Rust f32::acosh

Percentage Accurate: 53.2% → 98.8%

Time: 4.2s

Alternatives: 4

Speedup: 2.0×

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: 53.2% 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: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right) + \frac{0.0625}{{x}^{5}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log
  (-
   (* x 2.0)
   (+ (+ (/ 0.5 x) (/ 0.125 (pow x 3.0))) (/ 0.0625 (pow x 5.0))))))

C

float code(float x) {
	return logf(((x * 2.0f) - (((0.5f / x) + (0.125f / powf(x, 3.0f))) + (0.0625f / powf(x, 5.0f)))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log(((x * 2.0e0) - (((0.5e0 / x) + (0.125e0 / (x ** 3.0e0))) + (0.0625e0 / (x ** 5.0e0)))))
end function

Julia

function code(x)
	return log(Float32(Float32(x * Float32(2.0)) - Float32(Float32(Float32(Float32(0.5) / x) + Float32(Float32(0.125) / (x ^ Float32(3.0)))) + Float32(Float32(0.0625) / (x ^ Float32(5.0))))))
end

MATLAB

function tmp = code(x)
	tmp = log(((x * single(2.0)) - (((single(0.5) / x) + (single(0.125) / (x ^ single(3.0)))) + (single(0.0625) / (x ^ single(5.0))))));
end

Derivation

1. Initial program 51.8%

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

2. Taylor expanded in x around inf 99.1%

   \[\leadsto \log \color{blue}{\left(2 \cdot x - \left(0.0625 \cdot \frac{1}{{x}^{5}} + \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative 99.1%

      \[\leadsto \log \left(\color{blue}{x \cdot 2} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)\right)\right) \]

   2. +-commutative 99.1%

      \[\leadsto \log \left(x \cdot 2 - \color{blue}{\left(\left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right) + 0.0625 \cdot \frac{1}{{x}^{5}}\right)}\right) \]

   3. +-commutative 99.1%

      \[\leadsto \log \left(x \cdot 2 - \left(\color{blue}{\left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]

   4. associate-*r/ 99.1%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\color{blue}{\frac{0.5 \cdot 1}{x}} + 0.125 \cdot \frac{1}{{x}^{3}}\right) + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]

   5. metadata-eval 99.1%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{\color{blue}{0.5}}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right) + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]

   6. associate-*r/ 99.1%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \color{blue}{\frac{0.125 \cdot 1}{{x}^{3}}}\right) + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]

   7. metadata-eval 99.1%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \frac{\color{blue}{0.125}}{{x}^{3}}\right) + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right) \]

   8. associate-*r/ 99.1%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right) + \color{blue}{\frac{0.0625 \cdot 1}{{x}^{5}}}\right)\right) \]

   9. metadata-eval 99.1%

      \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right) + \frac{\color{blue}{0.0625}}{{x}^{5}}\right)\right) \]

4. Simplified 99.1%

   \[\leadsto \log \color{blue}{\left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right) + \frac{0.0625}{{x}^{5}}\right)\right)} \]

5. Final simplification 99.1%

   \[\leadsto \log \left(x \cdot 2 - \left(\left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right) + \frac{0.0625}{{x}^{5}}\right)\right) \]

Alternative 2: 98.2% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (log (- (* x 2.0) (/ 0.5 x))))

C

float code(float x) {
	return logf(((x * 2.0f) - (0.5f / x)));
}

Fortran

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

Julia

function code(x)
	return log(Float32(Float32(x * Float32(2.0)) - Float32(Float32(0.5) / x)))
end

MATLAB

function tmp = code(x)
	tmp = log(((x * single(2.0)) - (single(0.5) / x)));
end

Derivation

1. Initial program 51.8%

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

2. Taylor expanded in x around inf 98.7%

   \[\leadsto \log \color{blue}{\left(2 \cdot x - 0.5 \cdot \frac{1}{x}\right)} \]
3. Step-by-step derivation

   1. *-commutative 98.7%

      \[\leadsto \log \left(\color{blue}{x \cdot 2} - 0.5 \cdot \frac{1}{x}\right) \]

   2. associate-*r/ 98.7%

      \[\leadsto \log \left(x \cdot 2 - \color{blue}{\frac{0.5 \cdot 1}{x}}\right) \]

   3. metadata-eval 98.7%

      \[\leadsto \log \left(x \cdot 2 - \frac{\color{blue}{0.5}}{x}\right) \]

4. Simplified 98.7%

   \[\leadsto \log \color{blue}{\left(x \cdot 2 - \frac{0.5}{x}\right)} \]

5. Final simplification 98.7%

   \[\leadsto \log \left(x \cdot 2 - \frac{0.5}{x}\right) \]

Alternative 3: 96.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

function code(x)
	return log(Float32(x + x))
end

MATLAB

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

Derivation

1. Initial program 51.8%

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

2. Taylor expanded in x around inf 97.7%

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

3. Final simplification 97.7%

   \[\leadsto \log \left(x + x\right) \]

Alternative 4: 3.0% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -6 \end{array} \]

FPCore

(FPCore (x) :precision binary32 -6.0)

C

float code(float x) {
	return -6.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = -6.0e0
end function

Julia

function code(x)
	return Float32(-6.0)
end

MATLAB

function tmp = code(x)
	tmp = single(-6.0);
end

Derivation

1. Initial program 51.8%

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

2. Taylor expanded in x around inf 97.7%

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

   1. count-2 97.7%

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

   2. *-commutative 97.7%

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

   3. add-sqr-sqrt 97.7%

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

   4. log-prod 97.7%

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

   5. *-commutative 97.7%

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

   6. count-2 97.7%

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

   7. flip-+ -0.0%

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

   8. difference-of-squares -0.0%

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

   9. count-2 -0.0%

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

   10. *-commutative -0.0%

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

   11. associate-*r/ -0.0%

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

   12. +-inverses -0.0%

       \[\leadsto \log \left(\sqrt{\left(x \cdot 2\right) \cdot \frac{\color{blue}{0}}{x - x}}\right) + \log \left(\sqrt{x \cdot 2}\right) \]

   13. +-inverses -0.0%

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

   14. flip-+ 17.9%

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

   15. count-2 17.9%

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

   16. *-commutative 17.9%

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

   17. sqrt-unprod 30.4%

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

   18. add-sqr-sqrt 30.4%

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

   19. *-commutative 30.4%

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

   20. sum-log 30.4%

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

   21. *-commutative 30.4%

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

   22. count-2 30.4%

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

   23. flip-+ -0.0%

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

4. Applied egg-rr -0.0%

   \[\leadsto \color{blue}{\log \left(\frac{0}{0}\right) + \log \left(\frac{0}{0}\right)} \]

6. Simplified 3.0%

   \[\leadsto \color{blue}{-6} \]

7. Final simplification 3.0%

   \[\leadsto -6 \]

Developer target: 99.2% accurate, 0.7× 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 2023339 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

  :herbie-target
  (log (+ x (* (sqrt (- x 1.0)) (sqrt (+ x 1.0)))))

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