Rust f32::asinh

Percentage Accurate: 38.7% → 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: 38.7% 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

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

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(Float32(-x)), x)
end

MATLAB

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

Derivation

1. Initial program 38.7%

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

   1. lift-+.f32 N/A

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

   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. rem-sqrt-square-rev N/A

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

   4. sqr-neg-rev N/A

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

   5. sqrt-prod N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-sqrt.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-sqrt.f32 N/A

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

   10. lower-neg.f32 19.5

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

   11. lift-+.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lower-fma.f32 19.5

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

3. Applied rewrites 19.5%

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

   1. lift-fma.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. sqrt-unprod N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. sqr-neg-rev N/A

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

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

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

   9. lift-fabs.f32 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f32 38.7

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

   12. lift-fabs.f32 N/A

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

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

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

   14. sqr-neg-rev N/A

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

   15. lift-neg.f32 N/A

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

   16. lift-neg.f32 N/A

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

   17. sqrt-unprod N/A

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

   18. rem-square-sqrt 28.3

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

5. Applied rewrites 28.3%

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

   1. lift-log.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-sqrt.f32 N/A

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

   5. lift-fma.f32 N/A

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

   6. sqr-neg-rev N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-neg.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

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

   10. lower-asinh.f32 99.5

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

7. Applied rewrites 99.5%

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

Alternative 2: 45.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.30000001192092896:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<=
      (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)
      0.30000001192092896)
   (copysign (log (+ (fabs x) 1.0)) x)
   (copysign (log (+ x x)) x)))

C

float code(float x) {
	float tmp;
	if (copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x) <= 0.30000001192092896f) {
		tmp = copysignf(logf((fabsf(x) + 1.0f)), 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(0.30000001192092896))
		tmp = copysign(log(Float32(abs(x) + Float32(1.0))), 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(0.30000001192092896))
		tmp = sign(x) * abs(log((abs(x) + single(1.0))));
	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) < 0.300000012

   1. Initial program 38.7%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 31.8%

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

   if 0.300000012 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 38.7%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-fabs.f32 29.8

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

   4. Applied rewrites 29.8%

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

      1. lift-*.f32 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f32 29.8

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-+.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. associate-*r/ N/A

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

      10. sum-to-mult-rev N/A

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

      11. mul-1-neg N/A

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

      12. lower-+.f32 N/A

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

      13. lower-neg.f32 29.8

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

   6. Applied rewrites 29.8%

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

      1. lift-+.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. sub-flip-reverse N/A

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

      4. lower--.f32 29.8

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

      5. lift-fabs.f32 N/A

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

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

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

      7. sqr-neg-rev N/A

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

      8. lift-neg.f32 N/A

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

      9. lift-neg.f32 N/A

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

      10. sqrt-unprod N/A

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

      11. rem-square-sqrt 27.0

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

   8. Applied rewrites 27.0%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-negate N/A

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

      4. flip-- N/A

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

      5. lift-neg.f32 N/A

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

      6. lift-neg.f32 N/A

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

      7. sqr-neg-rev N/A

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

      8. sqr-neg-rev N/A

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

      9. lift-neg.f32 N/A

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

      10. lift-neg.f32 N/A

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

      11. +-commutative N/A

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

      12. flip-- N/A

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

      13. lift-neg.f32 N/A

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

      14. add-flip-rev N/A

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

      15. lower-+.f32 27.4

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

   10. Applied rewrites 27.4%

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

Alternative 3: 27.4% 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 38.7%

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

2. Taylor expanded in x around -inf

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fabs.f32 29.8

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

4. Applied rewrites 29.8%

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

   1. lift-*.f32 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f32 29.8

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. associate-*r/ N/A

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

   10. sum-to-mult-rev N/A

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

   11. mul-1-neg N/A

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

   12. lower-+.f32 N/A

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

   13. lower-neg.f32 29.8

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

6. Applied rewrites 29.8%

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

   1. lift-+.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. sub-flip-reverse N/A

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

   4. lower--.f32 29.8

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

   5. lift-fabs.f32 N/A

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

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

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

   7. sqr-neg-rev N/A

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

   8. lift-neg.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. sqrt-unprod N/A

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

   11. rem-square-sqrt 27.0

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

8. Applied rewrites 27.0%

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

   1. lift-neg.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-negate N/A

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

   4. flip-- N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. sqr-neg-rev N/A

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

   8. sqr-neg-rev N/A

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

   9. lift-neg.f32 N/A

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

   10. lift-neg.f32 N/A

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

   11. +-commutative N/A

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

   12. flip-- N/A

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

   13. lift-neg.f32 N/A

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

   14. add-flip-rev N/A

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

   15. lower-+.f32 27.4

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

10. Applied rewrites 27.4%

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

Alternative 4: 13.7% accurate, 2.0× speedup

Math

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

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(Float32(-x)), x)
end

MATLAB

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

Derivation

1. Initial program 38.7%

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

2. Taylor expanded in x around -inf

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

   1. lower-*.f32 13.7

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

4. Applied rewrites 13.7%

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

   1. lift-*.f32 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f32 13.7

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

6. Applied rewrites 13.7%

   \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-x\right)}, x\right) \]
7. 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 2025156 
(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))