Rust f32::asinh

Percentage Accurate: 37.5% → 99.5%

Time: 2.4s

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: 37.5% 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, 1.7× 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 37.5%

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

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

   11. unpow2 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 99.5

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

3. Applied rewrites 99.5%

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

Alternative 2: 75.0% 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 32.6%

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

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

      11. unpow2 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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in x around 0

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

      1. unpow1 67.7

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

      2. metadata-eval 67.7

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

      3. sqrt-pow1 67.7

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

      4. pow2 67.7

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

      5. rem-sqrt-square-rev 67.7

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

      6. asinh-def-rev 67.7

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

      7. unpow2 67.7

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

      8. pow2-fabs-rev 67.7

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

      9. +-commutative 67.7

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

      10. +-commutative 67.7

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

      11. pow2 67.7

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

   6. Applied rewrites 67.7%

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

   if 1 < x

   1. Initial program 52.0%

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

   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 44.1%

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

   5. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. metadata-eval N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. unpow-prod-down N/A

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

      8. mult-flip N/A

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

      9. inv-pow N/A

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

      10. pow-pow N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

   7. Applied rewrites 96.7%

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

Alternative 3: 62.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 32.7%

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

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

      11. unpow2 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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in x around 0

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

      1. unpow1 67.6

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

      2. metadata-eval 67.6

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

      3. sqrt-pow1 67.6

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

      4. pow2 67.6

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

      5. rem-sqrt-square-rev 67.6

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

      6. asinh-def-rev 67.6

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

      7. unpow2 67.6

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

      8. pow2-fabs-rev 67.6

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

      9. +-commutative 67.6

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

      10. +-commutative 67.6

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

      11. pow2 67.6

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

   6. Applied rewrites 67.6%

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

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

   1. Initial program 51.9%

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

   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 44.2%

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

   5. Taylor expanded in x around inf

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

      1. add-flip N/A

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

      2. division-flip N/A

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

      3. metadata-eval N/A

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

      4. fabs-lhs-div N/A

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

      5. mult-flip N/A

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

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

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

      7. pow2 N/A

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

      8. sqrt-pow1 N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. times-frac N/A

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

      12. inv-pow N/A

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

      13. pow-neg N/A

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

      14. metadata-eval N/A

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

      15. unpow1 N/A

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

      16. inv-pow N/A

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

      17. pow-plus N/A

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

      18. metadata-eval N/A

          \[\leadsto \mathsf{copysign}\left(-1 \cdot \log \left(\frac{1}{x}\right) - \left(\mathsf{neg}\left({x}^{0}\right)\right), x\right) \]

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

   7. Applied rewrites 48.0%

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

Alternative 4: 61.7% 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 32.9%

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

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

      11. unpow2 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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in x around 0

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

      1. unpow1 67.5

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

      2. metadata-eval 67.5

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

      3. sqrt-pow1 67.5

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

      4. pow2 67.5

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

      5. rem-sqrt-square-rev 67.5

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

      6. asinh-def-rev 67.5

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

      7. unpow2 67.5

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

      8. pow2-fabs-rev 67.5

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

      9. +-commutative 67.5

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

      10. +-commutative 67.5

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

      11. pow2 67.5

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

   6. Applied rewrites 67.5%

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

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

   1. Initial program 51.3%

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

   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 44.4%

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

Alternative 5: 53.4% 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 37.5%

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

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

   11. unpow2 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 99.5

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in x around 0

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

   1. unpow1 53.4

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

   2. metadata-eval 53.4

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

   3. sqrt-pow1 53.4

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

   4. pow2 53.4

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

   5. rem-sqrt-square-rev 53.4

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

   6. asinh-def-rev 53.4

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

   7. unpow2 53.4

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

   8. pow2-fabs-rev 53.4

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

   9. +-commutative 53.4

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

   10. +-commutative 53.4

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

   11. pow2 53.4

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

6. Applied rewrites 53.4%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]
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 2025128 
(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))