Rust f32::acosh

Percentage Accurate: 52.8% → 98.8%

Time: 8.2s

Alternatives: 7

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: 52.8% 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: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \log \left(\left(x \cdot 2 - \frac{0.0625}{{x}^{5}}\right) - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log
  (-
   (- (* x 2.0) (/ 0.0625 (pow x 5.0)))
   (+ (/ 0.5 x) (/ 0.125 (pow x 3.0))))))

C

float code(float x) {
	return logf((((x * 2.0f) - (0.0625f / powf(x, 5.0f))) - ((0.5f / x) + (0.125f / powf(x, 3.0f)))));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log((((x * 2.0e0) - (0.0625e0 / (x ** 5.0e0))) - ((0.5e0 / x) + (0.125e0 / (x ** 3.0e0)))))
end function

Julia

function code(x)
	return log(Float32(Float32(Float32(x * Float32(2.0)) - Float32(Float32(0.0625) / (x ^ Float32(5.0)))) - Float32(Float32(Float32(0.5) / x) + Float32(Float32(0.125) / (x ^ Float32(3.0))))))
end

MATLAB

function tmp = code(x)
	tmp = log((((x * single(2.0)) - (single(0.0625) / (x ^ single(5.0)))) - ((single(0.5) / x) + (single(0.125) / (x ^ single(3.0))))));
end

Derivation

1. Initial program 52.1%

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

3. Taylor expanded in x around inf 99.0%

   \[\leadsto \log \color{blue}{\left(2 \cdot x - \left(0.0625 \cdot \frac{1}{{x}^{5}} + \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate--r+ 99.0%

      \[\leadsto \log \color{blue}{\left(\left(2 \cdot x - 0.0625 \cdot \frac{1}{{x}^{5}}\right) - \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)\right)} \]

   2. *-commutative 99.0%

      \[\leadsto \log \left(\left(\color{blue}{x \cdot 2} - 0.0625 \cdot \frac{1}{{x}^{5}}\right) - \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)\right) \]

   3. associate-*r/ 99.0%

      \[\leadsto \log \left(\left(x \cdot 2 - \color{blue}{\frac{0.0625 \cdot 1}{{x}^{5}}}\right) - \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)\right) \]

   4. metadata-eval 99.0%

      \[\leadsto \log \left(\left(x \cdot 2 - \frac{\color{blue}{0.0625}}{{x}^{5}}\right) - \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)\right) \]

   5. +-commutative 99.0%

      \[\leadsto \log \left(\left(x \cdot 2 - \frac{0.0625}{{x}^{5}}\right) - \color{blue}{\left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)}\right) \]

   6. associate-*r/ 99.0%

      \[\leadsto \log \left(\left(x \cdot 2 - \frac{0.0625}{{x}^{5}}\right) - \left(\color{blue}{\frac{0.5 \cdot 1}{x}} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]

   7. metadata-eval 99.0%

      \[\leadsto \log \left(\left(x \cdot 2 - \frac{0.0625}{{x}^{5}}\right) - \left(\frac{\color{blue}{0.5}}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]

   8. associate-*r/ 99.0%

      \[\leadsto \log \left(\left(x \cdot 2 - \frac{0.0625}{{x}^{5}}\right) - \left(\frac{0.5}{x} + \color{blue}{\frac{0.125 \cdot 1}{{x}^{3}}}\right)\right) \]

   9. metadata-eval 99.0%

      \[\leadsto \log \left(\left(x \cdot 2 - \frac{0.0625}{{x}^{5}}\right) - \left(\frac{0.5}{x} + \frac{\color{blue}{0.125}}{{x}^{3}}\right)\right) \]

5. Simplified 99.0%

   \[\leadsto \log \color{blue}{\left(\left(x \cdot 2 - \frac{0.0625}{{x}^{5}}\right) - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right)} \]

6. Final simplification 99.0%

   \[\leadsto \log \left(\left(x \cdot 2 - \frac{0.0625}{{x}^{5}}\right) - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right) \]
7. Add Preprocessing

Alternative 2: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(x \cdot 2 + \left(-1 - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log1p (+ (* x 2.0) (- -1.0 (+ (/ 0.5 x) (/ 0.125 (pow x 3.0)))))))

C

float code(float x) {
	return log1pf(((x * 2.0f) + (-1.0f - ((0.5f / x) + (0.125f / powf(x, 3.0f))))));
}

Julia

function code(x)
	return log1p(Float32(Float32(x * Float32(2.0)) + Float32(Float32(-1.0) - Float32(Float32(Float32(0.5) / x) + Float32(Float32(0.125) / (x ^ Float32(3.0)))))))
end

Derivation

1. Initial program 52.1%

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

   1. log1p-expm1-u 52.1%

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

   2. expm1-undefine 52.1%

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

   3. add-exp-log 52.1%

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

   4. fma-neg 52.1%

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

   5. metadata-eval 52.1%

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

4. Applied egg-rr 52.1%

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

5. Taylor expanded in x around inf 98.9%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{2 \cdot x - \left(1 + \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)\right)}\right) \]

6. Taylor expanded in x around 0 98.9%

   \[\leadsto \mathsf{log1p}\left(2 \cdot x - \left(1 + \color{blue}{\left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)}\right)\right) \]
7. Step-by-step derivation

   1. associate-*r/ 98.9%

      \[\leadsto \mathsf{log1p}\left(2 \cdot x - \left(1 + \left(\color{blue}{\frac{0.5 \cdot 1}{x}} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

   2. metadata-eval 98.9%

      \[\leadsto \mathsf{log1p}\left(2 \cdot x - \left(1 + \left(\frac{\color{blue}{0.5}}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

   3. associate-*r/ 98.9%

      \[\leadsto \mathsf{log1p}\left(2 \cdot x - \left(1 + \left(\frac{0.5}{x} + \color{blue}{\frac{0.125 \cdot 1}{{x}^{3}}}\right)\right)\right) \]

   4. metadata-eval 98.9%

      \[\leadsto \mathsf{log1p}\left(2 \cdot x - \left(1 + \left(\frac{0.5}{x} + \frac{\color{blue}{0.125}}{{x}^{3}}\right)\right)\right) \]

8. Simplified 98.9%

   \[\leadsto \mathsf{log1p}\left(2 \cdot x - \left(1 + \color{blue}{\left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)}\right)\right) \]

9. Final simplification 98.9%

   \[\leadsto \mathsf{log1p}\left(x \cdot 2 + \left(-1 - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right)\right) \]
10. Add Preprocessing

Alternative 3: 98.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.1%

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

3. Taylor expanded in x around inf 98.5%

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

   1. associate-*r/ 98.5%

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

   2. metadata-eval 98.5%

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

5. Simplified 98.5%

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

6. Final simplification 98.5%

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

Alternative 4: 98.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.1%

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

3. Taylor expanded in x around inf 98.5%

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

   1. *-commutative 98.5%

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

   2. associate-*r/ 98.5%

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

   3. metadata-eval 98.5%

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

5. Simplified 98.5%

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

6. Final simplification 98.5%

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

Alternative 5: 97.0% 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 52.1%

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

3. Taylor expanded in x around inf 97.0%

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

4. Final simplification 97.0%

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

Alternative 6: 44.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log x \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log x))

C

float code(float x) {
	return logf(x);
}

Fortran

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

Julia

function code(x)
	return log(x)
end

MATLAB

function tmp = code(x)
	tmp = log(x);
end

Derivation

1. Initial program 52.1%

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

3. Taylor expanded in x around inf 98.5%

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

   1. associate-*r/ 98.5%

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

   2. metadata-eval 98.5%

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

5. Simplified 98.5%

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

   1. *-un-lft-identity 98.5%

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

   2. fma-define 98.5%

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

   3. clear-num 98.5%

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

   4. div-inv 98.5%

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

   5. metadata-eval 98.5%

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

   6. *-commutative 98.5%

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

   7. count-2 98.5%

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

   8. flip-+ -0.0%

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

   9. +-inverses -0.0%

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

   10. +-inverses -0.0%

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

   11. +-inverses -0.0%

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

   12. +-inverses -0.0%

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

   13. clear-num -0.0%

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

   14. +-inverses -0.0%

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

   15. +-inverses -0.0%

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

7. Applied egg-rr -0.0%

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

9. Simplified 43.9%

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

10. Taylor expanded in x around inf 44.3%

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

    1. mul-1-neg 44.3%

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

    2. log-rec 44.3%

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

    3. remove-double-neg 44.3%

       \[\leadsto \color{blue}{\log x} \]

12. Simplified 44.3%

    \[\leadsto \color{blue}{\log x} \]

13. Final simplification 44.3%

    \[\leadsto \log x \]
14. Add Preprocessing

Alternative 7: 3.1% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 -1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.1%

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

3. Taylor expanded in x around inf 99.0%

   \[\leadsto \log \color{blue}{\left(2 \cdot x - \left(0.0625 \cdot \frac{1}{{x}^{5}} + \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate--r+ 99.0%

      \[\leadsto \log \color{blue}{\left(\left(2 \cdot x - 0.0625 \cdot \frac{1}{{x}^{5}}\right) - \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)\right)} \]

   2. *-commutative 99.0%

      \[\leadsto \log \left(\left(\color{blue}{x \cdot 2} - 0.0625 \cdot \frac{1}{{x}^{5}}\right) - \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)\right) \]

   3. associate-*r/ 99.0%

      \[\leadsto \log \left(\left(x \cdot 2 - \color{blue}{\frac{0.0625 \cdot 1}{{x}^{5}}}\right) - \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)\right) \]

   4. metadata-eval 99.0%

      \[\leadsto \log \left(\left(x \cdot 2 - \frac{\color{blue}{0.0625}}{{x}^{5}}\right) - \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)\right) \]

   5. +-commutative 99.0%

      \[\leadsto \log \left(\left(x \cdot 2 - \frac{0.0625}{{x}^{5}}\right) - \color{blue}{\left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)}\right) \]

   6. associate-*r/ 99.0%

      \[\leadsto \log \left(\left(x \cdot 2 - \frac{0.0625}{{x}^{5}}\right) - \left(\color{blue}{\frac{0.5 \cdot 1}{x}} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]

   7. metadata-eval 99.0%

      \[\leadsto \log \left(\left(x \cdot 2 - \frac{0.0625}{{x}^{5}}\right) - \left(\frac{\color{blue}{0.5}}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]

   8. associate-*r/ 99.0%

      \[\leadsto \log \left(\left(x \cdot 2 - \frac{0.0625}{{x}^{5}}\right) - \left(\frac{0.5}{x} + \color{blue}{\frac{0.125 \cdot 1}{{x}^{3}}}\right)\right) \]

   9. metadata-eval 99.0%

      \[\leadsto \log \left(\left(x \cdot 2 - \frac{0.0625}{{x}^{5}}\right) - \left(\frac{0.5}{x} + \frac{\color{blue}{0.125}}{{x}^{3}}\right)\right) \]

5. Simplified 99.0%

   \[\leadsto \log \color{blue}{\left(\left(x \cdot 2 - \frac{0.0625}{{x}^{5}}\right) - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right)} \]

6. Taylor expanded in x around inf 96.5%

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

8. Simplified 3.1%

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

9. Final simplification 3.1%

   \[\leadsto -1 \]
10. 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 2024039 
(FPCore (x)
  :name "Rust f32::acosh"
  :precision binary32
  :pre (>= x 1.0)

  :herbie-target
  (log (+ x (* (sqrt (- x 1.0)) (sqrt (+ x 1.0)))))

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