Rust f32::asinh

Percentage Accurate: 38.3% → 99.6%

Time: 9.2s

Alternatives: 3

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \sinh^{-1} x \end{array} \]

FPCore

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

C

float code(float x) {
	return asinhf(x);
}

Julia

function code(x)
	return asinh(x)
end

MATLAB

function tmp = code(x)
	tmp = asinh(x);
end

Initial Program: 38.3% 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, 1.1× speedup

Math

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

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 39.9%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Add Preprocessing
3. 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

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

4. Applied rewrites 99.5%

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

Alternative 2: 20.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 3:\\ \;\;\;\;\mathsf{copysign}\left(\left(\frac{0.5}{\left|x\right| - -1} \cdot x\right) \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) 3.0)
   (copysign (* (* (/ 0.5 (- (fabs x) -1.0)) x) x) x)
   (copysign (log x) x)))

C

float code(float x) {
	float tmp;
	if (copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x) <= 3.0f) {
		tmp = copysignf((((0.5f / (fabsf(x) - -1.0f)) * x) * 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(3.0))
		tmp = copysign(Float32(Float32(Float32(Float32(0.5) / Float32(abs(x) - Float32(-1.0))) * x) * 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(3.0))
		tmp = sign(x) * abs((((single(0.5) / (abs(x) - single(-1.0))) * x) * 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) < 3

   1. Initial program 35.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + \frac{1}{2} \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f32 N/A

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

   5. Applied rewrites 62.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(\frac{0.5}{\left|x\right| - -1} \cdot x, x, \mathsf{log1p}\left(\left|x\right|\right)\right)}, x\right) \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign}\left(\frac{1}{2} \cdot \color{blue}{\frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 12.6%

      \[\leadsto \mathsf{copysign}\left(\frac{0.5 \cdot \left(x \cdot x\right)}{\color{blue}{1 + \left|x\right|}}, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 13.0%

       \[\leadsto \mathsf{copysign}\left(\left(\frac{0.5}{\left|x\right| - -1} \cdot \left(-x\right)\right) \cdot \left(-x\right), x\right) \]

   if 3 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) #s(literal 1 binary32))))) x)

   1. Initial program 54.9%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

      4. lower-log.f32 44.3

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

   5. Applied rewrites 44.3%

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

4. Final simplification 19.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 3:\\ \;\;\;\;\mathsf{copysign}\left(\left(\frac{0.5}{\left|x\right| - -1} \cdot x\right) \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 12.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\left(0.5 \cdot \frac{x}{\left|x\right| - -1}\right) \cdot x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (copysign (* (* 0.5 (/ x (- (fabs x) -1.0))) x) x))

C

float code(float x) {
	return copysignf(((0.5f * (x / (fabsf(x) - -1.0f))) * x), x);
}

Julia

function code(x)
	return copysign(Float32(Float32(Float32(0.5) * Float32(x / Float32(abs(x) - Float32(-1.0)))) * x), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(((single(0.5) * (x / (abs(x) - single(-1.0)))) * x));
end

Derivation

1. Initial program 39.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + \frac{1}{2} \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f32 N/A

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

5. Applied rewrites 51.2%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(\frac{0.5}{\left|x\right| - -1} \cdot x, x, \mathsf{log1p}\left(\left|x\right|\right)\right)}, x\right) \]

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{copysign}\left(\frac{1}{2} \cdot \color{blue}{\frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
7. Step-by-step derivation

8. Applied rewrites 12.2%

   \[\leadsto \mathsf{copysign}\left(\frac{0.5 \cdot \left(x \cdot x\right)}{\color{blue}{1 + \left|x\right|}}, x\right) \]
9. Step-by-step derivation

10. Applied rewrites 12.6%

    \[\leadsto \mathsf{copysign}\left(\left(0.5 \cdot \frac{x}{\left|x\right| - -1}\right) \cdot x, x\right) \]
11. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.6× 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 2024357 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :alt
  (! :herbie-platform default (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))