Rust f32::acosh

Percentage Accurate: 53.1% → 99.1%

Time: 6.6s

Alternatives: 6

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: 53.1% 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: 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

Derivation

1. Initial program 51.4%

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

   1. pow1/2 51.4%

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

   2. difference-of-sqr-1 51.4%

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

   3. unpow-prod-down 99.2%

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

   4. sub-neg 99.2%

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

   5. metadata-eval 99.2%

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

4. Applied egg-rr 99.2%

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

   1. unpow1/2 99.2%

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

   2. unpow1/2 99.2%

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

6. Simplified 99.2%

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

Alternative 2: 98.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 51.4%

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

3. Taylor expanded in x around inf 98.4%

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

   1. associate-*r/ 98.4%

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

   2. metadata-eval 98.4%

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

5. Simplified 98.4%

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

   1. unpow2 98.4%

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

7. Applied egg-rr 98.4%

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

Alternative 3: 98.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 51.4%

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

3. Taylor expanded in x around inf 98.4%

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

   1. associate-*r/ 98.4%

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

   2. metadata-eval 98.4%

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

5. Simplified 98.4%

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

   1. unpow2 98.4%

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

7. Applied egg-rr 98.4%

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

   1. sub-neg 98.4%

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

   2. distribute-lft-in 98.4%

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

   3. metadata-eval 98.4%

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

   4. div-inv 98.4%

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

   5. add-cbrt-cube 35.9%

      \[\leadsto \log \left(\frac{\color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}}}{0.5} + x \cdot \left(-\frac{0.5}{x \cdot x}\right)\right) \]

   6. cbrt-prod 50.5%

      \[\leadsto \log \left(\frac{\color{blue}{\sqrt[3]{x \cdot x} \cdot \sqrt[3]{x}}}{0.5} + x \cdot \left(-\frac{0.5}{x \cdot x}\right)\right) \]

   7. associate-/l* 50.5%

      \[\leadsto \log \left(\color{blue}{\sqrt[3]{x \cdot x} \cdot \frac{\sqrt[3]{x}}{0.5}} + x \cdot \left(-\frac{0.5}{x \cdot x}\right)\right) \]

   8. fma-define 50.5%

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{x \cdot x}, \frac{\sqrt[3]{x}}{0.5}, x \cdot \left(-\frac{0.5}{x \cdot x}\right)\right)\right)} \]

   9. cbrt-prod 98.4%

      \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{x}}, \frac{\sqrt[3]{x}}{0.5}, x \cdot \left(-\frac{0.5}{x \cdot x}\right)\right)\right) \]

   10. pow2 98.4%

       \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}, \frac{\sqrt[3]{x}}{0.5}, x \cdot \left(-\frac{0.5}{x \cdot x}\right)\right)\right) \]

   11. div-inv 98.4%

       \[\leadsto \log \left(\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \frac{\sqrt[3]{x}}{0.5}, x \cdot \left(-\color{blue}{0.5 \cdot \frac{1}{x \cdot x}}\right)\right)\right) \]

   12. distribute-lft-neg-in 98.4%

       \[\leadsto \log \left(\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \frac{\sqrt[3]{x}}{0.5}, x \cdot \color{blue}{\left(\left(-0.5\right) \cdot \frac{1}{x \cdot x}\right)}\right)\right) \]

   13. metadata-eval 98.4%

       \[\leadsto \log \left(\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \frac{\sqrt[3]{x}}{0.5}, x \cdot \left(\color{blue}{-0.5} \cdot \frac{1}{x \cdot x}\right)\right)\right) \]

   14. pow2 98.4%

       \[\leadsto \log \left(\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \frac{\sqrt[3]{x}}{0.5}, x \cdot \left(-0.5 \cdot \frac{1}{\color{blue}{{x}^{2}}}\right)\right)\right) \]

   15. pow-flip 98.4%

       \[\leadsto \log \left(\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \frac{\sqrt[3]{x}}{0.5}, x \cdot \left(-0.5 \cdot \color{blue}{{x}^{\left(-2\right)}}\right)\right)\right) \]

   16. metadata-eval 98.4%

       \[\leadsto \log \left(\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \frac{\sqrt[3]{x}}{0.5}, x \cdot \left(-0.5 \cdot {x}^{\color{blue}{-2}}\right)\right)\right) \]

9. Applied egg-rr 98.4%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left({\left(\sqrt[3]{x}\right)}^{2}, \frac{\sqrt[3]{x}}{0.5}, x \cdot \left(-0.5 \cdot {x}^{-2}\right)\right)\right)} \]
10. Step-by-step derivation

    1. fma-undefine 98.4%

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

    2. associate-*r/ 98.4%

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

    3. unpow2 98.4%

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

    4. rem-3cbrt-lft 98.4%

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

    5. *-commutative 98.4%

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

    6. associate-*l* 98.4%

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

    7. metadata-eval 98.4%

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

    8. pow-plus 98.4%

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

    9. metadata-eval 98.4%

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

    10. unpow-1 98.4%

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

    11. distribute-lft-neg-in 98.4%

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

    12. associate-*r/ 98.4%

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

    13. metadata-eval 98.4%

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

    14. distribute-neg-frac 98.4%

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

    15. metadata-eval 98.4%

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

11. Simplified 98.4%

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

Alternative 4: 97.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 51.4%

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

   1. flip-+ 7.0%

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

   2. div-inv 7.0%

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

   3. log-prod 7.0%

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

   4. pow2 7.0%

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

   5. add-sqr-sqrt 7.0%

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

   6. fmm-def 7.0%

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

   7. metadata-eval 7.0%

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

   8. fmm-def 7.0%

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

   9. metadata-eval 7.0%

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

4. Applied egg-rr 7.0%

   \[\leadsto \color{blue}{\log \left({x}^{2} - \mathsf{fma}\left(x, x, -1\right)\right) + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right)} \]
5. Step-by-step derivation

   1. log-rec 7.0%

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

   2. sub-neg 7.0%

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

   3. fma-undefine 7.0%

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

   4. unpow2 7.0%

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

   5. associate--r+ 9.4%

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

   6. +-inverses 9.4%

      \[\leadsto \log \left(\color{blue}{0} - -1\right) - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) \]

   7. metadata-eval 9.4%

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

   8. metadata-eval 9.4%

      \[\leadsto \color{blue}{0} - \log \left(x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) \]

   9. neg-sub0 9.4%

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

6. Simplified 9.4%

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

7. Taylor expanded in x around inf 97.8%

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

Alternative 5: 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 51.4%

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

3. Taylor expanded in x around inf 97.1%

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

Alternative 6: 6.1% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary32 0.0)

C

float code(float x) {
	return 0.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 0.0e0
end function

Julia

function code(x)
	return Float32(0.0)
end

MATLAB

function tmp = code(x)
	tmp = single(0.0);
end

Derivation

1. Initial program 51.4%

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

3. Taylor expanded in x around inf 97.1%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
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. log-div -0.0%

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

   3. unpow2 0.0%

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

   4. unpow2 -0.0%

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

   5. +-inverses -0.0%

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

   6. +-inverses -0.0%

      \[\leadsto \log 0 - \log \color{blue}{0} \]

5. Applied egg-rr -0.0%

   \[\leadsto \color{blue}{\log 0 - \log 0} \]
6. Step-by-step derivation

   1. +-inverses 6.1%

      \[\leadsto \color{blue}{0} \]

7. Simplified 6.1%

   \[\leadsto \color{blue}{0} \]
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 2024180 
(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)))))