Rust f32::acosh

Percentage Accurate: 53.9% → 97.9%

Time: 4.1s

Alternatives: 12

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: 53.9% 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: 97.9% 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 54.0%

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

2. Taylor expanded in x around inf 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*r/ 98.8%

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

   3. metadata-eval 98.8%

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

4. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 2: 96.5% 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 54.0%

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

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 3: 3.1% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary32 -1.0)

C

float code(float x) {
	return -1.0f;
}

Fortran

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

Julia

function code(x)
	return Float32(-1.0)
end

MATLAB

function tmp = code(x)
	tmp = single(-1.0);
end

Derivation

1. Initial program 54.0%

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

2. Taylor expanded in x around inf 97.7%

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

   1. 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. count-2 -0.0%

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

   4. *-commutative -0.0%

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

   5. associate-*r/ -0.0%

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

   6. +-inverses -0.0%

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

   7. +-inverses -0.0%

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

   8. flip-+ 17.2%

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

   9. count-2 17.2%

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

   10. *-commutative 17.2%

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

   11. sum-log 28.0%

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

   12. *-commutative 28.0%

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

   13. count-2 28.0%

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

   14. flip-+ -0.0%

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

   15. +-inverses -0.0%

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

   16. +-inverses -0.0%

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

   17. *-commutative -0.0%

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

   18. count-2 -0.0%

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

   19. flip-+ -0.0%

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

   20. +-inverses -0.0%

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

   21. +-inverses -0.0%

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

4. Applied egg-rr -0.0%

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

6. Simplified 3.1%

   \[\leadsto \color{blue}{-1} \]

7. Final simplification 3.1%

   \[\leadsto -1 \]

Alternative 4: 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 54.0%

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

2. Taylor expanded in x around inf 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*r/ 98.8%

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

   3. metadata-eval 98.8%

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

4. Simplified 98.8%

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

5. Taylor expanded in x around inf 98.3%

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

7. Simplified 40.0%

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

9. Applied egg-rr 21.4%

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

10. Final simplification 21.4%

    \[\leadsto 1.3333333333333333 \]

Alternative 5: 21.7% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1.5 \end{array} \]

FPCore

(FPCore (x) :precision binary32 1.5)

C

float code(float x) {
	return 1.5f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.5e0
end function

Julia

function code(x)
	return Float32(1.5)
end

MATLAB

function tmp = code(x)
	tmp = single(1.5);
end

Derivation

1. Initial program 54.0%

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

2. Taylor expanded in x around inf 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*r/ 98.8%

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

   3. metadata-eval 98.8%

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

4. Simplified 98.8%

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

5. Taylor expanded in x around inf 98.3%

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

7. Simplified 40.0%

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

9. Applied egg-rr 21.5%

   \[\leadsto \color{blue}{1.5} \]

10. Final simplification 21.5%

    \[\leadsto 1.5 \]

Alternative 6: 22.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary32 2.0)

C

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

Fortran

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

Julia

function code(x)
	return Float32(2.0)
end

MATLAB

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

Derivation

1. Initial program 54.0%

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

2. Taylor expanded in x around inf 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*r/ 98.8%

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

   3. metadata-eval 98.8%

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

4. Simplified 98.8%

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

5. Taylor expanded in x around inf 98.3%

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

7. Simplified 40.0%

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

9. Applied egg-rr 22.2%

   \[\leadsto \color{blue}{2} \]

10. Final simplification 22.2%

    \[\leadsto 2 \]

Alternative 7: 22.6% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 2.5 \end{array} \]

FPCore

(FPCore (x) :precision binary32 2.5)

C

float code(float x) {
	return 2.5f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 2.5e0
end function

Julia

function code(x)
	return Float32(2.5)
end

MATLAB

function tmp = code(x)
	tmp = single(2.5);
end

Derivation

1. Initial program 54.0%

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

2. Taylor expanded in x around inf 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*r/ 98.8%

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

   3. metadata-eval 98.8%

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

4. Simplified 98.8%

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

5. Taylor expanded in x around inf 98.3%

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

7. Simplified 40.0%

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

9. Applied egg-rr 22.5%

   \[\leadsto \color{blue}{2.5} \]

10. Final simplification 22.5%

    \[\leadsto 2.5 \]

Alternative 8: 23.6% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 4.0)

C

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

Fortran

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

Julia

function code(x)
	return Float32(4.0)
end

MATLAB

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

Derivation

1. Initial program 54.0%

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

2. Taylor expanded in x around inf 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*r/ 98.8%

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

   3. metadata-eval 98.8%

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

4. Simplified 98.8%

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

5. Taylor expanded in x around inf 98.3%

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

7. Simplified 40.0%

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

9. Applied egg-rr 23.4%

   \[\leadsto \color{blue}{4} \]

10. Final simplification 23.4%

    \[\leadsto 4 \]

Alternative 9: 23.8% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 4.5 \end{array} \]

FPCore

(FPCore (x) :precision binary32 4.5)

C

float code(float x) {
	return 4.5f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 4.5e0
end function

Julia

function code(x)
	return Float32(4.5)
end

MATLAB

function tmp = code(x)
	tmp = single(4.5);
end

Derivation

1. Initial program 54.0%

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

2. Taylor expanded in x around inf 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*r/ 98.8%

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

   3. metadata-eval 98.8%

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

4. Simplified 98.8%

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

5. Taylor expanded in x around inf 98.3%

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

7. Simplified 40.0%

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

9. Applied egg-rr 23.6%

   \[\leadsto \color{blue}{4.5} \]

10. Final simplification 23.6%

    \[\leadsto 4.5 \]

Alternative 10: 24.0% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 5 \end{array} \]

FPCore

(FPCore (x) :precision binary32 5.0)

C

float code(float x) {
	return 5.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 5.0e0
end function

Julia

function code(x)
	return Float32(5.0)
end

MATLAB

function tmp = code(x)
	tmp = single(5.0);
end

Derivation

1. Initial program 54.0%

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

2. Taylor expanded in x around inf 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*r/ 98.8%

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

   3. metadata-eval 98.8%

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

4. Simplified 98.8%

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

5. Taylor expanded in x around inf 98.3%

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

7. Simplified 40.0%

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

9. Applied egg-rr 23.8%

   \[\leadsto \color{blue}{5} \]

10. Final simplification 23.8%

    \[\leadsto 5 \]

Alternative 11: 24.4% 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 54.0%

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

2. Taylor expanded in x around inf 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*r/ 98.8%

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

   3. metadata-eval 98.8%

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

4. Simplified 98.8%

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

5. Taylor expanded in x around inf 98.3%

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

7. Simplified 40.0%

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

9. Applied egg-rr 24.2%

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

10. Final simplification 24.2%

    \[\leadsto 6 \]

Alternative 12: 25.6% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 9 \end{array} \]

FPCore

(FPCore (x) :precision binary32 9.0)

C

float code(float x) {
	return 9.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 9.0e0
end function

Julia

function code(x)
	return Float32(9.0)
end

MATLAB

function tmp = code(x)
	tmp = single(9.0);
end

Derivation

1. Initial program 54.0%

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

2. Taylor expanded in x around inf 98.8%

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

   1. *-commutative 98.8%

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

   2. associate-*r/ 98.8%

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

   3. metadata-eval 98.8%

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

4. Simplified 98.8%

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

5. Taylor expanded in x around inf 98.3%

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

7. Simplified 40.0%

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

9. Applied egg-rr 25.5%

   \[\leadsto \color{blue}{9} \]

10. Final simplification 25.5%

    \[\leadsto 9 \]

Developer target: 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 2023325 
(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)))))