Rust f32::acosh

Percentage Accurate: 53.4% → 98.8%

Time: 9.3s

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: 53.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary32 (log (+ x (sqrt (- (* x x) 1.0)))))

C

float code(float x) {
	return logf((x + sqrtf(((x * x) - 1.0f))));
}

Fortran

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

Julia

function code(x)
	return log(Float32(x + sqrt(Float32(Float32(x * x) - Float32(1.0)))))
end

MATLAB

function tmp = code(x)
	tmp = log((x + sqrt(((x * x) - single(1.0)))));
end

Alternative 1: 98.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.2%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. log-rec N/A

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

   4. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\log \left(\frac{x - \sqrt{x \cdot x - 1}}{x \cdot x - \sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}}\right)\right) \]

   5. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{x - \sqrt{x \cdot x - 1}}{x \cdot x - \sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}}\right)\right)\right) \]

   6. clear-num N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{\frac{x \cdot x - \sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}}{x - \sqrt{x \cdot x - 1}}}\right)\right)\right) \]

   7. flip-+ N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{x + \sqrt{x \cdot x - 1}}\right)\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(x + \sqrt{x \cdot x - 1}\right)\right)\right)\right) \]

   9. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(x, \left(\sqrt{x \cdot x - 1}\right)\right)\right)\right)\right) \]

   10. pow1/2 N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(x, \left({\left(x \cdot x - 1\right)}^{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   11. rem-square-sqrt N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(x, \left({\left(\sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}\right)}^{\frac{1}{2}}\right)\right)\right)\right)\right) \]

4. Applied egg-rr 53.1%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{\left(x \cdot \left(2 - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right)\right) \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(x \cdot \left(2 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(x \cdot \left(\left(\mathsf{neg}\left(-2\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   3. distribute-neg-in N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(x \cdot \left(\mathsf{neg}\left(\left(-2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(x \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} + -2\right)\right)\right)\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(x \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(x \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} - 2\right)\right)\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} - 2\right)\right)\right)\right)\right)\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} + -2\right)\right)\right)\right)\right)\right)\right) \]

   10. distribute-neg-in N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) + \left(\mathsf{neg}\left(-2\right)\right)\right)\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) + 2\right)\right)\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(2 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   13. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   14. associate-*r/ N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   16. distribute-neg-frac N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \left(\frac{\frac{-1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

   18. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{-1}{2}, \left({x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

   19. unpow2 N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{-1}{2}, \left(x \cdot x\right)\right)\right)\right)\right)\right)\right) \]

   20. *-lowering-*.f32 97.4%

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 97.4%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{\left(2 + \frac{\frac{-1}{2}}{x \cdot x}\right) \cdot x}\right)\right)\right) \]

   2. associate-/r* N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{2 + \frac{\frac{-1}{2}}{x \cdot x}}}{x}\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{2 + \frac{\frac{-1}{2}}{x \cdot x}}\right), x\right)\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \left(2 + \frac{\frac{-1}{2}}{x \cdot x}\right)\right), x\right)\right)\right) \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \left(\frac{\frac{-1}{2}}{x \cdot x}\right)\right)\right), x\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{-1}{2}, \left(x \cdot x\right)\right)\right)\right), x\right)\right)\right) \]

   7. *-lowering-*.f32 98.5%

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right)\right), x\right)\right)\right) \]

9. Applied egg-rr 98.5%

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

Alternative 2: 98.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.2%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. log-rec N/A

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

   4. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\log \left(\frac{x - \sqrt{x \cdot x - 1}}{x \cdot x - \sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}}\right)\right) \]

   5. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{x - \sqrt{x \cdot x - 1}}{x \cdot x - \sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}}\right)\right)\right) \]

   6. clear-num N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{\frac{x \cdot x - \sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}}{x - \sqrt{x \cdot x - 1}}}\right)\right)\right) \]

   7. flip-+ N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{x + \sqrt{x \cdot x - 1}}\right)\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(x + \sqrt{x \cdot x - 1}\right)\right)\right)\right) \]

   9. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(x, \left(\sqrt{x \cdot x - 1}\right)\right)\right)\right)\right) \]

   10. pow1/2 N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(x, \left({\left(x \cdot x - 1\right)}^{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   11. rem-square-sqrt N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(x, \left({\left(\sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}\right)}^{\frac{1}{2}}\right)\right)\right)\right)\right) \]

4. Applied egg-rr 53.1%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \color{blue}{\left(x \cdot \left(2 - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right)\right) \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(x \cdot \left(2 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(x \cdot \left(\left(\mathsf{neg}\left(-2\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   3. distribute-neg-in N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(x \cdot \left(\mathsf{neg}\left(\left(-2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(x \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} + -2\right)\right)\right)\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(x \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(x \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} - 2\right)\right)\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} - 2\right)\right)\right)\right)\right)\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} + -2\right)\right)\right)\right)\right)\right)\right) \]

   10. distribute-neg-in N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) + \left(\mathsf{neg}\left(-2\right)\right)\right)\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) + 2\right)\right)\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \left(2 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   13. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   14. associate-*r/ N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   16. distribute-neg-frac N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \left(\frac{\frac{-1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

   18. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{-1}{2}, \left({x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

   19. unpow2 N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{-1}{2}, \left(x \cdot x\right)\right)\right)\right)\right)\right)\right) \]

   20. *-lowering-*.f32 97.4%

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 97.4%

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

   1. associate-/r* N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{\frac{1}{x}}{2 + \frac{\frac{-1}{2}}{x \cdot x}}\right)\right)\right) \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{x}\right), \left(2 + \frac{\frac{-1}{2}}{x \cdot x}\right)\right)\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, x\right), \left(2 + \frac{\frac{-1}{2}}{x \cdot x}\right)\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, x\right), \mathsf{+.f32}\left(2, \left(\frac{\frac{-1}{2}}{x \cdot x}\right)\right)\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, x\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{-1}{2}, \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f32 98.5%

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, x\right), \mathsf{+.f32}\left(2, \mathsf{/.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(x, x\right)\right)\right)\right)\right)\right) \]

9. Applied egg-rr 98.5%

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

Alternative 3: 98.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.2%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f32}\left(\color{blue}{\left(x \cdot \left(\left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]

5. Simplified 98.5%

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

   1. +-commutative N/A

      \[\leadsto \mathsf{log.f32}\left(\left(x \cdot \left(2 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) + 1 \cdot \frac{\frac{-1}{2}}{x}\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\left(2 \cdot x + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \cdot x\right) + 1 \cdot \frac{\frac{-1}{2}}{x}\right)\right) \]

   3. count-2 N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\left(\left(x + x\right) + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \cdot x\right) + 1 \cdot \frac{\frac{-1}{2}}{x}\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\left(x + \left(x + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \cdot x\right)\right) + 1 \cdot \frac{\frac{-1}{2}}{x}\right)\right) \]

   5. *-lft-identity N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\left(x + \left(1 \cdot x + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \cdot x\right)\right) + 1 \cdot \frac{\frac{-1}{2}}{x}\right)\right) \]

   6. distribute-rgt-in N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\left(x + x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right) + 1 \cdot \frac{\frac{-1}{2}}{x}\right)\right) \]

   7. associate-+r+ N/A

      \[\leadsto \mathsf{log.f32}\left(\left(x + \left(x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) + 1 \cdot \frac{\frac{-1}{2}}{x}\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{log.f32}\left(\left(x + \left(1 \cdot \frac{\frac{-1}{2}}{x} + x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{log.f32}\left(\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + x \cdot \left(1 + \frac{\frac{-1}{8} + \frac{\frac{-1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right) + x\right)\right) \]

7. Applied egg-rr 98.5%

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

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(\frac{-1}{2}, x\right), \mathsf{+.f32}\left(\color{blue}{x}, x\right)\right)\right) \]
9. Step-by-step derivation

10. Simplified 97.4%

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

Alternative 4: 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 53.2%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \color{blue}{\left(x \cdot \left(1 + -1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)}\right)\right) \]
4. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x \cdot 1 + x \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)\right)\right) \]

   2. *-rgt-identity N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x + x \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)\right)\right) \]

   3. cancel-sign-sub N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)\right)\right) \]

   4. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x - \left(\mathsf{neg}\left(x \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x - \left(\mathsf{neg}\left(x \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   6. distribute-rgt-neg-out N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   7. remove-double-neg N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x - x \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x - \frac{x \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}{{x}^{2}}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x - \frac{x \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}{x \cdot x}\right)\right)\right) \]

   10. times-frac N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x - \frac{x}{x} \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right) \]

   11. *-inverses N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x - 1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right) \]

   13. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x - \left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right)\right) \]

   14. neg-mul-1 N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right)\right)\right)\right) \]

   15. remove-double-neg N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \left(x - \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right) \]

5. Simplified 98.1%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{\_.f32}\left(x, \color{blue}{\left(\frac{\frac{1}{2}}{x}\right)}\right)\right)\right) \]
7. Step-by-step derivation

   1. /-lowering-/.f32 97.4%

      \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \mathsf{\_.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, x\right)\right)\right)\right) \]

8. Simplified 97.4%

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

Alternative 5: 97.5% 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 53.2%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. log-rec N/A

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

   4. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\log \left(\frac{x - \sqrt{x \cdot x - 1}}{x \cdot x - \sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}}\right)\right) \]

   5. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{x - \sqrt{x \cdot x - 1}}{x \cdot x - \sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}}\right)\right)\right) \]

   6. clear-num N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{\frac{x \cdot x - \sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}}{x - \sqrt{x \cdot x - 1}}}\right)\right)\right) \]

   7. flip-+ N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\left(\frac{1}{x + \sqrt{x \cdot x - 1}}\right)\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \left(x + \sqrt{x \cdot x - 1}\right)\right)\right)\right) \]

   9. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(x, \left(\sqrt{x \cdot x - 1}\right)\right)\right)\right)\right) \]

   10. pow1/2 N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(x, \left({\left(x \cdot x - 1\right)}^{\frac{1}{2}}\right)\right)\right)\right)\right) \]

   11. rem-square-sqrt N/A

       \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(x, \left({\left(\sqrt{x \cdot x - 1} \cdot \sqrt{x \cdot x - 1}\right)}^{\frac{1}{2}}\right)\right)\right)\right)\right) \]

4. Applied egg-rr 53.1%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(\frac{\frac{1}{2}}{x}\right)}\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 96.2%

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, x\right)\right)\right) \]

7. Simplified 96.2%

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

Alternative 6: 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 53.2%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f32}\left(\mathsf{+.f32}\left(x, \color{blue}{x}\right)\right) \]
4. Step-by-step derivation

5. Simplified 95.1%

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

Developer Target 1: 99.3% 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 2024139 
(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)))))