Rust f32::asinh

Percentage Accurate: 38.1% → 99.6%

Time: 3.2s

Alternatives: 4

Speedup: 3.1×

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: 38.1% 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, 1.6× 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 38.1%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
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 99.6

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

3. Applied rewrites 99.6%

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

   1. *-rgt-identity N/A

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

   2. lift-fabs.f32 N/A

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

   3. metadata-eval N/A

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

   4. lift-fabs.f32 N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. *-rgt-identity N/A

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

   11. metadata-eval N/A

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

   12. mult-flip N/A

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

   13. div-flip N/A

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

   14. lower-unsound-/.f32 N/A

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

   15. lower-unsound-/.f32 N/A

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

   16. lower-neg.f32 99.2

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

5. Applied rewrites 99.2%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r/ N/A

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

   4. metadata-eval N/A

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

   5. mul-1-neg N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-fabs.f32 N/A

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

   8. neg-fabs N/A

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

   9. lift-neg.f32 N/A

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

   10. rem-square-sqrt N/A

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

   11. lift-sqrt.f32 N/A

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

   12. lift-sqrt.f32 N/A

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

   13. fabs-sqr N/A

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

   14. distribute-rgt-neg-in N/A

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

   15. distribute-lft-neg-out N/A

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

   16. sqr-neg-rev N/A

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

   17. lift-sqrt.f32 N/A

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

   18. lift-sqrt.f32 N/A

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

   19. rem-square-sqrt 99.6

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

7. Applied rewrites 99.6%

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

Alternative 2: 76.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

float code(float x) {
	float tmp;
	if (x <= 2.0f) {
		tmp = copysignf((-1.0f * x), x);
	} else {
		tmp = copysignf(logf((fabsf(x) + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(2.0))
		tmp = copysign(Float32(Float32(-1.0) * x), x);
	else
		tmp = copysign(log(Float32(abs(x) + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(2.0))
		tmp = sign(x) * abs((single(-1.0) * x));
	else
		tmp = sign(x) * abs(log((abs(x) + x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2

   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 99.6

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

   3. Applied rewrites 99.6%

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

      1. *-rgt-identity N/A

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

      2. lift-fabs.f32 N/A

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

      3. metadata-eval N/A

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

      4. lift-fabs.f32 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. *-rgt-identity N/A

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

      11. metadata-eval N/A

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

      12. mult-flip N/A

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

      13. div-flip N/A

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

      14. lower-unsound-/.f32 N/A

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

      15. lower-unsound-/.f32 N/A

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

      16. lower-neg.f32 99.2

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

   5. Applied rewrites 99.2%

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

      1. lift-/.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. neg-fabs N/A

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

      9. lift-neg.f32 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-sqrt.f32 N/A

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

      12. lift-sqrt.f32 N/A

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

      13. fabs-sqr N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. distribute-lft-neg-out N/A

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

      16. sqr-neg-rev N/A

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

      17. lift-sqrt.f32 N/A

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

      18. lift-sqrt.f32 N/A

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

      19. rem-square-sqrt 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f32 54.1

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

   10. Applied rewrites 54.1%

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

   if 2 < x

   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

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-fabs.f32 30.2

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

   4. Applied rewrites 30.2%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f32 N/A

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

      4. lift-/.f32 N/A

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

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

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

      6. +-commutative N/A

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

      7. lower-+.f32 30.2

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

   6. Applied rewrites 30.2%

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

Alternative 3: 62.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float x) {
	float tmp;
	if (x <= 2.0f) {
		tmp = copysignf((-1.0f * x), x);
	} else {
		tmp = copysignf(logf(fabsf(x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(2.0))
		tmp = copysign(Float32(Float32(-1.0) * x), x);
	else
		tmp = copysign(log(abs(x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(2.0))
		tmp = sign(x) * abs((single(-1.0) * x));
	else
		tmp = sign(x) * abs(log(abs(x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2

   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 99.6

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

   3. Applied rewrites 99.6%

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

      1. *-rgt-identity N/A

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

      2. lift-fabs.f32 N/A

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

      3. metadata-eval N/A

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

      4. lift-fabs.f32 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. metadata-eval N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. *-rgt-identity N/A

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

      11. metadata-eval N/A

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

      12. mult-flip N/A

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

      13. div-flip N/A

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

      14. lower-unsound-/.f32 N/A

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

      15. lower-unsound-/.f32 N/A

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

      16. lower-neg.f32 99.2

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

   5. Applied rewrites 99.2%

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

      1. lift-/.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. neg-fabs N/A

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

      9. lift-neg.f32 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-sqrt.f32 N/A

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

      12. lift-sqrt.f32 N/A

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

      13. fabs-sqr N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. distribute-lft-neg-out N/A

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

      16. sqr-neg-rev N/A

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

      17. lift-sqrt.f32 N/A

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

      18. lift-sqrt.f32 N/A

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

      19. rem-square-sqrt 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f32 54.1

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

   10. Applied rewrites 54.1%

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

   if 2 < x

   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

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-fabs.f32 30.2

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

   4. Applied rewrites 30.2%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f32 N/A

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

      4. lift-/.f32 N/A

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

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

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

      6. +-commutative N/A

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

      7. lower-+.f32 30.2

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

   6. Applied rewrites 30.2%

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

   7. Taylor expanded in x around 0

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

      1. lower-fabs.f32 27.3

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

   9. Applied rewrites 27.3%

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

Alternative 4: 54.1% accurate, 3.1× speedup

Math

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

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. Step-by-step derivation

   1. lift-log.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. sqr-abs-rev N/A

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

   7. lift-fabs.f32 N/A

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

   8. lift-fabs.f32 N/A

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

   9. asinh-def-rev N/A

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

   10. lower-asinh.f32 99.6

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

3. Applied rewrites 99.6%

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

   1. *-rgt-identity N/A

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

   2. lift-fabs.f32 N/A

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

   3. metadata-eval N/A

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

   4. lift-fabs.f32 N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. *-rgt-identity N/A

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

   11. metadata-eval N/A

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

   12. mult-flip N/A

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

   13. div-flip N/A

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

   14. lower-unsound-/.f32 N/A

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

   15. lower-unsound-/.f32 N/A

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

   16. lower-neg.f32 99.2

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

5. Applied rewrites 99.2%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r/ N/A

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

   4. metadata-eval N/A

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

   5. mul-1-neg N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-fabs.f32 N/A

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

   8. neg-fabs N/A

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

   9. lift-neg.f32 N/A

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

   10. rem-square-sqrt N/A

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

   11. lift-sqrt.f32 N/A

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

   12. lift-sqrt.f32 N/A

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

   13. fabs-sqr N/A

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

   14. distribute-rgt-neg-in N/A

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

   15. distribute-lft-neg-out N/A

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

   16. sqr-neg-rev N/A

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

   17. lift-sqrt.f32 N/A

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

   18. lift-sqrt.f32 N/A

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

   19. rem-square-sqrt 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in x around 0

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

   1. lower-*.f32 54.1

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

10. Applied rewrites 54.1%

    \[\leadsto \mathsf{copysign}\left(\color{blue}{-1 \cdot x}, x\right) \]
11. 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 2025154 
(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))