Rust f32::asinh

Percentage Accurate: 37.4% → 99.6%

Time: 2.0s

Alternatives: 3

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: 37.4% 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

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

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(Float32(-asinh(x)), x)
end

MATLAB

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

Derivation

1. Initial program 37.4%

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

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

   7. associate-/l* N/A

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

   8. associate-*l/ N/A

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

   9. lift-/.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-/.f32 N/A

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

   12. associate-*r/ N/A

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

   13. lower-/.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 58.0

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

5. Applied rewrites 58.0%

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

7. Applied rewrites 99.6%

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

Alternative 2: 75.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(1.0))
		tmp = copysign(Float32(-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(1.0))
		tmp = sign(x) * abs(-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 < 1

   1. Initial program 37.4%

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

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f32 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 58.0

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

   5. Applied rewrites 58.0%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 54.7%

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

   if 1 < x

   1. Initial program 37.4%

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

   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 29.4

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

   4. Applied rewrites 29.4%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right)}, x\right) \]
   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 29.4

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

   6. Applied rewrites 29.4%

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

Alternative 3: 54.7% 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 37.4%

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

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

   7. associate-/l* N/A

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

   8. associate-*l/ N/A

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

   9. lift-/.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-/.f32 N/A

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

   12. associate-*r/ N/A

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

   13. lower-/.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 58.0

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

5. Applied rewrites 58.0%

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 54.7%

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

Developer Target 1: 99.5% 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 2025164 
(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))