Rust f32::asinh

Percentage Accurate: 37.8% → 99.5%

Time: 2.1s

Alternatives: 4

Speedup: 4.6×

Specification

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))

C

float code(float x) {
	return copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
}

Julia

function code(x)
	return copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
end

Initial Program: 37.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))

C

float code(float x) {
	return copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
}

Julia

function code(x)
	return copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
end

Alternative 1: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return copysign(asinh(Float32(-x)), x)
end

MATLAB

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

Derivation

1. Initial program 37.8%

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

   1. lift-log.f32 N/A

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

   2. lift-fabs.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. lift-sqrt.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. pow2 N/A

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

   8. +-commutative N/A

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

   9. +-commutative N/A

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

   10. pow2 N/A

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

   11. sqr-abs-rev N/A

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

   12. asinh-def-rev N/A

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

   13. lower-asinh.f32 N/A

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

   14. lift-fabs.f32 99.5

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

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. sqr-neg-rev N/A

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

   4. sqrt-prod N/A

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

   5. lower-*.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower-neg.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lower-neg.f32 48.6

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

5. Applied rewrites 48.6%

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

   1. lift-*.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. rem-square-sqrt 99.5

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

7. Applied rewrites 99.5%

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

Alternative 2: 62.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(-x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + 1\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) 1.25)
   (copysign (- x) x)
   (copysign (log (+ (fabs x) 1.0)) x)))

C

float code(float x) {
	float tmp;
	if (copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x) <= 1.25f) {
		tmp = copysignf(-x, x);
	} else {
		tmp = copysignf(logf((fabsf(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(1.25))
		tmp = copysign(Float32(-x), x);
	else
		tmp = copysign(log(Float32(abs(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(1.25))
		tmp = sign(x) * abs(-x);
	else
		tmp = sign(x) * abs(log((abs(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) < 1.25

   1. Initial program 37.8%

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

      1. lift-log.f32 N/A

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

      2. lift-fabs.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. lift-sqrt.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-+.f32 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. pow2 N/A

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

      11. sqr-abs-rev N/A

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

      12. asinh-def-rev N/A

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

      13. lower-asinh.f32 N/A

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

      14. lift-fabs.f32 99.5

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

   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. sqr-neg-rev N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f32 N/A

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

      6. lower-sqrt.f32 N/A

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

      7. lower-neg.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lower-neg.f32 48.6

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

   5. Applied rewrites 48.6%

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

      1. lift-*.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. rem-square-sqrt 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f32 53.9

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

   10. Applied rewrites 53.9%

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

   if 1.25 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 37.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 31.6%

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

Alternative 3: 62.5% 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(Float32(-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 37.8%

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

      1. lift-log.f32 N/A

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

      2. lift-fabs.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. lift-sqrt.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-+.f32 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. pow2 N/A

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

      11. sqr-abs-rev N/A

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

      12. asinh-def-rev N/A

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

      13. lower-asinh.f32 N/A

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

      14. lift-fabs.f32 99.5

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

   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. sqr-neg-rev N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f32 N/A

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

      6. lower-sqrt.f32 N/A

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

      7. lower-neg.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lower-neg.f32 48.6

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

   5. Applied rewrites 48.6%

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

      1. lift-*.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. rem-square-sqrt 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f32 53.9

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

   10. Applied rewrites 53.9%

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

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

   1. Initial program 37.8%

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

   2. Taylor expanded in x around inf

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

      1. log-pow-rev N/A

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

      2. unpow-1 N/A

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

      3. inv-pow N/A

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

      4. pow-neg N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-log.f32 14.0

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

   4. Applied rewrites 14.0%

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

Alternative 4: 53.9% accurate, 4.6× 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(Float32(-x), x)
end

MATLAB

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

Derivation

1. Initial program 37.8%

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

   1. lift-log.f32 N/A

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

   2. lift-fabs.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. lift-sqrt.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. pow2 N/A

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

   8. +-commutative N/A

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

   9. +-commutative N/A

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

   10. pow2 N/A

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

   11. sqr-abs-rev N/A

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

   12. asinh-def-rev N/A

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

   13. lower-asinh.f32 N/A

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

   14. lift-fabs.f32 99.5

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

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. sqr-neg-rev N/A

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

   4. sqrt-prod N/A

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

   5. lower-*.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lower-neg.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lower-neg.f32 48.6

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

5. Applied rewrites 48.6%

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

   1. lift-*.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. rem-square-sqrt 99.5

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lift-neg.f32 53.9

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

10. Applied rewrites 53.9%

    \[\leadsto \mathsf{copysign}\left(\color{blue}{-x}, x\right) \]
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 2025142 
(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))