Rust f32::acosh

Percentage Accurate: 52.7% → 97.9%

Time: 8.6s

Alternatives: 8

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: 52.7% 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: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log 2 + \left(\log x - \frac{0.25}{x \cdot x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (+ (log 2.0) (- (log x) (/ 0.25 (* x x)))))

C

float code(float x) {
	return logf(2.0f) + (logf(x) - (0.25f / (x * x)));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = log(2.0e0) + (log(x) - (0.25e0 / (x * x)))
end function

Julia

function code(x)
	return Float32(log(Float32(2.0)) + Float32(log(x) - Float32(Float32(0.25) / Float32(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = log(single(2.0)) + (log(x) - (single(0.25) / (x * x)));
end

Derivation

1. Initial program 47.0%

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

2. Taylor expanded in x around inf 97.5%

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]
3. Step-by-step derivation

   1. associate--l+ 97.5%

      \[\leadsto \color{blue}{\log 2 + \left(-1 \cdot \log \left(\frac{1}{x}\right) - 0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

   2. mul-1-neg 97.5%

      \[\leadsto \log 2 + \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} - 0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   3. log-rec 97.5%

      \[\leadsto \log 2 + \left(\left(-\color{blue}{\left(-\log x\right)}\right) - 0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   4. remove-double-neg 97.5%

      \[\leadsto \log 2 + \left(\color{blue}{\log x} - 0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   5. unpow2 97.5%

      \[\leadsto \log 2 + \left(\log x - 0.25 \cdot \frac{1}{\color{blue}{x \cdot x}}\right) \]

   6. associate-*r/ 97.5%

      \[\leadsto \log 2 + \left(\log x - \color{blue}{\frac{0.25 \cdot 1}{x \cdot x}}\right) \]

   7. metadata-eval 97.5%

      \[\leadsto \log 2 + \left(\log x - \frac{\color{blue}{0.25}}{x \cdot x}\right) \]

4. Simplified 97.5%

   \[\leadsto \color{blue}{\log 2 + \left(\log x - \frac{0.25}{x \cdot x}\right)} \]

5. Final simplification 97.5%

   \[\leadsto \log 2 + \left(\log x - \frac{0.25}{x \cdot x}\right) \]

Alternative 2: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\left(2 \cdot x - \left(\frac{0.5}{x} + \frac{0.25}{x \cdot x} \cdot \frac{0.5}{x}\right)\right) + -1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (log1p (+ (- (* 2.0 x) (+ (/ 0.5 x) (* (/ 0.25 (* x x)) (/ 0.5 x)))) -1.0)))

C

float code(float x) {
	return log1pf((((2.0f * x) - ((0.5f / x) + ((0.25f / (x * x)) * (0.5f / x)))) + -1.0f));
}

Julia

function code(x)
	return log1p(Float32(Float32(Float32(Float32(2.0) * x) - Float32(Float32(Float32(0.5) / x) + Float32(Float32(Float32(0.25) / Float32(x * x)) * Float32(Float32(0.5) / x)))) + Float32(-1.0)))
end

Derivation

1. Initial program 47.0%

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

   1. log1p-expm1-u 47.0%

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

   2. expm1-udef 47.0%

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

   3. add-exp-log 47.0%

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

   4. fma-neg 47.0%

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

   5. metadata-eval 47.0%

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

3. Applied egg-rr 47.0%

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

4. Taylor expanded in x around inf 97.4%

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

   1. *-commutative 97.4%

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

   2. +-commutative 97.4%

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

   3. associate-*r/ 97.4%

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

   4. metadata-eval 97.4%

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

   5. associate-*r/ 97.4%

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

   6. metadata-eval 97.4%

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

6. Simplified 97.4%

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

   1. metadata-eval 97.4%

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

   2. cube-div 97.4%

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

   3. cube-mult 97.4%

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

   4. frac-times 97.4%

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

   5. metadata-eval 97.4%

      \[\leadsto \mathsf{log1p}\left(\left(x \cdot 2 - \left(\frac{0.5}{x} + \frac{0.5}{x} \cdot \frac{\color{blue}{0.25}}{x \cdot x}\right)\right) - 1\right) \]

8. Applied egg-rr 97.4%

   \[\leadsto \mathsf{log1p}\left(\left(x \cdot 2 - \left(\frac{0.5}{x} + \color{blue}{\frac{0.5}{x} \cdot \frac{0.25}{x \cdot x}}\right)\right) - 1\right) \]

9. Final simplification 97.4%

   \[\leadsto \mathsf{log1p}\left(\left(2 \cdot x - \left(\frac{0.5}{x} + \frac{0.25}{x \cdot x} \cdot \frac{0.5}{x}\right)\right) + -1\right) \]

Alternative 3: 98.0% 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 47.0%

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

2. Taylor expanded in x around inf 97.0%

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

   1. associate-*r/ 97.0%

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

   2. metadata-eval 97.0%

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

4. Simplified 97.0%

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

5. Final simplification 97.0%

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

Alternative 4: 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 47.0%

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

2. Taylor expanded in x around inf 95.6%

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

3. Final simplification 95.6%

   \[\leadsto \log \left(x + x\right) \]

Alternative 5: 20.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log 2 \end{array} \]

FPCore

(FPCore (x) :precision binary32 (log 2.0))

C

float code(float x) {
	return logf(2.0f);
}

Fortran

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

Julia

function code(x)
	return log(Float32(2.0))
end

MATLAB

function tmp = code(x)
	tmp = log(single(2.0));
end

Derivation

1. Initial program 47.0%

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

2. Taylor expanded in x around inf 97.5%

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]
3. Step-by-step derivation

   1. associate--l+ 97.5%

      \[\leadsto \color{blue}{\log 2 + \left(-1 \cdot \log \left(\frac{1}{x}\right) - 0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

   2. mul-1-neg 97.5%

      \[\leadsto \log 2 + \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} - 0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   3. log-rec 97.5%

      \[\leadsto \log 2 + \left(\left(-\color{blue}{\left(-\log x\right)}\right) - 0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   4. remove-double-neg 97.5%

      \[\leadsto \log 2 + \left(\color{blue}{\log x} - 0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   5. unpow2 97.5%

      \[\leadsto \log 2 + \left(\log x - 0.25 \cdot \frac{1}{\color{blue}{x \cdot x}}\right) \]

   6. associate-*r/ 97.5%

      \[\leadsto \log 2 + \left(\log x - \color{blue}{\frac{0.25 \cdot 1}{x \cdot x}}\right) \]

   7. metadata-eval 97.5%

      \[\leadsto \log 2 + \left(\log x - \frac{\color{blue}{0.25}}{x \cdot x}\right) \]

4. Simplified 97.5%

   \[\leadsto \color{blue}{\log 2 + \left(\log x - \frac{0.25}{x \cdot x}\right)} \]
5. Step-by-step derivation

   1. expm1-log1p-u 95.7%

      \[\leadsto \log 2 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log x - \frac{0.25}{x \cdot x}\right)\right)} \]

   2. metadata-eval 95.7%

      \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x - \frac{\color{blue}{0.5 \cdot 0.5}}{x \cdot x}\right)\right) \]

   3. frac-times 95.7%

      \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x - \color{blue}{\frac{0.5}{x} \cdot \frac{0.5}{x}}\right)\right) \]

   4. cancel-sign-sub-inv 95.7%

      \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\log x + \left(-\frac{0.5}{x}\right) \cdot \frac{0.5}{x}}\right)\right) \]

   5. distribute-lft-neg-in 95.7%

      \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \color{blue}{\left(-\frac{0.5}{x} \cdot \frac{0.5}{x}\right)}\right)\right) \]

   6. frac-times 95.7%

      \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\color{blue}{\frac{0.5 \cdot 0.5}{x \cdot x}}\right)\right)\right) \]

   7. metadata-eval 95.7%

      \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{\color{blue}{0.25}}{x \cdot x}\right)\right)\right) \]

   8. associate-/r* 95.7%

      \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\color{blue}{\frac{\frac{0.25}{x}}{x}}\right)\right)\right) \]

   9. associate-/r* 95.7%

      \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\color{blue}{\frac{0.25}{x \cdot x}}\right)\right)\right) \]

   10. clear-num 95.7%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\color{blue}{\frac{1}{\frac{x \cdot x}{0.25}}}\right)\right)\right) \]

   11. div-inv 95.7%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \frac{1}{0.25}}}\right)\right)\right) \]

   12. metadata-eval 95.7%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\left(x \cdot x\right) \cdot \color{blue}{4}}\right)\right)\right) \]

   13. metadata-eval 95.7%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\left(x \cdot x\right) \cdot \color{blue}{\left(2 \cdot 2\right)}}\right)\right)\right) \]

   14. swap-sqr 95.7%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\color{blue}{\left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}}\right)\right)\right) \]

   15. *-commutative 95.7%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\left(x \cdot 2\right) \cdot \color{blue}{\left(2 \cdot x\right)}}\right)\right)\right) \]

   16. count-2 95.7%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\left(x \cdot 2\right) \cdot \color{blue}{\left(x + x\right)}}\right)\right)\right) \]

   17. flip-+ -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\left(x \cdot 2\right) \cdot \color{blue}{\frac{x \cdot x - x \cdot x}{x - x}}}\right)\right)\right) \]

   18. +-inverses -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\left(x \cdot 2\right) \cdot \frac{x \cdot x - x \cdot x}{\color{blue}{0}}}\right)\right)\right) \]

   19. +-inverses -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\left(x \cdot 2\right) \cdot \frac{x \cdot x - x \cdot x}{\color{blue}{x \cdot x - x \cdot x}}}\right)\right)\right) \]

   20. associate-*r/ -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\color{blue}{\frac{\left(x \cdot 2\right) \cdot \left(x \cdot x - x \cdot x\right)}{x \cdot x - x \cdot x}}}\right)\right)\right) \]

   21. *-commutative -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\frac{\color{blue}{\left(2 \cdot x\right)} \cdot \left(x \cdot x - x \cdot x\right)}{x \cdot x - x \cdot x}}\right)\right)\right) \]

   22. count-2 -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\frac{\color{blue}{\left(x + x\right)} \cdot \left(x \cdot x - x \cdot x\right)}{x \cdot x - x \cdot x}}\right)\right)\right) \]

   23. +-inverses -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\frac{\left(x + x\right) \cdot \color{blue}{0}}{x \cdot x - x \cdot x}}\right)\right)\right) \]

   24. +-inverses -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\frac{\left(x + x\right) \cdot \color{blue}{\left(x - x\right)}}{x \cdot x - x \cdot x}}\right)\right)\right) \]

   25. difference-of-squares -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\frac{\color{blue}{x \cdot x - x \cdot x}}{x \cdot x - x \cdot x}}\right)\right)\right) \]

   26. +-inverses -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\frac{x \cdot x - x \cdot x}{\color{blue}{0}}}\right)\right)\right) \]

   27. +-inverses -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\frac{x \cdot x - x \cdot x}{\color{blue}{x - x}}}\right)\right)\right) \]

6. Applied egg-rr 91.7%

   \[\leadsto \log 2 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \frac{-0.5}{x}\right)\right)} \]

7. Taylor expanded in x around 0 20.5%

   \[\leadsto \log 2 + \color{blue}{\frac{-0.5}{x}} \]

8. Taylor expanded in x around inf 20.6%

   \[\leadsto \color{blue}{\log 2} \]

9. Final simplification 20.6%

   \[\leadsto \log 2 \]

Alternative 6: 5.3% accurate, 41.4× speedup

Math

\[\begin{array}{l} \\ \frac{-0.25}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary32 (/ -0.25 (* x x)))

C

float code(float x) {
	return -0.25f / (x * x);
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = (-0.25e0) / (x * x)
end function

Julia

function code(x)
	return Float32(Float32(-0.25) / Float32(x * x))
end

MATLAB

function tmp = code(x)
	tmp = single(-0.25) / (x * x);
end

Derivation

1. Initial program 47.0%

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

   1. log1p-expm1-u 47.0%

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

   2. expm1-udef 47.0%

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

   3. add-exp-log 47.0%

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

   4. fma-neg 47.0%

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

   5. metadata-eval 47.0%

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

3. Applied egg-rr 47.0%

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

4. Taylor expanded in x around inf 97.5%

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]
5. Step-by-step derivation

   1. +-commutative 97.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right)} - 0.25 \cdot \frac{1}{{x}^{2}} \]

   2. mul-1-neg 97.5%

      \[\leadsto \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \log 2\right) - 0.25 \cdot \frac{1}{{x}^{2}} \]

   3. log-rec 97.5%

      \[\leadsto \left(\left(-\color{blue}{\left(-\log x\right)}\right) + \log 2\right) - 0.25 \cdot \frac{1}{{x}^{2}} \]

   4. remove-double-neg 97.5%

      \[\leadsto \left(\color{blue}{\log x} + \log 2\right) - 0.25 \cdot \frac{1}{{x}^{2}} \]

   5. log-prod 96.9%

      \[\leadsto \color{blue}{\log \left(x \cdot 2\right)} - 0.25 \cdot \frac{1}{{x}^{2}} \]

   6. unpow2 96.9%

      \[\leadsto \log \left(x \cdot 2\right) - 0.25 \cdot \frac{1}{\color{blue}{x \cdot x}} \]

   7. associate-*r/ 96.9%

      \[\leadsto \log \left(x \cdot 2\right) - \color{blue}{\frac{0.25 \cdot 1}{x \cdot x}} \]

   8. metadata-eval 96.9%

      \[\leadsto \log \left(x \cdot 2\right) - \frac{\color{blue}{0.25}}{x \cdot x} \]

   9. associate-/r* 96.9%

      \[\leadsto \log \left(x \cdot 2\right) - \color{blue}{\frac{\frac{0.25}{x}}{x}} \]

6. Simplified 96.9%

   \[\leadsto \color{blue}{\log \left(x \cdot 2\right) - \frac{\frac{0.25}{x}}{x}} \]

7. Taylor expanded in x around 0 5.4%

   \[\leadsto \color{blue}{\frac{-0.25}{{x}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 5.4%

      \[\leadsto \frac{-0.25}{\color{blue}{x \cdot x}} \]

9. Simplified 5.4%

   \[\leadsto \color{blue}{\frac{-0.25}{x \cdot x}} \]

10. Final simplification 5.4%

    \[\leadsto \frac{-0.25}{x \cdot x} \]

Alternative 7: 4.5% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 47.0%

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

2. Taylor expanded in x around inf 97.5%

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]
3. Step-by-step derivation

   1. associate--l+ 97.5%

      \[\leadsto \color{blue}{\log 2 + \left(-1 \cdot \log \left(\frac{1}{x}\right) - 0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

   2. mul-1-neg 97.5%

      \[\leadsto \log 2 + \left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} - 0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   3. log-rec 97.5%

      \[\leadsto \log 2 + \left(\left(-\color{blue}{\left(-\log x\right)}\right) - 0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   4. remove-double-neg 97.5%

      \[\leadsto \log 2 + \left(\color{blue}{\log x} - 0.25 \cdot \frac{1}{{x}^{2}}\right) \]

   5. unpow2 97.5%

      \[\leadsto \log 2 + \left(\log x - 0.25 \cdot \frac{1}{\color{blue}{x \cdot x}}\right) \]

   6. associate-*r/ 97.5%

      \[\leadsto \log 2 + \left(\log x - \color{blue}{\frac{0.25 \cdot 1}{x \cdot x}}\right) \]

   7. metadata-eval 97.5%

      \[\leadsto \log 2 + \left(\log x - \frac{\color{blue}{0.25}}{x \cdot x}\right) \]

4. Simplified 97.5%

   \[\leadsto \color{blue}{\log 2 + \left(\log x - \frac{0.25}{x \cdot x}\right)} \]
5. Step-by-step derivation

   1. expm1-log1p-u 95.7%

      \[\leadsto \log 2 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log x - \frac{0.25}{x \cdot x}\right)\right)} \]

   2. metadata-eval 95.7%

      \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x - \frac{\color{blue}{0.5 \cdot 0.5}}{x \cdot x}\right)\right) \]

   3. frac-times 95.7%

      \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x - \color{blue}{\frac{0.5}{x} \cdot \frac{0.5}{x}}\right)\right) \]

   4. cancel-sign-sub-inv 95.7%

      \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\log x + \left(-\frac{0.5}{x}\right) \cdot \frac{0.5}{x}}\right)\right) \]

   5. distribute-lft-neg-in 95.7%

      \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \color{blue}{\left(-\frac{0.5}{x} \cdot \frac{0.5}{x}\right)}\right)\right) \]

   6. frac-times 95.7%

      \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\color{blue}{\frac{0.5 \cdot 0.5}{x \cdot x}}\right)\right)\right) \]

   7. metadata-eval 95.7%

      \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{\color{blue}{0.25}}{x \cdot x}\right)\right)\right) \]

   8. associate-/r* 95.7%

      \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\color{blue}{\frac{\frac{0.25}{x}}{x}}\right)\right)\right) \]

   9. associate-/r* 95.7%

      \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\color{blue}{\frac{0.25}{x \cdot x}}\right)\right)\right) \]

   10. clear-num 95.7%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\color{blue}{\frac{1}{\frac{x \cdot x}{0.25}}}\right)\right)\right) \]

   11. div-inv 95.7%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \frac{1}{0.25}}}\right)\right)\right) \]

   12. metadata-eval 95.7%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\left(x \cdot x\right) \cdot \color{blue}{4}}\right)\right)\right) \]

   13. metadata-eval 95.7%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\left(x \cdot x\right) \cdot \color{blue}{\left(2 \cdot 2\right)}}\right)\right)\right) \]

   14. swap-sqr 95.7%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\color{blue}{\left(x \cdot 2\right) \cdot \left(x \cdot 2\right)}}\right)\right)\right) \]

   15. *-commutative 95.7%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\left(x \cdot 2\right) \cdot \color{blue}{\left(2 \cdot x\right)}}\right)\right)\right) \]

   16. count-2 95.7%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\left(x \cdot 2\right) \cdot \color{blue}{\left(x + x\right)}}\right)\right)\right) \]

   17. flip-+ -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\left(x \cdot 2\right) \cdot \color{blue}{\frac{x \cdot x - x \cdot x}{x - x}}}\right)\right)\right) \]

   18. +-inverses -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\left(x \cdot 2\right) \cdot \frac{x \cdot x - x \cdot x}{\color{blue}{0}}}\right)\right)\right) \]

   19. +-inverses -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\left(x \cdot 2\right) \cdot \frac{x \cdot x - x \cdot x}{\color{blue}{x \cdot x - x \cdot x}}}\right)\right)\right) \]

   20. associate-*r/ -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\color{blue}{\frac{\left(x \cdot 2\right) \cdot \left(x \cdot x - x \cdot x\right)}{x \cdot x - x \cdot x}}}\right)\right)\right) \]

   21. *-commutative -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\frac{\color{blue}{\left(2 \cdot x\right)} \cdot \left(x \cdot x - x \cdot x\right)}{x \cdot x - x \cdot x}}\right)\right)\right) \]

   22. count-2 -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\frac{\color{blue}{\left(x + x\right)} \cdot \left(x \cdot x - x \cdot x\right)}{x \cdot x - x \cdot x}}\right)\right)\right) \]

   23. +-inverses -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\frac{\left(x + x\right) \cdot \color{blue}{0}}{x \cdot x - x \cdot x}}\right)\right)\right) \]

   24. +-inverses -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\frac{\left(x + x\right) \cdot \color{blue}{\left(x - x\right)}}{x \cdot x - x \cdot x}}\right)\right)\right) \]

   25. difference-of-squares -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\frac{\color{blue}{x \cdot x - x \cdot x}}{x \cdot x - x \cdot x}}\right)\right)\right) \]

   26. +-inverses -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\frac{x \cdot x - x \cdot x}{\color{blue}{0}}}\right)\right)\right) \]

   27. +-inverses -0.0%

       \[\leadsto \log 2 + \mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \left(-\frac{1}{\frac{x \cdot x - x \cdot x}{\color{blue}{x - x}}}\right)\right)\right) \]

6. Applied egg-rr 91.7%

   \[\leadsto \log 2 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log x + \frac{-0.5}{x}\right)\right)} \]

7. Taylor expanded in x around 0 20.5%

   \[\leadsto \log 2 + \color{blue}{\frac{-0.5}{x}} \]

8. Taylor expanded in x around 0 4.5%

   \[\leadsto \color{blue}{\frac{-0.5}{x}} \]

9. Final simplification 4.5%

   \[\leadsto \frac{-0.5}{x} \]

Alternative 8: 3.2% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -0.03125 \end{array} \]

FPCore

(FPCore (x) :precision binary32 -0.03125)

C

float code(float x) {
	return -0.03125f;
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = -0.03125e0
end function

Julia

function code(x)
	return Float32(-0.03125)
end

MATLAB

function tmp = code(x)
	tmp = single(-0.03125);
end

Derivation

1. Initial program 47.0%

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

2. Taylor expanded in x around inf 97.0%

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

   1. associate-*r/ 97.0%

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

   2. metadata-eval 97.0%

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

4. Simplified 97.0%

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

5. Taylor expanded in x around inf 97.6%

   \[\leadsto \color{blue}{\left(\log 2 + -1 \cdot \log \left(\frac{1}{x}\right)\right) - \left(0.03125 \cdot \frac{1}{{x}^{4}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

7. Simplified 3.2%

   \[\leadsto \color{blue}{-0.03125} \]

8. Final simplification 3.2%

   \[\leadsto -0.03125 \]

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 2023275 
(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)))))