Rust f32::acosh

Percentage Accurate: 53.9% → 98.7%

Time: 6.7s

Alternatives: 4

Speedup: 1.2×

Specification

\[x \geq 1\]

Math

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

FPCore

(FPCore (x) :precision binary32 (acosh x))

C

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

Julia

function code(x)
	return acosh(x)
end

MATLAB

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

Initial Program: 53.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(x + \sqrt{x \cdot x - 1}\right) \end{array} \]

FPCore

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

C

float code(float x) {
	return logf((x + sqrtf(((x * x) - 1.0f))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + sqrt(((x * x) - 1.0e0))))
end function

Julia

function code(x)
	return log(Float32(x + sqrt(Float32(Float32(x * x) - Float32(1.0)))))
end

MATLAB

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

Alternative 1: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \log \left(\left(\left(2 - \frac{0.5}{x \cdot x}\right) - \frac{0.125 + \frac{0.0625}{x \cdot x}}{{x}^{4}}\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log
  (*
   (- (- 2.0 (/ 0.5 (* x x))) (/ (+ 0.125 (/ 0.0625 (* x x))) (pow x 4.0)))
   x)))

C

float code(float x) {
	return logf((((2.0f - (0.5f / (x * x))) - ((0.125f + (0.0625f / (x * x))) / powf(x, 4.0f))) * x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((((2.0e0 - (0.5e0 / (x * x))) - ((0.125e0 + (0.0625e0 / (x * x))) / (x ** 4.0e0))) * x))
end function

Julia

function code(x)
	return log(Float32(Float32(Float32(Float32(2.0) - Float32(Float32(0.5) / Float32(x * x))) - Float32(Float32(Float32(0.125) + Float32(Float32(0.0625) / Float32(x * x))) / (x ^ Float32(4.0)))) * x))
end

MATLAB

function tmp = code(x)
	tmp = log((((single(2.0) - (single(0.5) / (x * x))) - ((single(0.125) + (single(0.0625) / (x * x))) / (x ^ single(4.0)))) * x));
end

Derivation

1. Initial program 49.5%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]
4. Step-by-step derivation

   1. lower-*.f32 97.2

      \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]

5. Applied rewrites 97.2%

   \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \log \color{blue}{\left(x \cdot \left(\left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \log \color{blue}{\left(\left(\left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto \log \color{blue}{\left(\left(\left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)} \]

8. Applied rewrites 98.7%

   \[\leadsto \log \color{blue}{\left(\left(\left(2 - \frac{0.5}{x \cdot x}\right) - \frac{\frac{0.0625}{x \cdot x} + 0.125}{{x}^{4}}\right) \cdot x\right)} \]

9. Final simplification 98.7%

   \[\leadsto \log \left(\left(\left(2 - \frac{0.5}{x \cdot x}\right) - \frac{0.125 + \frac{0.0625}{x \cdot x}}{{x}^{4}}\right) \cdot x\right) \]
10. Add Preprocessing

Alternative 2: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\left(x - \frac{0.5}{x}\right) + x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log (+ (- x (/ 0.5 x)) x)))

C

float code(float x) {
	return logf(((x - (0.5f / x)) + x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log(((x - (0.5e0 / x)) + x))
end function

Julia

function code(x)
	return log(Float32(Float32(x - Float32(Float32(0.5) / x)) + x))
end

MATLAB

function tmp = code(x)
	tmp = log(((x - (single(0.5) / x)) + x));
end

Derivation

1. Initial program 49.5%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. distribute-rgt-neg-out N/A

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

   5. unsub-neg N/A

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

   6. remove-double-neg N/A

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

   7. distribute-rgt-neg-out N/A

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

   8. distribute-lft-neg-out N/A

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

   9. mul-1-neg N/A

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

   10. *-commutative N/A

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

   11. lower--.f32 N/A

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

   12. *-commutative N/A

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

   13. mul-1-neg N/A

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

   14. distribute-lft-neg-out N/A

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

   15. *-commutative N/A

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

   16. distribute-rgt-neg-in N/A

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

   17. metadata-eval N/A

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

   18. associate-*r* N/A

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

5. Applied rewrites 98.1%

   \[\leadsto \log \left(x + \color{blue}{\left(x - \frac{0.5}{x}\right)}\right) \]

6. Final simplification 98.1%

   \[\leadsto \log \left(\left(x - \frac{0.5}{x}\right) + x\right) \]
7. Add Preprocessing

Alternative 3: 97.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \log \left(2 \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log (* 2.0 x)))

C

float code(float x) {
	return logf((2.0f * x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((2.0e0 * x))
end function

Julia

function code(x)
	return log(Float32(Float32(2.0) * x))
end

MATLAB

function tmp = code(x)
	tmp = log((single(2.0) * x));
end

Derivation

1. Initial program 49.5%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]
4. Step-by-step derivation

   1. lower-*.f32 97.2

      \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]

5. Applied rewrites 97.2%

   \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]
6. Add Preprocessing

Alternative 4: 44.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \log x \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log x))

C

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

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log(x)
end function

Julia

function code(x)
	return log(x)
end

MATLAB

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

Derivation

1. Initial program 49.5%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. rem-square-sqrt N/A

      \[\leadsto \log \left(x + \sqrt{\color{blue}{\sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}}}\right) \]

   2. sqrt-unprod N/A

      \[\leadsto \log \left(x + \sqrt{\color{blue}{\sqrt{\left(x \cdot x - 1\right) \cdot \left(x \cdot x - 1\right)}}}\right) \]

   3. rem-sqrt-square N/A

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

   4. lower-fabs.f32 49.5

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

   5. lift--.f32 N/A

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

   6. sub-neg N/A

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

   7. lift-*.f32 N/A

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

   8. lower-fma.f32 N/A

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

   9. metadata-eval 43.1

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

4. Applied rewrites 43.1%

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

5. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. log-rec N/A

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

   3. remove-double-neg N/A

      \[\leadsto \color{blue}{\log x} \]

   4. lower-log.f32 44.2

      \[\leadsto \color{blue}{\log x} \]

7. Applied rewrites 44.2%

   \[\leadsto \color{blue}{\log x} \]
8. Add Preprocessing

Developer Target 1: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \log \left(x + \sqrt{x - 1} \cdot \sqrt{x + 1}\right) \end{array} \]

FPCore

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

C

float code(float x) {
	return logf((x + (sqrtf((x - 1.0f)) * sqrtf((x + 1.0f)))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((x + (sqrt((x - 1.0e0)) * sqrt((x + 1.0e0)))))
end function

Julia

function code(x)
	return log(Float32(x + Float32(sqrt(Float32(x - Float32(1.0))) * sqrt(Float32(x + Float32(1.0))))))
end

MATLAB

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

Reproduce

herbie shell --seed 2024250 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

  :alt
  (! :herbie-platform default (log (+ x (* (sqrt (- x 1)) (sqrt (+ x 1))))))

  (log (+ x (sqrt (- (* x x) 1.0)))))