Rust f32::asinh

Percentage Accurate: 38.2% → 99.5%

Time: 2.9s

Alternatives: 4

Speedup: 5.8×

Specification

Math

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

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: 38.2% accurate, 1.0× speedup

Math

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

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

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

FPCore

(FPCore (x) :precision binary32 (copysign (asinh (fabs x)) x))

C

float code(float x) {
	return copysignf(asinhf(fabsf(x)), x);
}

Julia

function code(x)
	return copysign(asinh(abs(x)), x)
end

MATLAB

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

Derivation

1. Initial program 38.2%

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

   1. lift-log.f32 N/A

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

   2. lift-+.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 99.5%

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

3. Applied rewrites 99.5%

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

Alternative 2: 97.1% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\left|x\right|, \left|x\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|\left|x\right|\right| + \left|x\right|\right), \left|x\right|\right)\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (*
  (copysign 1.0 x)
  (if (<= (fabs x) 0.5)
    (copysign (fabs x) (fabs x))
    (copysign (log (+ (fabs (fabs x)) (fabs x))) (fabs x)))))

C

float code(float x) {
	float tmp;
	if (fabsf(x) <= 0.5f) {
		tmp = copysignf(fabsf(x), fabsf(x));
	} else {
		tmp = copysignf(logf((fabsf(fabsf(x)) + fabsf(x))), fabsf(x));
	}
	return copysignf(1.0f, x) * tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(0.5))
		tmp = copysign(abs(x), abs(x));
	else
		tmp = copysign(log(Float32(abs(abs(x)) + abs(x))), abs(x));
	end
	return Float32(copysign(Float32(1.0), x) * tmp)
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (abs(x) <= single(0.5))
		tmp = sign(abs(x)) * abs(abs(x));
	else
		tmp = sign(abs(x)) * abs(log((abs(abs(x)) + abs(x))));
	end
	tmp_2 = (sign(x) * abs(single(1.0))) * tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 38.2%

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

      1. lift-log.f32 N/A

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

      2. lift-+.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 99.5%

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

   3. Applied rewrites 99.5%

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

      1. lift-fabs.f32 N/A

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

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

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. sqrt-div N/A

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

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

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

      10. lower-unsound-sqrt.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lower-unsound-sqrt.f32 57.6%

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.8%

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

   if 0.5 < x

   1. Initial program 38.2%

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

   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 29.6%

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

   4. Applied rewrites 29.6%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right)}, x\right) \]
   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 29.6%

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

   6. Applied rewrites 29.6%

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

Alternative 3: 70.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left|\left|x\right|\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(t\_0 + \sqrt{\left|x\right| \cdot \left|x\right| + 1}\right), \left|x\right|\right) \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(\left|x\right|, \left|x\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log t\_0, \left|x\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (fabs (fabs x))))
   (*
    (copysign 1.0 x)
    (if (<=
         (copysign (log (+ t_0 (sqrt (+ (* (fabs x) (fabs x)) 1.0)))) (fabs x))
         2.0)
      (copysign (fabs x) (fabs x))
      (copysign (log t_0) (fabs x))))))

C

float code(float x) {
	float t_0 = fabsf(fabsf(x));
	float tmp;
	if (copysignf(logf((t_0 + sqrtf(((fabsf(x) * fabsf(x)) + 1.0f)))), fabsf(x)) <= 2.0f) {
		tmp = copysignf(fabsf(x), fabsf(x));
	} else {
		tmp = copysignf(logf(t_0), fabsf(x));
	}
	return copysignf(1.0f, x) * tmp;
}

Julia

function code(x)
	t_0 = abs(abs(x))
	tmp = Float32(0.0)
	if (copysign(log(Float32(t_0 + sqrt(Float32(Float32(abs(x) * abs(x)) + Float32(1.0))))), abs(x)) <= Float32(2.0))
		tmp = copysign(abs(x), abs(x));
	else
		tmp = copysign(log(t_0), abs(x));
	end
	return Float32(copysign(Float32(1.0), x) * tmp)
end

MATLAB

function tmp_2 = code(x)
	t_0 = abs(abs(x));
	tmp = single(0.0);
	if ((sign(abs(x)) * abs(log((t_0 + sqrt(((abs(x) * abs(x)) + single(1.0))))))) <= single(2.0))
		tmp = sign(abs(x)) * abs(abs(x));
	else
		tmp = sign(abs(x)) * abs(log(t_0));
	end
	tmp_2 = (sign(x) * abs(single(1.0))) * 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 38.2%

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

      1. lift-log.f32 N/A

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

      2. lift-+.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 99.5%

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

   3. Applied rewrites 99.5%

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

      1. lift-fabs.f32 N/A

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

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

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. sqrt-div N/A

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

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

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

      10. lower-unsound-sqrt.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lower-unsound-sqrt.f32 57.6%

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.8%

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

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

   1. Initial program 38.2%

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

   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 30.0%

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

   4. Applied rewrites 30.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-fabs.f32 27.4%

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

   7. Applied rewrites 27.4%

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

Alternative 4: 53.8% accurate, 5.8× speedup

Math

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

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 38.2%

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

   1. lift-log.f32 N/A

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

   2. lift-+.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 99.5%

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

3. Applied rewrites 99.5%

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

   1. lift-fabs.f32 N/A

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

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

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

   3. *-rgt-identity N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. sqrt-div N/A

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

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

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

   10. lower-unsound-sqrt.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-unsound-sqrt.f32 57.6%

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

5. Applied rewrites 57.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 53.8%

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

Developer Target 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \]

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 2025183 
(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))