Rust f32::acosh

Percentage Accurate: 53.2% → 98.0%

Time: 5.7s

Alternatives: 7

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: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in x around inf 98.7%

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

   1. associate-*r/ 98.7%

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

   2. metadata-eval 98.7%

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

5. Simplified 98.7%

   \[\leadsto \log \color{blue}{\left(x \cdot \left(2 - \frac{0.5}{{x}^{2}}\right)\right)} \]
6. Add Preprocessing

Alternative 2: 96.8% 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 54.7%

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

3. Taylor expanded in x around inf 97.2%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
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 54.7%

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

3. Taylor expanded in x around inf 98.7%

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

   1. associate-*r/ 98.7%

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

   2. metadata-eval 98.7%

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0 -0.0%

   \[\leadsto \color{blue}{\log -0.5 + -1 \cdot \log x} \]

8. Simplified 44.0%

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

Alternative 4: 30.9% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 63.75 \end{array} \]

FPCore

(FPCore (x) :precision binary32 63.75)

C

float code(float x) {
	return 63.75f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 63.75e0
end function

Julia

function code(x)
	return Float32(63.75)
end

MATLAB

function tmp = code(x)
	tmp = single(63.75);
end

Derivation

1. Initial program 54.7%

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

   1. add-sqr-sqrt 54.6%

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

   2. pow2 54.6%

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

   3. log-pow 54.6%

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

   4. fma-neg 54.6%

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

   5. metadata-eval 54.6%

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

4. Applied egg-rr 54.6%

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

5. Taylor expanded in x around 0 -0.0%

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

7. Simplified 2.2%

   \[\leadsto \color{blue}{-4 + \frac{x}{-4}} \]
8. Step-by-step derivation

   1. flip-+ 2.0%

      \[\leadsto \color{blue}{\frac{-4 \cdot -4 - \frac{x}{-4} \cdot \frac{x}{-4}}{-4 - \frac{x}{-4}}} \]

   2. div-inv 2.0%

      \[\leadsto \color{blue}{\left(-4 \cdot -4 - \frac{x}{-4} \cdot \frac{x}{-4}\right) \cdot \frac{1}{-4 - \frac{x}{-4}}} \]

   3. metadata-eval 2.0%

      \[\leadsto \left(\color{blue}{16} - \frac{x}{-4} \cdot \frac{x}{-4}\right) \cdot \frac{1}{-4 - \frac{x}{-4}} \]

   4. pow2 2.0%

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

   5. div-inv 2.0%

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

   6. unpow-prod-down 1.9%

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

   7. metadata-eval 1.9%

      \[\leadsto \left(16 - {x}^{2} \cdot {\color{blue}{-0.25}}^{2}\right) \cdot \frac{1}{-4 - \frac{x}{-4}} \]

   8. metadata-eval 1.9%

      \[\leadsto \left(16 - {x}^{2} \cdot \color{blue}{0.0625}\right) \cdot \frac{1}{-4 - \frac{x}{-4}} \]

   9. div-inv 1.9%

      \[\leadsto \left(16 - {x}^{2} \cdot 0.0625\right) \cdot \frac{1}{-4 - \color{blue}{x \cdot \frac{1}{-4}}} \]

   10. metadata-eval 1.9%

       \[\leadsto \left(16 - {x}^{2} \cdot 0.0625\right) \cdot \frac{1}{-4 - x \cdot \color{blue}{-0.25}} \]

9. Applied egg-rr 1.9%

   \[\leadsto \color{blue}{\left(16 - {x}^{2} \cdot 0.0625\right) \cdot \frac{1}{-4 - x \cdot -0.25}} \]

11. Simplified 30.9%

    \[\leadsto \color{blue}{63.75} \]
12. Add Preprocessing

Alternative 5: 22.8% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 3 \end{array} \]

FPCore

(FPCore (x) :precision binary32 3.0)

C

float code(float x) {
	return 3.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 3.0e0
end function

Julia

function code(x)
	return Float32(3.0)
end

MATLAB

function tmp = code(x)
	tmp = single(3.0);
end

Derivation

1. Initial program 54.7%

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

   1. add-sqr-sqrt 54.6%

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

   2. pow2 54.6%

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

   3. log-pow 54.6%

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

   4. fma-neg 54.6%

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

   5. metadata-eval 54.6%

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

4. Applied egg-rr 54.6%

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

5. Taylor expanded in x around inf 97.2%

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

   1. pow1/2 97.2%

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

   2. flip-+ -0.0%

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

   3. difference-of-squares -0.0%

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

   4. +-inverses -0.0%

      \[\leadsto 2 \cdot \log \left({\left(\frac{\left(x + x\right) \cdot \color{blue}{0}}{x - x}\right)}^{0.5}\right) \]

   5. +-inverses -0.0%

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

   6. +-inverses -0.0%

      \[\leadsto 2 \cdot \log \left({\left(\frac{\left(x + x\right) \cdot \left(x \cdot x - x \cdot x\right)}{\color{blue}{0}}\right)}^{0.5}\right) \]

   7. +-inverses -0.0%

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

   8. associate-*r/ -0.0%

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

   9. difference-of-squares -0.0%

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

   10. +-inverses -0.0%

       \[\leadsto 2 \cdot \log \left({\left(\left(x + x\right) \cdot \frac{\left(x + x\right) \cdot \color{blue}{0}}{x \cdot x - x \cdot x}\right)}^{0.5}\right) \]

   11. +-inverses -0.0%

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

   12. associate-*r/ -0.0%

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

   13. +-inverses -0.0%

       \[\leadsto 2 \cdot \log \left({\left(\left(x + x\right) \cdot \left(\left(x + x\right) \cdot \frac{x \cdot x - x \cdot x}{\color{blue}{0}}\right)\right)}^{0.5}\right) \]

   14. +-inverses -0.0%

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

   15. flip-+ 12.9%

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

   16. associate-*l* 12.9%

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

   17. pow3 12.9%

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

   18. pow-pow 19.9%

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

   19. metadata-eval 19.9%

       \[\leadsto 2 \cdot \log \left({\left(x + x\right)}^{\color{blue}{1.5}}\right) \]

   20. metadata-eval 19.9%

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

7. Applied egg-rr -0.0%

   \[\leadsto 2 \cdot \color{blue}{\left(1.5 \cdot \log \left(\frac{0}{0}\right)\right)} \]

9. Simplified 22.9%

   \[\leadsto 2 \cdot \color{blue}{1.5} \]
10. Step-by-step derivation

    1. metadata-eval 22.9%

       \[\leadsto \color{blue}{3} \]

11. Applied egg-rr 22.9%

    \[\leadsto \color{blue}{3} \]
12. Add Preprocessing

Alternative 6: 21.5% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1.3333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary32 1.3333333333333333)

C

float code(float x) {
	return 1.3333333333333333f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.3333333333333333e0
end function

Julia

function code(x)
	return Float32(1.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = single(1.3333333333333333);
end

Derivation

1. Initial program 54.7%

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

   1. add-sqr-sqrt 54.6%

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

   2. pow2 54.6%

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

   3. log-pow 54.6%

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

   4. fma-neg 54.6%

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

   5. metadata-eval 54.6%

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

4. Applied egg-rr 54.6%

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

5. Taylor expanded in x around inf 97.2%

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

   1. pow1 97.2%

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

   2. pow1 97.2%

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

   3. flip-+ -0.0%

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

   4. difference-of-squares -0.0%

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

   5. +-inverses -0.0%

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

   6. +-inverses -0.0%

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

   7. +-inverses -0.0%

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

   8. +-inverses -0.0%

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

   9. associate-*r/ -0.0%

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

   10. +-inverses -0.0%

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

   11. +-inverses -0.0%

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

   12. flip-+ 17.4%

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

   13. sqrt-unprod 28.0%

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

   14. add-sqr-sqrt 28.0%

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

   15. add-cbrt-cube 13.6%

       \[\leadsto 2 \cdot \log \color{blue}{\left(\sqrt[3]{\left(\left(x + x\right) \cdot \left(x + x\right)\right) \cdot \left(x + x\right)}\right)} \]

   16. pow1/3 13.6%

       \[\leadsto 2 \cdot \log \color{blue}{\left({\left(\left(\left(x + x\right) \cdot \left(x + x\right)\right) \cdot \left(x + x\right)\right)}^{0.3333333333333333}\right)} \]

7. Applied egg-rr -0.0%

   \[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(\frac{0}{0}\right)\right)} \]

9. Simplified 21.5%

   \[\leadsto 2 \cdot \color{blue}{0.6666666666666666} \]
10. Step-by-step derivation

    1. metadata-eval 21.5%

       \[\leadsto \color{blue}{1.3333333333333333} \]

11. Applied egg-rr 21.5%

    \[\leadsto \color{blue}{1.3333333333333333} \]
12. Add Preprocessing

Alternative 7: 21.4% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1.25 \end{array} \]

FPCore

(FPCore (x) :precision binary32 1.25)

C

float code(float x) {
	return 1.25f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.25e0
end function

Julia

function code(x)
	return Float32(1.25)
end

MATLAB

function tmp = code(x)
	tmp = single(1.25);
end

Derivation

1. Initial program 54.7%

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

   1. add-sqr-sqrt 54.6%

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

   2. pow2 54.6%

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

   3. log-pow 54.6%

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

   4. fma-neg 54.6%

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

   5. metadata-eval 54.6%

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

4. Applied egg-rr 54.6%

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

5. Taylor expanded in x around 0 -0.0%

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

7. Simplified 2.2%

   \[\leadsto \color{blue}{-4 + \frac{x}{-4}} \]

8. Taylor expanded in x around inf 2.2%

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(0.25 + 4 \cdot \frac{1}{x}\right)\right)} \]

10. Simplified 21.4%

    \[\leadsto \color{blue}{1.25} \]
11. 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 2024137 
(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)))))