Rust f32::acosh

Percentage Accurate: 51.8% → 97.6%

Time: 4.7s

Alternatives: 14

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: 51.8% 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: 97.6% 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 49.6%

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

   1. flip-+ 7.5%

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

   2. div-inv 7.5%

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

   3. log-prod 7.5%

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

   4. pow2 7.5%

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

   5. add-sqr-sqrt 7.1%

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

   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 97.0%

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

Alternative 2: 96.6% 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 49.6%

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

3. Taylor expanded in x around inf 95.2%

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

Alternative 3: 25.5% 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 49.6%

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

   1. flip-+ 7.5%

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

   2. div-inv 7.5%

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

   3. log-prod 7.5%

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

   4. pow2 7.5%

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

   5. add-sqr-sqrt 7.1%

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

   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 97.0%

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

9. Applied egg-rr 25.3%

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

Alternative 4: 25.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 8 \end{array} \]

FPCore

(FPCore (x) :precision binary32 8.0)

C

float code(float x) {
	return 8.0f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 8.0e0
end function

Julia

function code(x)
	return Float32(8.0)
end

MATLAB

function tmp = code(x)
	tmp = single(8.0);
end

Derivation

1. Initial program 49.6%

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

   1. flip-+ 7.5%

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

   2. div-inv 7.5%

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

   3. log-prod 7.5%

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

   4. pow2 7.5%

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

   5. add-sqr-sqrt 7.1%

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

   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 97.0%

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

9. Applied egg-rr 24.8%

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

Alternative 5: 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 49.6%

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

   1. flip-+ 7.5%

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

   2. div-inv 7.5%

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

   3. log-prod 7.5%

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

   4. pow2 7.5%

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

   5. add-sqr-sqrt 7.1%

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

   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 97.0%

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

9. Applied egg-rr 24.0%

   \[\leadsto \color{blue}{6} \]
10. Add Preprocessing

Alternative 6: 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 49.6%

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

   1. flip-+ 7.5%

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

   2. div-inv 7.5%

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

   3. log-prod 7.5%

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

   4. pow2 7.5%

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

   5. add-sqr-sqrt 7.1%

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

   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 97.0%

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

9. Applied egg-rr 23.7%

   \[\leadsto \color{blue}{5} \]
10. Add Preprocessing

Alternative 7: 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 49.6%

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

   1. flip-+ 7.5%

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

   2. div-inv 7.5%

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

   3. log-prod 7.5%

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

   4. pow2 7.5%

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

   5. add-sqr-sqrt 7.1%

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

   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 97.0%

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

9. Applied egg-rr 23.3%

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

Alternative 8: 22.9% 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 49.6%

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

   1. flip-+ 7.5%

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

   2. div-inv 7.5%

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

   3. log-prod 7.5%

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

   4. pow2 7.5%

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

   5. add-sqr-sqrt 7.1%

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

   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 97.0%

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

9. Applied egg-rr 22.7%

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

Alternative 9: 22.5% 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 49.6%

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

   1. flip-+ 7.5%

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

   2. div-inv 7.5%

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

   3. log-prod 7.5%

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

   4. pow2 7.5%

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

   5. add-sqr-sqrt 7.1%

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

   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 97.0%

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

9. Applied egg-rr 22.6%

   \[\leadsto \color{blue}{2.5} \]
10. Add Preprocessing

Alternative 10: 22.2% 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 49.6%

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

   1. flip-+ 7.5%

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

   2. div-inv 7.5%

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

   3. log-prod 7.5%

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

   4. pow2 7.5%

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

   5. add-sqr-sqrt 7.1%

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

   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 97.0%

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

9. Applied egg-rr 22.3%

   \[\leadsto \color{blue}{2} \]
10. Add Preprocessing

Alternative 11: 21.6% 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 49.6%

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

   1. flip-+ 7.5%

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

   2. div-inv 7.5%

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

   3. log-prod 7.5%

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

   4. pow2 7.5%

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

   5. add-sqr-sqrt 7.1%

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

   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 97.0%

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

9. Applied egg-rr 21.6%

   \[\leadsto \color{blue}{1.5} \]
10. Add Preprocessing

Alternative 12: 21.1% 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 49.6%

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

   1. flip-+ 7.5%

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

   2. div-inv 7.5%

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

   3. log-prod 7.5%

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

   4. pow2 7.5%

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

   5. add-sqr-sqrt 7.1%

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

   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 97.0%

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

9. Applied egg-rr 21.0%

   \[\leadsto \color{blue}{1} \]
10. Add Preprocessing

Alternative 13: 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 49.6%

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

   1. flip-+ 7.5%

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

   2. div-inv 7.5%

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

   3. log-prod 7.5%

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

   4. pow2 7.5%

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

   5. add-sqr-sqrt 7.1%

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

   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 97.0%

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

9. Applied egg-rr 20.1%

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

Alternative 14: 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 49.6%

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

3. Taylor expanded in x around inf 95.2%

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

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

   4. +-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: 99.0% 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 2024176 
(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)))))