Rust f32::acosh

Percentage Accurate: 52.2% → 96.9%

Time: 5.4s

Alternatives: 5

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: 52.2% 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: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log 2 + \log x \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 46.7%

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

3. Taylor expanded in x around inf 97.0%

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

   1. mul-1-neg 97.0%

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

   2. log-rec 97.0%

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

   3. remove-double-neg 97.0%

      \[\leadsto \log 2 + \color{blue}{\log x} \]

5. Simplified 97.0%

   \[\leadsto \color{blue}{\log 2 + \log x} \]
6. Add Preprocessing

Alternative 2: 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 46.7%

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

3. Taylor expanded in x around inf 96.5%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
4. Add Preprocessing

Alternative 3: 44.2% accurate, 2.0× 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 46.7%

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

3. Taylor expanded in x around inf 96.5%

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

4. Taylor expanded in x around 0 97.0%

   \[\leadsto \color{blue}{\log 2 + \log x} \]

6. Simplified 44.5%

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

Alternative 4: 21.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log 4 \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log 4.0))

C

float code(float x) {
	return logf(4.0f);
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log(4.0e0)
end function

Julia

function code(x)
	return log(Float32(4.0))
end

MATLAB

function tmp = code(x)
	tmp = log(single(4.0));
end

Derivation

1. Initial program 46.7%

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

3. Taylor expanded in x around inf 96.5%

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

   1. flip-+ -0.0%

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

   2. difference-of-squares -0.0%

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

   3. +-inverses -0.0%

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

   4. +-inverses -0.0%

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

   5. +-inverses -0.0%

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

   6. +-inverses -0.0%

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

   7. associate-*r/ -0.0%

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

   8. +-inverses -0.0%

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

   9. +-inverses -0.0%

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

   10. flip-+ 15.8%

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

   11. pow2 15.8%

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

5. Applied egg-rr -0.0%

   \[\leadsto \log \color{blue}{\left(4 \cdot {\left(x \cdot \frac{0}{0}\right)}^{2}\right)} \]

7. Simplified 21.3%

   \[\leadsto \log \color{blue}{4} \]
8. Add Preprocessing

Alternative 5: 20.2% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary32 0.5)

C

float code(float x) {
	return 0.5f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.5e0
end function

Julia

function code(x)
	return Float32(0.5)
end

MATLAB

function tmp = code(x)
	tmp = single(0.5);
end

Derivation

1. Initial program 46.7%

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

3. Taylor expanded in x around inf 96.5%

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

   1. flip-+ -0.0%

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

   2. +-inverses -0.0%

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

   3. +-inverses -0.0%

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

   4. diff-log -0.0%

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

   5. +-inverses -0.0%

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

   6. metadata-eval -0.0%

      \[\leadsto \log \color{blue}{\left({0}^{0.5}\right)} - \log \left(x \cdot x - x \cdot x\right) \]

   7. +-inverses -0.0%

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

   8. pow1/2 -0.0%

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

   9. +-inverses -0.0%

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

   10. metadata-eval -0.0%

       \[\leadsto \log \left(\sqrt{x \cdot x - x \cdot x}\right) - \log \color{blue}{\left({0}^{0.5}\right)} \]

   11. +-inverses -0.0%

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

   12. pow1/2 -0.0%

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

   13. log-div -0.0%

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

   14. +-inverses -0.0%

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

   15. +-inverses -0.0%

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

   16. sqrt-div -0.0%

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

   17. flip-+ 27.8%

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

   18. pow1/2 27.8%

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

   19. log-pow 27.8%

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

5. Applied egg-rr -0.0%

   \[\leadsto \color{blue}{0.5 \cdot \log \left(\frac{0}{0}\right)} \]

7. Simplified 20.1%

   \[\leadsto \color{blue}{0.5} \]
8. Add Preprocessing

Developer Target 1: 99.1% 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 2024152 
(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)))))