5s

Alternatives: 5

Speedup: 5.8×

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: 37.2% 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 37.2%

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

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

   14. pow2 N/A

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

   15. sqrt-pow1 N/A

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

   16. metadata-eval N/A

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

   17. unpow1 N/A

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

   18. lower-asinh.f32 99.6

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

3. Applied rewrites 99.6%

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

Alternative 2: 75.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float x) {
	float tmp;
	if (x <= 1.0f) {
		tmp = copysignf(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(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(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 32.1%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

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

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

      8. pow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lower-+.f32 N/A

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

      13. lower-log.f32 N/A

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

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

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

      15. pow2 N/A

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

      16. sqrt-pow1 N/A

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

      17. metadata-eval N/A

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

      18. unpow1 N/A

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

      19. lower-+.f32 14.5

          \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(\frac{x \cdot x}{1 + x}, 0.5, \log \left(1 + x\right)\right), x\right) \]

   4. Applied rewrites 14.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 69.0%

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

   if 1 < x

   1. Initial program 52.8%

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

      1. lift-fabs.f32 N/A

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

      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. +-commutative N/A

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

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

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

      8. pow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lower-+.f32 N/A

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

      13. lower-sqrt.f32 N/A

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

      14. lower-fma.f32 52.8

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

   3. Applied rewrites 52.8%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 96.5%

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

Alternative 3: 63.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(1 + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) 2.0)
   (copysign x x)
   (copysign (log (+ 1.0 x)) x)))

C

float code(float x) {
	float tmp;
	if (copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x) <= 2.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((1.0f + 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(2.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(Float32(1.0) + 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(2.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((single(1.0) + 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) < 2

   1. Initial program 32.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

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

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

      8. pow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lower-+.f32 N/A

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

      13. lower-log.f32 N/A

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

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

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

      15. pow2 N/A

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

      16. sqrt-pow1 N/A

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

      17. metadata-eval N/A

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

      18. unpow1 N/A

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

      19. lower-+.f32 14.6

          \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(\frac{x \cdot x}{1 + x}, 0.5, \log \left(1 + x\right)\right), x\right) \]

   4. Applied rewrites 14.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 68.8%

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

   if 2 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 51.7%

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

   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. rem-sqrt-square-rev N/A

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

      2. pow2 N/A

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

      3. sqrt-pow1 N/A

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

      4. metadata-eval N/A

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

      5. unpow1 N/A

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

      6. lower-+.f32 44.5

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

   4. Applied rewrites 44.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + x\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 63.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) 2.0)
   (copysign x x)
   (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x) <= 2.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf(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(2.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(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(2.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(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) < 2

   1. Initial program 32.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

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

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

      8. pow2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. unpow1 N/A

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

      12. lower-+.f32 N/A

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

      13. lower-log.f32 N/A

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

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

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

      15. pow2 N/A

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

      16. sqrt-pow1 N/A

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

      17. metadata-eval N/A

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

      18. unpow1 N/A

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

      19. lower-+.f32 14.6

          \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(\frac{x \cdot x}{1 + x}, 0.5, \log \left(1 + x\right)\right), x\right) \]

   4. Applied rewrites 14.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 68.8%

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

   if 2 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 51.7%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 44.5%

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

Alternative 5: 55.0% accurate, 5.8× 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 37.2%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. pow2 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. pow2 N/A

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

   9. sqrt-pow1 N/A

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

   10. metadata-eval N/A

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

   11. unpow1 N/A

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

   12. lower-+.f32 N/A

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

   13. lower-log.f32 N/A

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

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

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

   15. pow2 N/A

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

   16. sqrt-pow1 N/A

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

   17. metadata-eval N/A

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

   18. unpow1 N/A

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

   19. lower-+.f32 13.6

       \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(\frac{x \cdot x}{1 + x}, 0.5, \log \left(1 + x\right)\right), x\right) \]

4. Applied rewrites 13.6%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 55.0%

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

Developer Target 1: 99.5% 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 2025122 
(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))