Rust f32::acosh

Percentage Accurate: 52.1% → 99.3%

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: 52.1% 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: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ -\log \left(\frac{0.5 + \left(\frac{0.0625}{{x}^{4}} + 0.125 \cdot \frac{1}{{x}^{2}}\right)}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (-
  (log
   (/ (+ 0.5 (+ (/ 0.0625 (pow x 4.0)) (* 0.125 (/ 1.0 (pow x 2.0))))) x))))

C

float code(float x) {
	return -logf(((0.5f + ((0.0625f / powf(x, 4.0f)) + (0.125f * (1.0f / powf(x, 2.0f))))) / x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = -log(((0.5e0 + ((0.0625e0 / (x ** 4.0e0)) + (0.125e0 * (1.0e0 / (x ** 2.0e0))))) / x))
end function

Julia

function code(x)
	return Float32(-log(Float32(Float32(Float32(0.5) + Float32(Float32(Float32(0.0625) / (x ^ Float32(4.0))) + Float32(Float32(0.125) * Float32(Float32(1.0) / (x ^ Float32(2.0)))))) / x)))
end

MATLAB

function tmp = code(x)
	tmp = -log(((single(0.5) + ((single(0.0625) / (x ^ single(4.0))) + (single(0.125) * (single(1.0) / (x ^ single(2.0)))))) / x));
end

Derivation

1. Initial program 57.6%

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

   1. flip-+ 7.1%

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

   2. div-inv 7.1%

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

   3. log-prod 7.1%

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

   4. pow2 7.1%

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

   5. add-sqr-sqrt 6.8%

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

   6. fma-neg 6.6%

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

   7. metadata-eval 6.6%

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

   8. fma-neg 6.6%

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

   9. metadata-eval 6.6%

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

4. Applied egg-rr 6.6%

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

   1. log-rec 6.6%

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

   2. sub-neg 6.6%

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

   3. fma-undefine 6.7%

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

   4. unpow2 6.7%

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

   5. associate--r+ 9.5%

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

   6. +-inverses 9.5%

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

   7. metadata-eval 9.5%

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

   8. metadata-eval 9.5%

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

   9. neg-sub0 9.5%

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

6. Simplified 9.5%

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

7. Taylor expanded in x around inf 99.2%

   \[\leadsto -\log \color{blue}{\left(\frac{0.5 + \left(\frac{0.0625}{{x}^{4}} + 0.125 \cdot \frac{1}{{x}^{2}}\right)}{x}\right)} \]
8. Add Preprocessing

Alternative 2: 98.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(x \cdot \left(2 - \frac{\frac{0.5 + x}{x}}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log1p (* x (- 2.0 (/ (/ (+ 0.5 x) x) x)))))

C

float code(float x) {
	return log1pf((x * (2.0f - (((0.5f + x) / x) / x))));
}

Julia

function code(x)
	return log1p(Float32(x * Float32(Float32(2.0) - Float32(Float32(Float32(Float32(0.5) + x) / x) / x))))
end

Derivation

1. Initial program 57.6%

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

   1. log1p-expm1-u 57.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(x + \sqrt{x \cdot x - 1}\right)\right)\right)} \]

   2. expm1-undefine 57.6%

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

   3. add-exp-log 57.6%

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

   4. fma-neg 57.6%

      \[\leadsto \mathsf{log1p}\left(\left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right) - 1\right) \]

   5. metadata-eval 57.6%

      \[\leadsto \mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right) - 1\right) \]

4. Applied egg-rr 57.6%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) - 1\right)} \]

5. Taylor expanded in x around inf 97.8%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(2 + -1 \cdot \frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right) \]
6. Step-by-step derivation

   1. un-div-inv 97.8%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + -1 \cdot \frac{1 + \color{blue}{\frac{0.5}{x}}}{x}\right)\right) \]

   2. frac-2neg 97.8%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + -1 \cdot \frac{1 + \color{blue}{\frac{-0.5}{-x}}}{x}\right)\right) \]

   3. metadata-eval 97.8%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + -1 \cdot \frac{1 + \frac{\color{blue}{-0.5}}{-x}}{x}\right)\right) \]

7. Applied egg-rr 97.8%

   \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + -1 \cdot \frac{1 + \color{blue}{\frac{-0.5}{-x}}}{x}\right)\right) \]

8. Taylor expanded in x around 0 97.8%

   \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + -1 \cdot \frac{\color{blue}{\frac{0.5 + x}{x}}}{x}\right)\right) \]
9. Step-by-step derivation

   1. +-commutative 97.8%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + -1 \cdot \frac{\frac{\color{blue}{x + 0.5}}{x}}{x}\right)\right) \]

10. Simplified 97.8%

    \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + -1 \cdot \frac{\color{blue}{\frac{x + 0.5}{x}}}{x}\right)\right) \]

11. Final simplification 97.8%

    \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 - \frac{\frac{0.5 + x}{x}}{x}\right)\right) \]
12. Add Preprocessing

Alternative 3: 98.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(x \cdot \left(2 - \frac{1 + \frac{0.5}{x}}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log1p (* x (- 2.0 (/ (+ 1.0 (/ 0.5 x)) x)))))

C

float code(float x) {
	return log1pf((x * (2.0f - ((1.0f + (0.5f / x)) / x))));
}

Julia

function code(x)
	return log1p(Float32(x * Float32(Float32(2.0) - Float32(Float32(Float32(1.0) + Float32(Float32(0.5) / x)) / x))))
end

Derivation

1. Initial program 57.6%

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

   1. log1p-expm1-u 57.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(x + \sqrt{x \cdot x - 1}\right)\right)\right)} \]

   2. expm1-undefine 57.6%

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

   3. add-exp-log 57.6%

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

   4. fma-neg 57.6%

      \[\leadsto \mathsf{log1p}\left(\left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right) - 1\right) \]

   5. metadata-eval 57.6%

      \[\leadsto \mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right) - 1\right) \]

4. Applied egg-rr 57.6%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) - 1\right)} \]

5. Taylor expanded in x around inf 97.8%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(2 + -1 \cdot \frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right) \]
6. Step-by-step derivation

   1. un-div-inv 97.8%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + -1 \cdot \frac{1 + \color{blue}{\frac{0.5}{x}}}{x}\right)\right) \]

   2. frac-2neg 97.8%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + -1 \cdot \frac{1 + \color{blue}{\frac{-0.5}{-x}}}{x}\right)\right) \]

   3. metadata-eval 97.8%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + -1 \cdot \frac{1 + \frac{\color{blue}{-0.5}}{-x}}{x}\right)\right) \]

7. Applied egg-rr 97.8%

   \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + -1 \cdot \frac{1 + \color{blue}{\frac{-0.5}{-x}}}{x}\right)\right) \]

8. Taylor expanded in x around inf 97.8%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(2 + -1 \cdot \frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right) \]
9. Step-by-step derivation

   1. mul-1-neg 97.8%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 + \color{blue}{\left(-\frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right)\right) \]

   2. unsub-neg 97.8%

      \[\leadsto \mathsf{log1p}\left(x \cdot \color{blue}{\left(2 - \frac{1 + 0.5 \cdot \frac{1}{x}}{x}\right)}\right) \]

   3. associate-*r/ 97.8%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 - \frac{1 + \color{blue}{\frac{0.5 \cdot 1}{x}}}{x}\right)\right) \]

   4. metadata-eval 97.8%

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(2 - \frac{1 + \frac{\color{blue}{0.5}}{x}}{x}\right)\right) \]

10. Simplified 97.8%

    \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(2 - \frac{1 + \frac{0.5}{x}}{x}\right)}\right) \]
11. Add Preprocessing

Alternative 4: 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 57.6%

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

   1. flip-+ 7.1%

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

   2. div-inv 7.1%

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

   3. log-prod 7.1%

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

   4. pow2 7.1%

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

   5. add-sqr-sqrt 6.8%

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

   6. fma-neg 6.6%

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

   7. metadata-eval 6.6%

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

   8. fma-neg 6.6%

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

   9. metadata-eval 6.6%

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

4. Applied egg-rr 6.6%

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

   1. log-rec 6.6%

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

   2. sub-neg 6.6%

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

   3. fma-undefine 6.7%

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

   4. unpow2 6.7%

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

   5. associate--r+ 9.5%

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

   6. +-inverses 9.5%

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

   7. metadata-eval 9.5%

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

   8. metadata-eval 9.5%

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

   9. neg-sub0 9.5%

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

6. Simplified 9.5%

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

7. Taylor expanded in x around inf 99.2%

   \[\leadsto -\log \color{blue}{\left(\frac{0.5 + \left(\frac{0.0625}{{x}^{4}} + 0.125 \cdot \frac{1}{{x}^{2}}\right)}{x}\right)} \]

8. Taylor expanded in x around inf 97.6%

   \[\leadsto -\log \color{blue}{\left(\frac{0.5}{x}\right)} \]
9. 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 57.6%

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

3. Taylor expanded in x around inf 96.5%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
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 57.6%

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

3. Taylor expanded in x around inf 96.5%

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

   1. add-sqr-sqrt 96.5%

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

   2. log-prod 96.5%

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

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. Add Preprocessing

Developer target: 99.2% 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 2024103 
(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)))))