Rust f32::acosh

Percentage Accurate: 52.3% → 98.8%

Time: 7.9s

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: 52.3% 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, 1.9× speedup

Math

\[\begin{array}{l} \\ -\log \left(\frac{0.5 + \frac{\frac{0.125}{x}}{x}}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (- (log (/ (+ 0.5 (/ (/ 0.125 x) x)) x))))

C

float code(float x) {
	return -logf(((0.5f + ((0.125f / x) / x)) / x));
}

Fortran

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

Julia

function code(x)
	return Float32(-log(Float32(Float32(Float32(0.5) + Float32(Float32(Float32(0.125) / x) / x)) / x)))
end

MATLAB

function tmp = code(x)
	tmp = -log(((single(0.5) + ((single(0.125) / x) / x)) / x));
end

Derivation

1. Initial program 50.2%

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

   1. flip-+ 7.9%

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

   2. div-inv 7.9%

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

   3. log-prod 7.9%

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

   4. pow2 7.9%

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

   5. add-sqr-sqrt 7.6%

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

   6. fma-neg 7.6%

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

   7. metadata-eval 7.6%

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

   8. fma-neg 7.6%

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

   9. metadata-eval 7.6%

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

4. Applied egg-rr 7.6%

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

   1. log-rec 7.6%

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

   2. sub-neg 7.6%

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

   3. fma-undefine 7.6%

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

   4. unpow2 7.6%

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

   5. associate--r+ 9.9%

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

   6. +-inverses 9.9%

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

   7. metadata-eval 9.9%

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

   8. metadata-eval 9.9%

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

   9. neg-sub0 9.9%

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

6. Simplified 9.9%

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

7. Taylor expanded in x around inf 99.1%

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

   1. un-div-inv 99.1%

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

   2. unpow2 99.1%

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

   3. associate-/r* 99.1%

      \[\leadsto -\log \left(\frac{0.5 + \color{blue}{\frac{\frac{0.125}{x}}{x}}}{x}\right) \]

9. Applied egg-rr 99.1%

   \[\leadsto -\log \left(\frac{0.5 + \color{blue}{\frac{\frac{0.125}{x}}{x}}}{x}\right) \]

10. Final simplification 99.1%

    \[\leadsto -\log \left(\frac{0.5 + \frac{\frac{0.125}{x}}{x}}{x}\right) \]
11. Add Preprocessing

Alternative 2: 97.6% 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 50.2%

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

   1. flip-+ 7.9%

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

   2. div-inv 7.9%

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

   3. log-prod 7.9%

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

   4. pow2 7.9%

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

   5. add-sqr-sqrt 7.6%

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

   6. fma-neg 7.6%

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

   7. metadata-eval 7.6%

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

   8. fma-neg 7.6%

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

   9. metadata-eval 7.6%

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

4. Applied egg-rr 7.6%

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

   1. log-rec 7.6%

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

   2. sub-neg 7.6%

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

   3. fma-undefine 7.6%

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

   4. unpow2 7.6%

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

   5. associate--r+ 9.9%

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

   6. +-inverses 9.9%

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

   7. metadata-eval 9.9%

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

   8. metadata-eval 9.9%

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

   9. neg-sub0 9.9%

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

6. Simplified 9.9%

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

7. Taylor expanded in x around inf 97.7%

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

8. Final simplification 97.7%

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

Alternative 3: 96.7% 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 50.2%

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

3. Taylor expanded in x around inf 96.3%

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

4. Final simplification 96.3%

   \[\leadsto \log \left(x + x\right) \]
5. Add Preprocessing

Alternative 4: 6.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -\mathsf{log1p}\left(-1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (- (log1p -1.0)))

C

float code(float x) {
	return -log1pf(-1.0f);
}

Julia

function code(x)
	return Float32(-log1p(Float32(-1.0)))
end

Derivation

1. Initial program 50.2%

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

   1. flip-+ 7.9%

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

   2. div-inv 7.9%

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

   3. log-prod 7.9%

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

   4. pow2 7.9%

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

   5. add-sqr-sqrt 7.6%

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

   6. fma-neg 7.6%

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

   7. metadata-eval 7.6%

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

   8. fma-neg 7.6%

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

   9. metadata-eval 7.6%

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

4. Applied egg-rr 7.6%

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

   1. log-rec 7.6%

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

   2. sub-neg 7.6%

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

   3. fma-undefine 7.6%

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

   4. unpow2 7.6%

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

   5. associate--r+ 9.9%

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

   6. +-inverses 9.9%

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

   7. metadata-eval 9.9%

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

   8. metadata-eval 9.9%

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

   9. neg-sub0 9.9%

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

6. Simplified 9.9%

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

7. Taylor expanded in x around inf 97.7%

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

   1. log1p-expm1-u 18.2%

      \[\leadsto -\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{0.5}{x}\right)\right)\right)} \]

   2. expm1-undefine 18.2%

      \[\leadsto -\mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{0.5}{x}\right)} - 1}\right) \]

   3. add-exp-log 18.2%

      \[\leadsto -\mathsf{log1p}\left(\color{blue}{\frac{0.5}{x}} - 1\right) \]

9. Applied egg-rr 18.2%

   \[\leadsto -\color{blue}{\mathsf{log1p}\left(\frac{0.5}{x} - 1\right)} \]

10. Taylor expanded in x around inf 6.4%

    \[\leadsto -\mathsf{log1p}\left(\color{blue}{-1}\right) \]

11. Final simplification 6.4%

    \[\leadsto -\mathsf{log1p}\left(-1\right) \]
12. Add Preprocessing

Alternative 5: 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 50.2%

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

3. Taylor expanded in x around inf 96.3%

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

   1. add-sqr-sqrt 96.3%

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

   2. log-prod 96.3%

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

   3. flip-+ -0.0%

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

   4. difference-of-squares -0.0%

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

   5. +-inverses -0.0%

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

   6. metadata-eval -0.0%

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

   7. +-inverses -0.0%

      \[\leadsto \log \left(\sqrt{\frac{\left(x + x\right) \cdot \log 1}{\color{blue}{0}}}\right) + \log \left(\sqrt{x + x}\right) \]

   8. metadata-eval -0.0%

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

   9. associate-*r/ -0.0%

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

   10. metadata-eval -0.0%

       \[\leadsto \log \left(\sqrt{\left(x + x\right) \cdot \frac{\color{blue}{0}}{\log 1}}\right) + \log \left(\sqrt{x + x}\right) \]

   11. +-inverses -0.0%

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

   12. unpow2 -0.0%

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

   13. unpow2 -0.0%

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

   14. metadata-eval -0.0%

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

   15. +-inverses -0.0%

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

   16. flip-+ 17.6%

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

   17. sqrt-unprod 30.2%

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

   18. add-sqr-sqrt 30.2%

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

   19. count-2 30.2%

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

   20. sum-log 30.2%

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

   21. flip-+ -0.0%

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

5. Applied egg-rr -0.0%

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

7. Simplified 6.1%

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

8. Final simplification 6.1%

   \[\leadsto 0 \]
9. Add Preprocessing

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 2024078 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

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

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