Rust f32::acosh

Percentage Accurate: 52.8% → 98.1%

Time: 8.6s

Alternatives: 13

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: 98.1% 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 52.1%

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

3. Taylor expanded in x around inf

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

   1. associate--l+ N/A

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

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

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

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

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

   4. sub-neg N/A

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

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

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

   6. mul-1-neg N/A

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

   7. log-rec N/A

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

   8. remove-double-neg N/A

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

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

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

   10. associate-*r/ N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{4} \cdot 1}{{x}^{2}}\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{4}}{{x}^{2}}\right)\right)\right)\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \left(\frac{\mathsf{neg}\left(\frac{1}{4}\right)}{\color{blue}{{x}^{2}}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \mathsf{/.f32}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

   14. metadata-eval N/A

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

5. Simplified 98.3%

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

Alternative 2: 98.9% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (log
  (+ (/ (+ -0.125 (/ -0.0625 (* x x))) (* x (* x x))) (+ x (- x (/ 0.5 x))))))

C

float code(float x) {
	return logf((((-0.125f + (-0.0625f / (x * x))) / (x * (x * x))) + (x + (x - (0.5f / x)))));
}

Fortran

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = log((((single(-0.125) + (single(-0.0625) / (x * x))) / (x * (x * x))) + (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

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

5. Simplified 97.6%

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

   1. +-commutative N/A

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

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

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

7. Applied egg-rr 97.6%

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

   1. +-commutative N/A

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

   2. div-sub N/A

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

   3. associate-+l- N/A

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

   4. associate-+l- N/A

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

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

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

   6. associate-/l/ N/A

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

   7. associate-/l/ N/A

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

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

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

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

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

   10. /-lowering-/.f32 N/A

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

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

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

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

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

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

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

   14. --lowering--.f32 N/A

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

   15. --lowering--.f32 N/A

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

   16. /-lowering-/.f32 97.6%

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

9. Applied egg-rr 97.6%

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

10. Final simplification 97.6%

    \[\leadsto \log \left(\frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot x\right)} + \left(x + \left(x - \frac{0.5}{x}\right)\right)\right) \]
11. Add Preprocessing

Alternative 3: 98.9% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (log (+ x (+ x (/ (- (/ (/ (+ -0.125 (/ -0.0625 (* x x))) x) x) 0.5) x)))))

C

float code(float x) {
	return logf((x + (x + (((((-0.125f + (-0.0625f / (x * x))) / x) / x) - 0.5f) / x))));
}

Fortran

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = log((x + (x + (((((single(-0.125) + (single(-0.0625) / (x * x))) / 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

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

5. Simplified 97.6%

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

   1. +-commutative N/A

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

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

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

7. Applied egg-rr 97.6%

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

8. Final simplification 97.6%

   \[\leadsto \log \left(x + \left(x + \frac{\frac{\frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x}}{x} - 0.5}{x}\right)\right) \]
9. Add Preprocessing

Alternative 4: 98.7% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (log (+ (/ -0.125 (* x (* x x))) (+ x (- x (/ 0.5 x))))))

C

float code(float x) {
	return logf(((-0.125f / (x * (x * x))) + (x + (x - (0.5f / x)))));
}

Fortran

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

Julia

function code(x)
	return log(Float32(Float32(Float32(-0.125) / Float32(x * Float32(x * x))) + Float32(x + Float32(x - Float32(Float32(0.5) / x)))))
end

MATLAB

function tmp = code(x)
	tmp = log(((single(-0.125) / (x * (x * x))) + (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

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

5. Simplified 97.6%

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

   1. +-commutative N/A

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

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

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

7. Applied egg-rr 97.6%

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

   1. +-commutative N/A

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

   2. div-sub N/A

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

   3. associate-+l- N/A

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

   4. associate-+l- N/A

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

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

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

   6. associate-/l/ N/A

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

   7. associate-/l/ N/A

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

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

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

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

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

   10. /-lowering-/.f32 N/A

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

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

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

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

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

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

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

   14. --lowering--.f32 N/A

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

   15. --lowering--.f32 N/A

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

   16. /-lowering-/.f32 97.6%

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

9. Applied egg-rr 97.6%

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

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{log.f32}\left(\mathsf{\_.f32}\left(\color{blue}{\left(\frac{\frac{-1}{8}}{{x}^{3}}\right)}, \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, x\right), x\right), x\right)\right)\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{-1}{8}, \left({x}^{3}\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, x\right), x\right), x\right)\right)\right) \]

    2. cube-mult N/A

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

    3. unpow2 N/A

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

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

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

    5. unpow2 N/A

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

    6. *-lowering-*.f32 97.5%

       \[\leadsto \mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{-1}{8}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, x\right), x\right), x\right)\right)\right) \]

12. Simplified 97.5%

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

13. Final simplification 97.5%

    \[\leadsto \log \left(\frac{-0.125}{x \cdot \left(x \cdot x\right)} + \left(x + \left(x - \frac{0.5}{x}\right)\right)\right) \]
14. Add Preprocessing

Alternative 5: 98.7% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (log (+ (/ (/ -0.125 (* x x)) x) (+ (/ -0.5 x) (/ x 0.5)))))

C

float code(float x) {
	return logf((((-0.125f / (x * x)) / x) + ((-0.5f / x) + (x / 0.5f))));
}

Fortran

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

Julia

function code(x)
	return log(Float32(Float32(Float32(Float32(-0.125) / Float32(x * x)) / x) + Float32(Float32(Float32(-0.5) / x) + Float32(x / Float32(0.5)))))
end

MATLAB

function tmp = code(x)
	tmp = log((((single(-0.125) / (x * x)) / x) + ((single(-0.5) / x) + (x / single(0.5)))));
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

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

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

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

   4. *-commutative N/A

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

   5. mul-1-neg N/A

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

   6. distribute-neg-frac2 N/A

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

   7. associate-*r/ N/A

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

   8. unpow2 N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. mul-1-neg N/A

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

   11. times-frac N/A

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

   12. *-inverses N/A

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

   13. mul-1-neg N/A

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

   14. distribute-neg-frac2 N/A

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

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

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

   16. metadata-eval N/A

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

5. Simplified 97.5%

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

   1. associate-+r+ N/A

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

   2. count-2 N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-frac2 N/A

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

   8. +-commutative N/A

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

   9. clear-num N/A

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

   10. associate-/r/ N/A

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

   11. +-commutative N/A

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

   12. distribute-rgt-in N/A

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

   13. div-inv N/A

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

   14. associate-+l+ N/A

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

   15. distribute-neg-frac2 N/A

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

   16. metadata-eval N/A

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

   17. div-inv N/A

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

   18. metadata-eval N/A

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

   19. *-commutative N/A

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

   20. count-2 N/A

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

   21. associate-+l+ N/A

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

   22. +-commutative N/A

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

7. Applied egg-rr 97.5%

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

Alternative 6: 98.7% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary32
 (log (+ x (+ x (/ (+ (/ -0.125 (* x x)) -0.5) x)))))

C

float code(float x) {
	return logf((x + (x + (((-0.125f / (x * x)) + -0.5f) / x))));
}

Fortran

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

Julia

function code(x)
	return log(Float32(x + Float32(x + Float32(Float32(Float32(Float32(-0.125) / Float32(x * x)) + Float32(-0.5)) / x))))
end

MATLAB

function tmp = code(x)
	tmp = log((x + (x + (((single(-0.125) / (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

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

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

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

   4. *-commutative N/A

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

   5. mul-1-neg N/A

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

   6. distribute-neg-frac2 N/A

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

   7. associate-*r/ N/A

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

   8. unpow2 N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. mul-1-neg N/A

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

   11. times-frac N/A

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

   12. *-inverses N/A

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

   13. mul-1-neg N/A

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

   14. distribute-neg-frac2 N/A

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

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

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

   16. metadata-eval N/A

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

5. Simplified 97.5%

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

6. Final simplification 97.5%

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

Alternative 7: 97.7% 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 52.1%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 95.6%

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

   1. count-2 N/A

      \[\leadsto \log \left(2 \cdot x\right) \]

   2. *-commutative N/A

      \[\leadsto \log \left(x \cdot 2\right) \]

   3. metadata-eval N/A

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

   4. div-inv N/A

      \[\leadsto \log \left(\frac{x}{\frac{1}{2}}\right) \]

   5. metadata-eval N/A

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

   6. distribute-neg-frac2 N/A

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

   7. clear-num N/A

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

   8. distribute-neg-frac2 N/A

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

   9. log-rec N/A

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

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

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

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

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

   14. /-lowering-/.f32 97.4%

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

7. Applied egg-rr 97.4%

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

Alternative 8: 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

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

5. Simplified 95.6%

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

Alternative 9: 5.4% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   1. associate--l+ N/A

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

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

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

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

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

   4. sub-neg N/A

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

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

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

   6. mul-1-neg N/A

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

   7. log-rec N/A

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

   8. remove-double-neg N/A

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

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

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

   10. associate-*r/ N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{4} \cdot 1}{{x}^{2}}\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{4}}{{x}^{2}}\right)\right)\right)\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \left(\frac{\mathsf{neg}\left(\frac{1}{4}\right)}{\color{blue}{{x}^{2}}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \mathsf{/.f32}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

   14. metadata-eval N/A

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

5. Simplified 98.3%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{4}}{{x}^{2}}} \]
7. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \color{blue}{\left({x}^{2}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \left(x \cdot \color{blue}{x}\right)\right) \]

   3. *-lowering-*.f32 5.4%

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right) \]

8. Simplified 5.4%

   \[\leadsto \color{blue}{\frac{-0.25}{x \cdot x}} \]
9. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{-1}{4}}{x}}{\color{blue}{x}} \]

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

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{-1}{4}}{x}\right), \color{blue}{x}\right) \]

   3. /-lowering-/.f32 5.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{-1}{4}, x\right), x\right) \]

10. Applied egg-rr 5.4%

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

    1. associate-/l/ N/A

       \[\leadsto \frac{\frac{-1}{4}}{\color{blue}{x \cdot x}} \]

    2. pow2 N/A

       \[\leadsto \frac{\frac{-1}{4}}{{x}^{\color{blue}{2}}} \]

    3. metadata-eval N/A

       \[\leadsto \frac{\frac{-1}{4}}{{x}^{\left(3 + \color{blue}{-1}\right)}} \]

    4. pow-prod-up N/A

       \[\leadsto \frac{\frac{-1}{4}}{{x}^{3} \cdot \color{blue}{{x}^{-1}}} \]

    5. cube-unmult N/A

       \[\leadsto \frac{\frac{-1}{4}}{\left(x \cdot \left(x \cdot x\right)\right) \cdot {\color{blue}{x}}^{-1}} \]

    6. inv-pow N/A

       \[\leadsto \frac{\frac{-1}{4}}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{\color{blue}{x}}} \]

    7. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{-1}{4}}{x \cdot \left(x \cdot x\right)}}{\color{blue}{\frac{1}{x}}} \]

    8. clear-num N/A

       \[\leadsto \frac{\frac{1}{\frac{x \cdot \left(x \cdot x\right)}{\frac{-1}{4}}}}{\frac{\color{blue}{1}}{x}} \]

    9. associate-/l/ N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{x} \cdot \frac{x \cdot \left(x \cdot x\right)}{\frac{-1}{4}}}} \]

    10. associate-/r* N/A

        \[\leadsto \frac{\frac{1}{\frac{1}{x}}}{\color{blue}{\frac{x \cdot \left(x \cdot x\right)}{\frac{-1}{4}}}} \]

    11. remove-double-div N/A

        \[\leadsto \frac{x}{\frac{\color{blue}{x \cdot \left(x \cdot x\right)}}{\frac{-1}{4}}} \]

    12. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{/.f32}\left(x, \color{blue}{\left(\frac{x \cdot \left(x \cdot x\right)}{\frac{-1}{4}}\right)}\right) \]

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

        \[\leadsto \mathsf{/.f32}\left(x, \mathsf{/.f32}\left(\left(x \cdot \left(x \cdot x\right)\right), \color{blue}{\frac{-1}{4}}\right)\right) \]

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

        \[\leadsto \mathsf{/.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \left(x \cdot x\right)\right), \frac{-1}{4}\right)\right) \]

    15. *-lowering-*.f32 5.5%

        \[\leadsto \mathsf{/.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right), \frac{-1}{4}\right)\right) \]

12. Applied egg-rr 5.5%

    \[\leadsto \color{blue}{\frac{x}{\frac{x \cdot \left(x \cdot x\right)}{-0.25}}} \]
13. Add Preprocessing

Alternative 10: 5.4% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   1. associate--l+ N/A

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

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

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

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

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

   4. sub-neg N/A

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

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

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

   6. mul-1-neg N/A

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

   7. log-rec N/A

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

   8. remove-double-neg N/A

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

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

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

   10. associate-*r/ N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{4} \cdot 1}{{x}^{2}}\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{4}}{{x}^{2}}\right)\right)\right)\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \left(\frac{\mathsf{neg}\left(\frac{1}{4}\right)}{\color{blue}{{x}^{2}}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \mathsf{/.f32}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

   14. metadata-eval N/A

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

5. Simplified 98.3%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{4}}{{x}^{2}}} \]
7. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \color{blue}{\left({x}^{2}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \left(x \cdot \color{blue}{x}\right)\right) \]

   3. *-lowering-*.f32 5.4%

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right) \]

8. Simplified 5.4%

   \[\leadsto \color{blue}{\frac{-0.25}{x \cdot x}} \]
9. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \left({x}^{\color{blue}{2}}\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \left({x}^{\left(3 + \color{blue}{-1}\right)}\right)\right) \]

   3. pow-prod-up N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \left({x}^{3} \cdot \color{blue}{{x}^{-1}}\right)\right) \]

   4. cube-unmult N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot {\color{blue}{x}}^{-1}\right)\right) \]

   5. inv-pow N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{\color{blue}{x}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{*.f32}\left(\left(x \cdot \left(x \cdot x\right)\right), \color{blue}{\left(\frac{1}{x}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, \left(x \cdot x\right)\right), \left(\frac{\color{blue}{1}}{x}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right), \left(\frac{1}{x}\right)\right)\right) \]

   9. /-lowering-/.f32 5.5%

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right), \mathsf{/.f32}\left(1, \color{blue}{x}\right)\right)\right) \]

10. Applied egg-rr 5.5%

    \[\leadsto \frac{-0.25}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{x}}} \]
11. Step-by-step derivation

    1. un-div-inv N/A

       \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \left(\frac{x \cdot \left(x \cdot x\right)}{\color{blue}{x}}\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{/.f32}\left(\left(x \cdot \left(x \cdot x\right)\right), \color{blue}{x}\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \left(x \cdot x\right)\right), x\right)\right) \]

    4. *-lowering-*.f32 5.5%

       \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right), x\right)\right) \]

12. Applied egg-rr 5.5%

    \[\leadsto \frac{-0.25}{\color{blue}{\frac{x \cdot \left(x \cdot x\right)}{x}}} \]
13. Add Preprocessing

Alternative 11: 5.4% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   1. associate--l+ N/A

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

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

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

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

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

   4. sub-neg N/A

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

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

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

   6. mul-1-neg N/A

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

   7. log-rec N/A

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

   8. remove-double-neg N/A

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

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

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

   10. associate-*r/ N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{4} \cdot 1}{{x}^{2}}\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{4}}{{x}^{2}}\right)\right)\right)\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \left(\frac{\mathsf{neg}\left(\frac{1}{4}\right)}{\color{blue}{{x}^{2}}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \mathsf{/.f32}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

   14. metadata-eval N/A

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

5. Simplified 98.3%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{4}}{{x}^{2}}} \]
7. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \color{blue}{\left({x}^{2}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \left(x \cdot \color{blue}{x}\right)\right) \]

   3. *-lowering-*.f32 5.4%

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right) \]

8. Simplified 5.4%

   \[\leadsto \color{blue}{\frac{-0.25}{x \cdot x}} \]
9. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{-1}{4}}{x}}{\color{blue}{x}} \]

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

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{-1}{4}}{x}\right), \color{blue}{x}\right) \]

   3. /-lowering-/.f32 5.4%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{-1}{4}, x\right), x\right) \]

10. Applied egg-rr 5.4%

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

    1. associate-/l/ N/A

       \[\leadsto \frac{\frac{-1}{4}}{\color{blue}{x \cdot x}} \]

    2. pow2 N/A

       \[\leadsto \frac{\frac{-1}{4}}{{x}^{\color{blue}{2}}} \]

    3. metadata-eval N/A

       \[\leadsto \frac{\frac{-1}{4}}{{x}^{\left(3 + \color{blue}{-1}\right)}} \]

    4. pow-prod-up N/A

       \[\leadsto \frac{\frac{-1}{4}}{{x}^{3} \cdot \color{blue}{{x}^{-1}}} \]

    5. cube-unmult N/A

       \[\leadsto \frac{\frac{-1}{4}}{\left(x \cdot \left(x \cdot x\right)\right) \cdot {\color{blue}{x}}^{-1}} \]

    6. inv-pow N/A

       \[\leadsto \frac{\frac{-1}{4}}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{\color{blue}{x}}} \]

    7. un-div-inv N/A

       \[\leadsto \frac{\frac{-1}{4}}{\frac{x \cdot \left(x \cdot x\right)}{\color{blue}{x}}} \]

    8. associate-/r/ N/A

       \[\leadsto \frac{\frac{-1}{4}}{x \cdot \left(x \cdot x\right)} \cdot \color{blue}{x} \]

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

       \[\leadsto \mathsf{*.f32}\left(\left(\frac{\frac{-1}{4}}{x \cdot \left(x \cdot x\right)}\right), \color{blue}{x}\right) \]

    10. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{-1}{4}, \left(x \cdot \left(x \cdot x\right)\right)\right), x\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{*.f32}\left(x, \left(x \cdot x\right)\right)\right), x\right) \]

    12. *-lowering-*.f32 5.5%

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right)\right), x\right) \]

12. Applied egg-rr 5.5%

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

13. Final simplification 5.5%

    \[\leadsto x \cdot \frac{-0.25}{x \cdot \left(x \cdot x\right)} \]
14. Add Preprocessing

Alternative 12: 5.3% accurate, 29.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{x}{\frac{-0.25}{x}}} \end{array} \]

FPCore

(FPCore (x) :precision binary32 (/ 1.0 (/ x (/ -0.25 x))))

C

float code(float x) {
	return 1.0f / (x / (-0.25f / x));
}

Fortran

real(4) function code(x)
    real(4), intent (in) :: x
    code = 1.0e0 / (x / ((-0.25e0) / x))
end function

Julia

function code(x)
	return Float32(Float32(1.0) / Float32(x / Float32(Float32(-0.25) / x)))
end

MATLAB

function tmp = code(x)
	tmp = single(1.0) / (x / (single(-0.25) / 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

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

   1. associate--l+ N/A

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

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

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

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

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

   4. sub-neg N/A

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

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

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

   6. mul-1-neg N/A

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

   7. log-rec N/A

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

   8. remove-double-neg N/A

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

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

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

   10. associate-*r/ N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{4} \cdot 1}{{x}^{2}}\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{4}}{{x}^{2}}\right)\right)\right)\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \left(\frac{\mathsf{neg}\left(\frac{1}{4}\right)}{\color{blue}{{x}^{2}}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \mathsf{/.f32}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

   14. metadata-eval N/A

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

5. Simplified 98.3%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{4}}{{x}^{2}}} \]
7. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \color{blue}{\left({x}^{2}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \left(x \cdot \color{blue}{x}\right)\right) \]

   3. *-lowering-*.f32 5.4%

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right) \]

8. Simplified 5.4%

   \[\leadsto \color{blue}{\frac{-0.25}{x \cdot x}} \]
9. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{\frac{-1}{4}}}} \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{x \cdot x}{\frac{-1}{4}}\right)}\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{1}{\color{blue}{\frac{\frac{-1}{4}}{x \cdot x}}}\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{1}{\frac{\frac{\frac{-1}{4}}{x}}{\color{blue}{x}}}\right)\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{x}{\color{blue}{\frac{\frac{-1}{4}}{x}}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(x, \color{blue}{\left(\frac{\frac{-1}{4}}{x}\right)}\right)\right) \]

   7. /-lowering-/.f32 5.4%

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{4}, \color{blue}{x}\right)\right)\right) \]

10. Applied egg-rr 5.4%

    \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{-0.25}{x}}}} \]
11. Add Preprocessing

Alternative 13: 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 52.1%

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

3. Taylor expanded in x around inf

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

   1. associate--l+ N/A

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

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

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

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

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

   4. sub-neg N/A

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

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

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

   6. mul-1-neg N/A

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

   7. log-rec N/A

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

   8. remove-double-neg N/A

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

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

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

   10. associate-*r/ N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{4} \cdot 1}{{x}^{2}}\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{4}}{{x}^{2}}\right)\right)\right)\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \left(\frac{\mathsf{neg}\left(\frac{1}{4}\right)}{\color{blue}{{x}^{2}}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \mathsf{/.f32}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

   14. metadata-eval N/A

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

   15. unpow2 N/A

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

   16. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{+.f32}\left(\mathsf{log.f32}\left(2\right), \mathsf{+.f32}\left(\mathsf{log.f32}\left(x\right), \mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

5. Simplified 98.3%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{4}}{{x}^{2}}} \]
7. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \color{blue}{\left({x}^{2}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \left(x \cdot \color{blue}{x}\right)\right) \]

   3. *-lowering-*.f32 5.4%

      \[\leadsto \mathsf{/.f32}\left(\frac{-1}{4}, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right) \]

8. Simplified 5.4%

   \[\leadsto \color{blue}{\frac{-0.25}{x \cdot x}} \]
9. 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 2024164 
(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)))))