Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.8% → 98.2%

Time: 10.6s

Alternatives: 16

Speedup: 1.0×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(0.25 \leq u \land u \leq 1\right)\]

Math

\[\begin{array}{l} \\ \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))

C

float code(float s, float u) {
	return (3.0f * s) * logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * log((1.0e0 / (1.0e0 - ((u - 0.25e0) / 0.75e0))))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * log((single(1.0) / (single(1.0) - ((u - single(0.25)) / single(0.75)))));
end

Initial Program: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))

C

float code(float s, float u) {
	return (3.0f * s) * logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * log((1.0e0 / (1.0e0 - ((u - 0.25e0) / 0.75e0))))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * log((single(1.0) / (single(1.0) - ((u - single(0.25)) / single(0.75)))));
end

Alternative 1: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot \mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right)\right) \cdot -3 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* s (log1p (/ (- 0.25 u) 0.75))) -3.0))

C

float code(float s, float u) {
	return (s * log1pf(((0.25f - u) / 0.75f))) * -3.0f;
}

Julia

function code(s, u)
	return Float32(Float32(s * log1p(Float32(Float32(Float32(0.25) - u) / Float32(0.75)))) * Float32(-3.0))
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \]

   2. neg-mul-1 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-1 \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(3 \cdot s\right) \cdot -1\right) \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)} \]

   4. *-commutative N/A

      \[\leadsto \log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right) \cdot \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)} \]

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

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right), \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   7. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   8. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   10. div-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   11. associate--r- N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(0 - \frac{u}{\frac{3}{4}}\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}} + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right), \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   16. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(s \cdot 3\right) \cdot -1\right)\right) \]

   20. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(s \cdot \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

   22. metadata-eval 96.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

3. Simplified 96.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\log \left(1 + \left(\frac{1}{3} + \frac{u}{\frac{-3}{4}}\right)\right) \cdot s\right) \cdot \color{blue}{-3} \]

   2. *-commutative N/A

      \[\leadsto \left(s \cdot \log \left(1 + \left(\frac{1}{3} + \frac{u}{\frac{-3}{4}}\right)\right)\right) \cdot -3 \]

   3. +-commutative N/A

      \[\leadsto \left(s \cdot \log \left(1 + \left(\frac{u}{\frac{-3}{4}} + \frac{1}{3}\right)\right)\right) \cdot -3 \]

   4. associate-+r+ N/A

      \[\leadsto \left(s \cdot \log \left(\left(1 + \frac{u}{\frac{-3}{4}}\right) + \frac{1}{3}\right)\right) \cdot -3 \]

   5. metadata-eval N/A

      \[\leadsto \left(s \cdot \log \left(\left(1 + \frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right) + \frac{1}{3}\right)\right) \cdot -3 \]

   6. distribute-neg-frac2 N/A

      \[\leadsto \left(s \cdot \log \left(\left(1 + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right) + \frac{1}{3}\right)\right) \cdot -3 \]

   7. sub-neg N/A

      \[\leadsto \left(s \cdot \log \left(\left(1 - \frac{u}{\frac{3}{4}}\right) + \frac{1}{3}\right)\right) \cdot -3 \]

   8. associate--r- N/A

      \[\leadsto \left(s \cdot \log \left(1 - \left(\frac{u}{\frac{3}{4}} - \frac{1}{3}\right)\right)\right) \cdot -3 \]

   9. metadata-eval N/A

      \[\leadsto \left(s \cdot \log \left(1 - \left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \cdot -3 \]

   10. div-sub N/A

       \[\leadsto \left(s \cdot \log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) \cdot -3 \]

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

       \[\leadsto \mathsf{*.f32}\left(\left(s \cdot \log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right), \color{blue}{-3}\right) \]

6. Applied egg-rr 96.4%

   \[\leadsto \color{blue}{\left(s \cdot \mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right)\right) \cdot -3} \]
7. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{log1p.f32}\left(\left(\frac{1}{3} + \frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right)\right)\right), -3\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{log1p.f32}\left(\left(\frac{1}{3} + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right)\right), -3\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{log1p.f32}\left(\left(\frac{1}{3} - \frac{u}{\frac{3}{4}}\right)\right)\right), -3\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{log1p.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}} - \frac{u}{\frac{3}{4}}\right)\right)\right), -3\right) \]

   5. div-sub N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{log1p.f32}\left(\left(\frac{\frac{1}{4} - u}{\frac{3}{4}}\right)\right)\right), -3\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} - u\right), \frac{3}{4}\right)\right)\right), -3\right) \]

   7. --lowering--.f32 98.2%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{4}, u\right), \frac{3}{4}\right)\right)\right), -3\right) \]

8. Applied egg-rr 98.2%

   \[\leadsto \left(s \cdot \mathsf{log1p}\left(\color{blue}{\frac{0.25 - u}{0.75}}\right)\right) \cdot -3 \]
9. Add Preprocessing

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right) \cdot \left(s \cdot -3\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (log1p (/ (- 0.25 u) 0.75)) (* s -3.0)))

C

float code(float s, float u) {
	return log1pf(((0.25f - u) / 0.75f)) * (s * -3.0f);
}

Julia

function code(s, u)
	return Float32(log1p(Float32(Float32(Float32(0.25) - u) / Float32(0.75))) * Float32(s * Float32(-3.0)))
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \]

   2. neg-mul-1 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-1 \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(3 \cdot s\right) \cdot -1\right) \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)} \]

   4. *-commutative N/A

      \[\leadsto \log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right) \cdot \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)} \]

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

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right), \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   7. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   8. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   10. div-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   11. associate--r- N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(0 - \frac{u}{\frac{3}{4}}\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}} + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right), \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   16. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(s \cdot 3\right) \cdot -1\right)\right) \]

   20. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(s \cdot \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

   22. metadata-eval 96.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

3. Simplified 96.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{-1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{\frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u - \frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   8. frac-2neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\mathsf{neg}\left(\frac{-3}{4}\right)}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\frac{3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) + u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} - u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   17. --lowering--.f32 98.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{4}, u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

6. Applied egg-rr 98.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{0.25 - u}{0.75}}\right) \cdot \left(s \cdot -3\right) \]
7. Add Preprocessing

Alternative 3: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\left(0.25 - u\right) \cdot 1.3333333333333333\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* s -3.0) (log1p (* (- 0.25 u) 1.3333333333333333))))

C

float code(float s, float u) {
	return (s * -3.0f) * log1pf(((0.25f - u) * 1.3333333333333333f));
}

Julia

function code(s, u)
	return Float32(Float32(s * Float32(-3.0)) * log1p(Float32(Float32(Float32(0.25) - u) * Float32(1.3333333333333333))))
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \]

   2. neg-mul-1 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-1 \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(3 \cdot s\right) \cdot -1\right) \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)} \]

   4. *-commutative N/A

      \[\leadsto \log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right) \cdot \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)} \]

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

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right), \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   7. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   8. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   10. div-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   11. associate--r- N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(0 - \frac{u}{\frac{3}{4}}\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}} + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right), \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   16. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(s \cdot 3\right) \cdot -1\right)\right) \]

   20. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(s \cdot \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

   22. metadata-eval 96.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

3. Simplified 96.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{-1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{\frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u - \frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   8. frac-2neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\mathsf{neg}\left(\frac{-3}{4}\right)}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\frac{3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   10. div-inv N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right) \cdot \frac{1}{\frac{3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right), \left(\frac{1}{\frac{3}{4}}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{neg}\left(\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right), \left(\frac{1}{\frac{3}{4}}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) + u\right)\right)\right), \left(\frac{1}{\frac{3}{4}}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   14. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \left(\frac{1}{\frac{3}{4}}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \left(\frac{1}{\frac{3}{4}}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\left(\frac{1}{4} + \left(\mathsf{neg}\left(u\right)\right)\right), \left(\frac{1}{\frac{3}{4}}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   17. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\left(\frac{1}{4} - u\right), \left(\frac{1}{\frac{3}{4}}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(\frac{1}{4}, u\right), \left(\frac{1}{\frac{3}{4}}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   19. metadata-eval 97.9%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(\frac{1}{4}, u\right), \frac{4}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

6. Applied egg-rr 97.9%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(0.25 - u\right) \cdot 1.3333333333333333}\right) \cdot \left(s \cdot -3\right) \]

7. Final simplification 97.9%

   \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\left(0.25 - u\right) \cdot 1.3333333333333333\right) \]
8. Add Preprocessing

Alternative 4: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot -3\right) \cdot \log \left(1.3333333333333333 + \frac{u}{-0.75}\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* s -3.0) (log (+ 1.3333333333333333 (/ u -0.75)))))

C

float code(float s, float u) {
	return (s * -3.0f) * logf((1.3333333333333333f + (u / -0.75f)));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (s * (-3.0e0)) * log((1.3333333333333333e0 + (u / (-0.75e0))))
end function

Julia

function code(s, u)
	return Float32(Float32(s * Float32(-3.0)) * log(Float32(Float32(1.3333333333333333) + Float32(u / Float32(-0.75)))))
end

MATLAB

function tmp = code(s, u)
	tmp = (s * single(-3.0)) * log((single(1.3333333333333333) + (u / single(-0.75))));
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \]

   2. neg-mul-1 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-1 \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(3 \cdot s\right) \cdot -1\right) \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)} \]

   4. *-commutative N/A

      \[\leadsto \log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right) \cdot \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)} \]

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

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right), \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   7. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   8. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   10. div-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   11. associate--r- N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(0 - \frac{u}{\frac{3}{4}}\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}} + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right), \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   16. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(s \cdot 3\right) \cdot -1\right)\right) \]

   20. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(s \cdot \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

   22. metadata-eval 96.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

3. Simplified 96.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 + \left(\frac{u}{\frac{-3}{4}} + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. associate-+r+ N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(\left(1 + \frac{u}{\frac{-3}{4}}\right) + \frac{1}{3}\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(\left(1 + \frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right) + \frac{1}{3}\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   4. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(\left(1 + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right) + \frac{1}{3}\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(\left(1 - \frac{u}{\frac{3}{4}}\right) + \frac{1}{3}\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   6. associate--r- N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - \left(\frac{u}{\frac{3}{4}} - \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - \left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   8. div-sub N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log.f32}\left(\left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right), \mathsf{*.f32}\left(\color{blue}{s}, -3\right)\right) \]

   10. div-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log.f32}\left(\left(1 - \left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log.f32}\left(\left(1 - \left(\frac{u}{\frac{3}{4}} - \frac{1}{3}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   12. associate--r- N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log.f32}\left(\left(\left(1 - \frac{u}{\frac{3}{4}}\right) + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right) + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   14. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right) + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log.f32}\left(\left(\left(1 + \frac{u}{\frac{-3}{4}}\right) + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   16. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log.f32}\left(\left(\left(\frac{u}{\frac{-3}{4}} + 1\right) + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   17. associate-+l+ N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(1 + \frac{1}{3}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \frac{4}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \frac{1}{\frac{3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\frac{u}{\frac{-3}{4}}\right), \left(\frac{1}{\frac{3}{4}}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   21. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \frac{-3}{4}\right), \left(\frac{1}{\frac{3}{4}}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   22. metadata-eval 94.9%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \frac{-3}{4}\right), \frac{4}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

6. Applied egg-rr 94.9%

   \[\leadsto \color{blue}{\log \left(\frac{u}{-0.75} + 1.3333333333333333\right)} \cdot \left(s \cdot -3\right) \]

7. Final simplification 94.9%

   \[\leadsto \left(s \cdot -3\right) \cdot \log \left(1.3333333333333333 + \frac{u}{-0.75}\right) \]
8. Add Preprocessing

Alternative 5: 37.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot -3\right) \cdot \left(u \cdot \left(\left(u \cdot u\right) \cdot \left(u \cdot \left(-0.7901234567901234 - \frac{0.7901234567901234 + \frac{0.8888888888888888}{u}}{u}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  (* s -3.0)
  (*
   u
   (*
    (* u u)
    (*
     u
     (-
      -0.7901234567901234
      (/ (+ 0.7901234567901234 (/ 0.8888888888888888 u)) u)))))))

C

float code(float s, float u) {
	return (s * -3.0f) * (u * ((u * u) * (u * (-0.7901234567901234f - ((0.7901234567901234f + (0.8888888888888888f / u)) / u)))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (s * (-3.0e0)) * (u * ((u * u) * (u * ((-0.7901234567901234e0) - ((0.7901234567901234e0 + (0.8888888888888888e0 / u)) / u)))))
end function

Julia

function code(s, u)
	return Float32(Float32(s * Float32(-3.0)) * Float32(u * Float32(Float32(u * u) * Float32(u * Float32(Float32(-0.7901234567901234) - Float32(Float32(Float32(0.7901234567901234) + Float32(Float32(0.8888888888888888) / u)) / u))))))
end

MATLAB

function tmp = code(s, u)
	tmp = (s * single(-3.0)) * (u * ((u * u) * (u * (single(-0.7901234567901234) - ((single(0.7901234567901234) + (single(0.8888888888888888) / u)) / u)))));
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \]

   2. neg-mul-1 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-1 \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(3 \cdot s\right) \cdot -1\right) \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)} \]

   4. *-commutative N/A

      \[\leadsto \log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right) \cdot \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)} \]

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

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right), \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   7. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   8. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   10. div-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   11. associate--r- N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(0 - \frac{u}{\frac{3}{4}}\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}} + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right), \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   16. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(s \cdot 3\right) \cdot -1\right)\right) \]

   20. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(s \cdot \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

   22. metadata-eval 96.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

3. Simplified 96.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{-1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{\frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u - \frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   8. frac-2neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\mathsf{neg}\left(\frac{-3}{4}\right)}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\frac{3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) + u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} - u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   17. --lowering--.f32 98.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{4}, u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

6. Applied egg-rr 98.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{0.25 - u}{0.75}}\right) \cdot \left(s \cdot -3\right) \]

7. Taylor expanded in u around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot u\right)}, \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(u\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. neg-lowering-neg.f32 19.3%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

9. Simplified 19.3%

   \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{-u}}{0.75}\right) \cdot \left(s \cdot -3\right) \]

10. Taylor expanded in u around 0

    \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(u \cdot \left(u \cdot \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right) - \frac{8}{9}\right) - \frac{4}{3}\right)\right)}, \mathsf{*.f32}\left(s, -3\right)\right) \]
11. Step-by-step derivation

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \left(u \cdot \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right) - \frac{8}{9}\right) - \frac{4}{3}\right)\right), \mathsf{*.f32}\left(\color{blue}{s}, -3\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \left(u \cdot \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right) - \frac{8}{9}\right) + \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \left(u \cdot \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right) - \frac{8}{9}\right) + \frac{-4}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    4. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \left(\frac{-4}{3} + u \cdot \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right) - \frac{8}{9}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \left(u \cdot \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right) - \frac{8}{9}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right) - \frac{8}{9}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    7. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right) + \left(\mathsf{neg}\left(\frac{8}{9}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right) + \frac{-8}{9}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    9. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \left(\frac{-8}{9} + u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-8}{9}, \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-8}{9}, \mathsf{*.f32}\left(u, \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    12. sub-neg N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-8}{9}, \mathsf{*.f32}\left(u, \left(\frac{-64}{81} \cdot u + \left(\mathsf{neg}\left(\frac{64}{81}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    13. metadata-eval N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-8}{9}, \mathsf{*.f32}\left(u, \left(\frac{-64}{81} \cdot u + \frac{-64}{81}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    14. +-commutative N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-8}{9}, \mathsf{*.f32}\left(u, \left(\frac{-64}{81} + \frac{-64}{81} \cdot u\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-8}{9}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-64}{81}, \left(\frac{-64}{81} \cdot u\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    16. *-commutative N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-8}{9}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-64}{81}, \left(u \cdot \frac{-64}{81}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    17. *-lowering-*.f32 27.3%

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-8}{9}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-64}{81}, \mathsf{*.f32}\left(u, \frac{-64}{81}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

12. Simplified 27.3%

    \[\leadsto \color{blue}{\left(u \cdot \left(-1.3333333333333333 + u \cdot \left(-0.8888888888888888 + u \cdot \left(-0.7901234567901234 + u \cdot -0.7901234567901234\right)\right)\right)\right)} \cdot \left(s \cdot -3\right) \]

13. Taylor expanded in u around inf

    \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \color{blue}{\left({u}^{3} \cdot \left(-1 \cdot \frac{\frac{64}{81} + \frac{8}{9} \cdot \frac{1}{u}}{u} - \frac{64}{81}\right)\right)}\right), \mathsf{*.f32}\left(s, -3\right)\right) \]
14. Step-by-step derivation

    1. unpow3 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \left(\left(\left(u \cdot u\right) \cdot u\right) \cdot \left(-1 \cdot \frac{\frac{64}{81} + \frac{8}{9} \cdot \frac{1}{u}}{u} - \frac{64}{81}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \left(\left({u}^{2} \cdot u\right) \cdot \left(-1 \cdot \frac{\frac{64}{81} + \frac{8}{9} \cdot \frac{1}{u}}{u} - \frac{64}{81}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \left({u}^{2} \cdot \left(u \cdot \left(-1 \cdot \frac{\frac{64}{81} + \frac{8}{9} \cdot \frac{1}{u}}{u} - \frac{64}{81}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\left({u}^{2}\right), \left(u \cdot \left(-1 \cdot \frac{\frac{64}{81} + \frac{8}{9} \cdot \frac{1}{u}}{u} - \frac{64}{81}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\left(u \cdot u\right), \left(u \cdot \left(-1 \cdot \frac{\frac{64}{81} + \frac{8}{9} \cdot \frac{1}{u}}{u} - \frac{64}{81}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \left(u \cdot \left(-1 \cdot \frac{\frac{64}{81} + \frac{8}{9} \cdot \frac{1}{u}}{u} - \frac{64}{81}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{*.f32}\left(u, \left(-1 \cdot \frac{\frac{64}{81} + \frac{8}{9} \cdot \frac{1}{u}}{u} - \frac{64}{81}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    8. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{*.f32}\left(u, \left(-1 \cdot \frac{\frac{64}{81} + \frac{8}{9} \cdot \frac{1}{u}}{u} + \left(\mathsf{neg}\left(\frac{64}{81}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    9. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{*.f32}\left(u, \left(-1 \cdot \frac{\frac{64}{81} + \frac{8}{9} \cdot \frac{1}{u}}{u} + \frac{-64}{81}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    10. +-commutative N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{*.f32}\left(u, \left(\frac{-64}{81} + -1 \cdot \frac{\frac{64}{81} + \frac{8}{9} \cdot \frac{1}{u}}{u}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    11. mul-1-neg N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{*.f32}\left(u, \left(\frac{-64}{81} + \left(\mathsf{neg}\left(\frac{\frac{64}{81} + \frac{8}{9} \cdot \frac{1}{u}}{u}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    12. unsub-neg N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{*.f32}\left(u, \left(\frac{-64}{81} - \frac{\frac{64}{81} + \frac{8}{9} \cdot \frac{1}{u}}{u}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\frac{-64}{81}, \left(\frac{\frac{64}{81} + \frac{8}{9} \cdot \frac{1}{u}}{u}\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\frac{-64}{81}, \mathsf{/.f32}\left(\left(\frac{64}{81} + \frac{8}{9} \cdot \frac{1}{u}\right), u\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\frac{-64}{81}, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{64}{81}, \left(\frac{8}{9} \cdot \frac{1}{u}\right)\right), u\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    16. associate-*r/ N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\frac{-64}{81}, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{64}{81}, \left(\frac{\frac{8}{9} \cdot 1}{u}\right)\right), u\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    17. metadata-eval N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\frac{-64}{81}, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{64}{81}, \left(\frac{\frac{8}{9}}{u}\right)\right), u\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    18. /-lowering-/.f32 37.8%

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, u\right), \mathsf{*.f32}\left(u, \mathsf{\_.f32}\left(\frac{-64}{81}, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{64}{81}, \mathsf{/.f32}\left(\frac{8}{9}, u\right)\right), u\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

15. Simplified 37.8%

    \[\leadsto \left(u \cdot \color{blue}{\left(\left(u \cdot u\right) \cdot \left(u \cdot \left(-0.7901234567901234 - \frac{0.7901234567901234 + \frac{0.8888888888888888}{u}}{u}\right)\right)\right)}\right) \cdot \left(s \cdot -3\right) \]

16. Final simplification 37.8%

    \[\leadsto \left(s \cdot -3\right) \cdot \left(u \cdot \left(\left(u \cdot u\right) \cdot \left(u \cdot \left(-0.7901234567901234 - \frac{0.7901234567901234 + \frac{0.8888888888888888}{u}}{u}\right)\right)\right)\right) \]
17. Add Preprocessing

Alternative 6: 30.0% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot -3\right) \cdot \left(u \cdot \left(-1.3333333333333333 + -0.7901234567901234 \cdot \left(u \cdot \left(u \cdot u\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  (* s -3.0)
  (* u (+ -1.3333333333333333 (* -0.7901234567901234 (* u (* u u)))))))

C

float code(float s, float u) {
	return (s * -3.0f) * (u * (-1.3333333333333333f + (-0.7901234567901234f * (u * (u * u)))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (s * (-3.0e0)) * (u * ((-1.3333333333333333e0) + ((-0.7901234567901234e0) * (u * (u * u)))))
end function

Julia

function code(s, u)
	return Float32(Float32(s * Float32(-3.0)) * Float32(u * Float32(Float32(-1.3333333333333333) + Float32(Float32(-0.7901234567901234) * Float32(u * Float32(u * u))))))
end

MATLAB

function tmp = code(s, u)
	tmp = (s * single(-3.0)) * (u * (single(-1.3333333333333333) + (single(-0.7901234567901234) * (u * (u * u)))));
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \]

   2. neg-mul-1 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-1 \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(3 \cdot s\right) \cdot -1\right) \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)} \]

   4. *-commutative N/A

      \[\leadsto \log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right) \cdot \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)} \]

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

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right), \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   7. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   8. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   10. div-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   11. associate--r- N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(0 - \frac{u}{\frac{3}{4}}\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}} + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right), \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   16. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(s \cdot 3\right) \cdot -1\right)\right) \]

   20. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(s \cdot \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

   22. metadata-eval 96.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

3. Simplified 96.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{-1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{\frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u - \frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   8. frac-2neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\mathsf{neg}\left(\frac{-3}{4}\right)}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\frac{3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) + u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} - u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   17. --lowering--.f32 98.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{4}, u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

6. Applied egg-rr 98.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{0.25 - u}{0.75}}\right) \cdot \left(s \cdot -3\right) \]

7. Taylor expanded in u around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot u\right)}, \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(u\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. neg-lowering-neg.f32 19.3%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

9. Simplified 19.3%

   \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{-u}}{0.75}\right) \cdot \left(s \cdot -3\right) \]

10. Taylor expanded in u around 0

    \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(u \cdot \left(u \cdot \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right) - \frac{8}{9}\right) - \frac{4}{3}\right)\right)}, \mathsf{*.f32}\left(s, -3\right)\right) \]
11. Step-by-step derivation

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \left(u \cdot \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right) - \frac{8}{9}\right) - \frac{4}{3}\right)\right), \mathsf{*.f32}\left(\color{blue}{s}, -3\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \left(u \cdot \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right) - \frac{8}{9}\right) + \left(\mathsf{neg}\left(\frac{4}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \left(u \cdot \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right) - \frac{8}{9}\right) + \frac{-4}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    4. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \left(\frac{-4}{3} + u \cdot \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right) - \frac{8}{9}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \left(u \cdot \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right) - \frac{8}{9}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right) - \frac{8}{9}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    7. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right) + \left(\mathsf{neg}\left(\frac{8}{9}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right) + \frac{-8}{9}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    9. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \left(\frac{-8}{9} + u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-8}{9}, \left(u \cdot \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-8}{9}, \mathsf{*.f32}\left(u, \left(\frac{-64}{81} \cdot u - \frac{64}{81}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    12. sub-neg N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-8}{9}, \mathsf{*.f32}\left(u, \left(\frac{-64}{81} \cdot u + \left(\mathsf{neg}\left(\frac{64}{81}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    13. metadata-eval N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-8}{9}, \mathsf{*.f32}\left(u, \left(\frac{-64}{81} \cdot u + \frac{-64}{81}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    14. +-commutative N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-8}{9}, \mathsf{*.f32}\left(u, \left(\frac{-64}{81} + \frac{-64}{81} \cdot u\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-8}{9}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-64}{81}, \left(\frac{-64}{81} \cdot u\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    16. *-commutative N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-8}{9}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-64}{81}, \left(u \cdot \frac{-64}{81}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    17. *-lowering-*.f32 27.3%

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-8}{9}, \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-64}{81}, \mathsf{*.f32}\left(u, \frac{-64}{81}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

12. Simplified 27.3%

    \[\leadsto \color{blue}{\left(u \cdot \left(-1.3333333333333333 + u \cdot \left(-0.8888888888888888 + u \cdot \left(-0.7901234567901234 + u \cdot -0.7901234567901234\right)\right)\right)\right)} \cdot \left(s \cdot -3\right) \]

13. Taylor expanded in u around inf

    \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \color{blue}{\left(\frac{-64}{81} \cdot {u}^{3}\right)}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]
14. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \left({u}^{3} \cdot \frac{-64}{81}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(\left({u}^{3}\right), \frac{-64}{81}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    3. cube-mult N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(\left(u \cdot \left(u \cdot u\right)\right), \frac{-64}{81}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(\left(u \cdot {u}^{2}\right), \frac{-64}{81}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \left({u}^{2}\right)\right), \frac{-64}{81}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \left(u \cdot u\right)\right), \frac{-64}{81}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    7. *-lowering-*.f32 30.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(u, u\right)\right), \frac{-64}{81}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

15. Simplified 30.2%

    \[\leadsto \left(u \cdot \left(-1.3333333333333333 + \color{blue}{\left(u \cdot \left(u \cdot u\right)\right) \cdot -0.7901234567901234}\right)\right) \cdot \left(s \cdot -3\right) \]

16. Final simplification 30.2%

    \[\leadsto \left(s \cdot -3\right) \cdot \left(u \cdot \left(-1.3333333333333333 + -0.7901234567901234 \cdot \left(u \cdot \left(u \cdot u\right)\right)\right)\right) \]
17. Add Preprocessing

Alternative 7: 29.5% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{0.75}{u}}{s \cdot 3}} \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (/ 1.0 (/ (/ 0.75 u) (* s 3.0))))

C

float code(float s, float u) {
	return 1.0f / ((0.75f / u) / (s * 3.0f));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = 1.0e0 / ((0.75e0 / u) / (s * 3.0e0))
end function

Julia

function code(s, u)
	return Float32(Float32(1.0) / Float32(Float32(Float32(0.75) / u) / Float32(s * Float32(3.0))))
end

MATLAB

function tmp = code(s, u)
	tmp = single(1.0) / ((single(0.75) / u) / (s * single(3.0)));
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \]

   2. neg-mul-1 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-1 \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(3 \cdot s\right) \cdot -1\right) \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)} \]

   4. *-commutative N/A

      \[\leadsto \log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right) \cdot \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)} \]

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

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right), \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   7. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   8. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   10. div-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   11. associate--r- N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(0 - \frac{u}{\frac{3}{4}}\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}} + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right), \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   16. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(s \cdot 3\right) \cdot -1\right)\right) \]

   20. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(s \cdot \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

   22. metadata-eval 96.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

3. Simplified 96.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{-1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{\frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u - \frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   8. frac-2neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\mathsf{neg}\left(\frac{-3}{4}\right)}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\frac{3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) + u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} - u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   17. --lowering--.f32 98.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{4}, u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

6. Applied egg-rr 98.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{0.25 - u}{0.75}}\right) \cdot \left(s \cdot -3\right) \]

7. Taylor expanded in u around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot u\right)}, \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(u\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. neg-lowering-neg.f32 19.3%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

9. Simplified 19.3%

   \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{-u}}{0.75}\right) \cdot \left(s \cdot -3\right) \]

10. Taylor expanded in u around 0

    \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right)} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(s \cdot u\right) \cdot \color{blue}{4} \]

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

       \[\leadsto \mathsf{*.f32}\left(\left(s \cdot u\right), \color{blue}{4}\right) \]

    3. *-lowering-*.f32 29.7%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, u\right), 4\right) \]

12. Simplified 29.7%

    \[\leadsto \color{blue}{\left(s \cdot u\right) \cdot 4} \]
13. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \left(s \cdot u\right) \cdot \frac{-3}{\color{blue}{\frac{-3}{4}}} \]

    2. associate-/l* N/A

       \[\leadsto \frac{\left(s \cdot u\right) \cdot -3}{\color{blue}{\frac{-3}{4}}} \]

    3. *-commutative N/A

       \[\leadsto \frac{\left(u \cdot s\right) \cdot -3}{\frac{-3}{4}} \]

    4. associate-*r* N/A

       \[\leadsto \frac{u \cdot \left(s \cdot -3\right)}{\frac{-3}{4}} \]

    5. *-commutative N/A

       \[\leadsto \frac{\left(s \cdot -3\right) \cdot u}{\frac{-3}{4}} \]

    6. frac-2neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(s \cdot -3\right) \cdot u\right)}{\color{blue}{\mathsf{neg}\left(\frac{-3}{4}\right)}} \]

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

       \[\leadsto \frac{\left(s \cdot -3\right) \cdot \left(\mathsf{neg}\left(u\right)\right)}{\mathsf{neg}\left(\color{blue}{\frac{-3}{4}}\right)} \]

    8. metadata-eval N/A

       \[\leadsto \frac{\left(s \cdot -3\right) \cdot \left(\mathsf{neg}\left(u\right)\right)}{\frac{3}{4}} \]

    9. neg-mul-1 N/A

       \[\leadsto \frac{\left(s \cdot -3\right) \cdot \left(-1 \cdot u\right)}{\frac{3}{4}} \]

    10. associate-*r* N/A

        \[\leadsto \frac{\left(\left(s \cdot -3\right) \cdot -1\right) \cdot u}{\frac{3}{4}} \]

    11. associate-*r/ N/A

        \[\leadsto \left(\left(s \cdot -3\right) \cdot -1\right) \cdot \color{blue}{\frac{u}{\frac{3}{4}}} \]

    12. clear-num N/A

        \[\leadsto \left(\left(s \cdot -3\right) \cdot -1\right) \cdot \frac{1}{\color{blue}{\frac{\frac{3}{4}}{u}}} \]

    13. div-inv N/A

        \[\leadsto \frac{\left(s \cdot -3\right) \cdot -1}{\color{blue}{\frac{\frac{3}{4}}{u}}} \]

    14. clear-num N/A

        \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\frac{3}{4}}{u}}{\left(s \cdot -3\right) \cdot -1}}} \]

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

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

    16. /-lowering-/.f32 N/A

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

    17. /-lowering-/.f32 N/A

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

    18. associate-*l* N/A

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

    19. metadata-eval N/A

        \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{3}{4}, u\right), \left(s \cdot 3\right)\right)\right) \]

    20. *-lowering-*.f32 29.8%

        \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{3}{4}, u\right), \mathsf{*.f32}\left(s, \color{blue}{3}\right)\right)\right) \]

14. Applied egg-rr 29.8%

    \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.75}{u}}{s \cdot 3}}} \]
15. Add Preprocessing

Alternative 8: 29.5% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{0.25}{u}}{s}} \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (/ 1.0 (/ (/ 0.25 u) s)))

C

float code(float s, float u) {
	return 1.0f / ((0.25f / u) / s);
}

Fortran

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

Julia

function code(s, u)
	return Float32(Float32(1.0) / Float32(Float32(Float32(0.25) / u) / s))
end

MATLAB

function tmp = code(s, u)
	tmp = single(1.0) / ((single(0.25) / u) / s);
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \]

   2. neg-mul-1 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-1 \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(3 \cdot s\right) \cdot -1\right) \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)} \]

   4. *-commutative N/A

      \[\leadsto \log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right) \cdot \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)} \]

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

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right), \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   7. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   8. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   10. div-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   11. associate--r- N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(0 - \frac{u}{\frac{3}{4}}\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}} + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right), \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   16. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(s \cdot 3\right) \cdot -1\right)\right) \]

   20. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(s \cdot \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

   22. metadata-eval 96.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

3. Simplified 96.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{-1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{\frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u - \frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   8. frac-2neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\mathsf{neg}\left(\frac{-3}{4}\right)}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\frac{3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) + u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} - u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   17. --lowering--.f32 98.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{4}, u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

6. Applied egg-rr 98.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{0.25 - u}{0.75}}\right) \cdot \left(s \cdot -3\right) \]

7. Taylor expanded in u around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot u\right)}, \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(u\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. neg-lowering-neg.f32 19.3%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

9. Simplified 19.3%

   \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{-u}}{0.75}\right) \cdot \left(s \cdot -3\right) \]

10. Taylor expanded in u around 0

    \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right)} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(s \cdot u\right) \cdot \color{blue}{4} \]

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

       \[\leadsto \mathsf{*.f32}\left(\left(s \cdot u\right), \color{blue}{4}\right) \]

    3. *-lowering-*.f32 29.7%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, u\right), 4\right) \]

12. Simplified 29.7%

    \[\leadsto \color{blue}{\left(s \cdot u\right) \cdot 4} \]
13. Step-by-step derivation

    1. associate-*l* N/A

       \[\leadsto s \cdot \color{blue}{\left(u \cdot 4\right)} \]

    2. metadata-eval N/A

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

    3. div-inv N/A

       \[\leadsto s \cdot \frac{u}{\color{blue}{\frac{1}{4}}} \]

    4. clear-num N/A

       \[\leadsto s \cdot \frac{1}{\color{blue}{\frac{\frac{1}{4}}{u}}} \]

    5. un-div-inv N/A

       \[\leadsto \frac{s}{\color{blue}{\frac{\frac{1}{4}}{u}}} \]

    6. clear-num N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\frac{1}{4}}{u}}{s}}} \]

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

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

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

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

    9. /-lowering-/.f32 29.8%

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

14. Applied egg-rr 29.8%

    \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.25}{u}}{s}}} \]
15. Add Preprocessing

Alternative 9: 29.5% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{0.25}{s \cdot u}} \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (/ 1.0 (/ 0.25 (* s u))))

C

float code(float s, float u) {
	return 1.0f / (0.25f / (s * u));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = 1.0e0 / (0.25e0 / (s * u))
end function

Julia

function code(s, u)
	return Float32(Float32(1.0) / Float32(Float32(0.25) / Float32(s * u)))
end

MATLAB

function tmp = code(s, u)
	tmp = single(1.0) / (single(0.25) / (s * u));
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \]

   2. neg-mul-1 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-1 \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(3 \cdot s\right) \cdot -1\right) \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)} \]

   4. *-commutative N/A

      \[\leadsto \log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right) \cdot \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)} \]

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

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right), \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   7. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   8. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   10. div-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   11. associate--r- N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(0 - \frac{u}{\frac{3}{4}}\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}} + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right), \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   16. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(s \cdot 3\right) \cdot -1\right)\right) \]

   20. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(s \cdot \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

   22. metadata-eval 96.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

3. Simplified 96.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{-1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{\frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u - \frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   8. frac-2neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\mathsf{neg}\left(\frac{-3}{4}\right)}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\frac{3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) + u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} - u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   17. --lowering--.f32 98.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{4}, u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

6. Applied egg-rr 98.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{0.25 - u}{0.75}}\right) \cdot \left(s \cdot -3\right) \]

7. Taylor expanded in u around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot u\right)}, \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(u\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. neg-lowering-neg.f32 19.3%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

9. Simplified 19.3%

   \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{-u}}{0.75}\right) \cdot \left(s \cdot -3\right) \]

10. Taylor expanded in u around 0

    \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right)} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(s \cdot u\right) \cdot \color{blue}{4} \]

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

       \[\leadsto \mathsf{*.f32}\left(\left(s \cdot u\right), \color{blue}{4}\right) \]

    3. *-lowering-*.f32 29.7%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, u\right), 4\right) \]

12. Simplified 29.7%

    \[\leadsto \color{blue}{\left(s \cdot u\right) \cdot 4} \]
13. Step-by-step derivation

    1. metadata-eval N/A

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

    2. div-inv N/A

       \[\leadsto \frac{s \cdot u}{\color{blue}{\frac{1}{4}}} \]

    3. clear-num N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{4}}{s \cdot u}}} \]

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

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

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

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

    6. *-commutative N/A

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

    7. *-lowering-*.f32 29.8%

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

14. Applied egg-rr 29.8%

    \[\leadsto \color{blue}{\frac{1}{\frac{0.25}{u \cdot s}}} \]

15. Final simplification 29.8%

    \[\leadsto \frac{1}{\frac{0.25}{s \cdot u}} \]
16. Add Preprocessing

Alternative 10: 29.5% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \frac{s}{-0.75} \cdot \left(u \cdot -3\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (/ s -0.75) (* u -3.0)))

C

float code(float s, float u) {
	return (s / -0.75f) * (u * -3.0f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (s / (-0.75e0)) * (u * (-3.0e0))
end function

Julia

function code(s, u)
	return Float32(Float32(s / Float32(-0.75)) * Float32(u * Float32(-3.0)))
end

MATLAB

function tmp = code(s, u)
	tmp = (s / single(-0.75)) * (u * single(-3.0));
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \]

   2. neg-mul-1 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-1 \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(3 \cdot s\right) \cdot -1\right) \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)} \]

   4. *-commutative N/A

      \[\leadsto \log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right) \cdot \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)} \]

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

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right), \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   7. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   8. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   10. div-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   11. associate--r- N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(0 - \frac{u}{\frac{3}{4}}\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}} + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right), \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   16. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(s \cdot 3\right) \cdot -1\right)\right) \]

   20. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(s \cdot \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

   22. metadata-eval 96.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

3. Simplified 96.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{-1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{\frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u - \frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   8. frac-2neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\mathsf{neg}\left(\frac{-3}{4}\right)}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\frac{3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) + u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} - u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   17. --lowering--.f32 98.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{4}, u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

6. Applied egg-rr 98.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{0.25 - u}{0.75}}\right) \cdot \left(s \cdot -3\right) \]

7. Taylor expanded in u around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot u\right)}, \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(u\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. neg-lowering-neg.f32 19.3%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

9. Simplified 19.3%

   \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{-u}}{0.75}\right) \cdot \left(s \cdot -3\right) \]

10. Taylor expanded in u around 0

    \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right)} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(s \cdot u\right) \cdot \color{blue}{4} \]

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

       \[\leadsto \mathsf{*.f32}\left(\left(s \cdot u\right), \color{blue}{4}\right) \]

    3. *-lowering-*.f32 29.7%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, u\right), 4\right) \]

12. Simplified 29.7%

    \[\leadsto \color{blue}{\left(s \cdot u\right) \cdot 4} \]
13. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \left(s \cdot u\right) \cdot \frac{-3}{\color{blue}{\frac{-3}{4}}} \]

    2. associate-/l* N/A

       \[\leadsto \frac{\left(s \cdot u\right) \cdot -3}{\color{blue}{\frac{-3}{4}}} \]

    3. *-commutative N/A

       \[\leadsto \frac{\left(u \cdot s\right) \cdot -3}{\frac{-3}{4}} \]

    4. associate-*r* N/A

       \[\leadsto \frac{u \cdot \left(s \cdot -3\right)}{\frac{-3}{4}} \]

    5. associate-*l/ N/A

       \[\leadsto \frac{u}{\frac{-3}{4}} \cdot \color{blue}{\left(s \cdot -3\right)} \]

    6. clear-num N/A

       \[\leadsto \frac{1}{\frac{\frac{-3}{4}}{u}} \cdot \left(\color{blue}{s} \cdot -3\right) \]

    7. associate-*l/ N/A

       \[\leadsto \frac{1 \cdot \left(s \cdot -3\right)}{\color{blue}{\frac{\frac{-3}{4}}{u}}} \]

    8. *-lft-identity N/A

       \[\leadsto \frac{s \cdot -3}{\frac{\color{blue}{\frac{-3}{4}}}{u}} \]

    9. div-inv N/A

       \[\leadsto \frac{s \cdot -3}{\frac{-3}{4} \cdot \color{blue}{\frac{1}{u}}} \]

    10. times-frac N/A

        \[\leadsto \frac{s}{\frac{-3}{4}} \cdot \color{blue}{\frac{-3}{\frac{1}{u}}} \]

    11. un-div-inv N/A

        \[\leadsto \frac{s}{\frac{-3}{4}} \cdot \left(-3 \cdot \color{blue}{\frac{1}{\frac{1}{u}}}\right) \]

    12. remove-double-div N/A

        \[\leadsto \frac{s}{\frac{-3}{4}} \cdot \left(-3 \cdot u\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\left(\frac{s}{\frac{-3}{4}}\right), \color{blue}{\left(-3 \cdot u\right)}\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, \frac{-3}{4}\right), \left(\color{blue}{-3} \cdot u\right)\right) \]

    15. *-commutative N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, \frac{-3}{4}\right), \left(u \cdot \color{blue}{-3}\right)\right) \]

    16. *-lowering-*.f32 29.7%

        \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, \frac{-3}{4}\right), \mathsf{*.f32}\left(u, \color{blue}{-3}\right)\right) \]

14. Applied egg-rr 29.7%

    \[\leadsto \color{blue}{\frac{s}{-0.75} \cdot \left(u \cdot -3\right)} \]
15. Add Preprocessing

Alternative 11: 29.5% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(1.3333333333333333 \cdot \left(s \cdot 3\right)\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* u (* 1.3333333333333333 (* s 3.0))))

C

float code(float s, float u) {
	return u * (1.3333333333333333f * (s * 3.0f));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = u * (1.3333333333333333e0 * (s * 3.0e0))
end function

Julia

function code(s, u)
	return Float32(u * Float32(Float32(1.3333333333333333) * Float32(s * Float32(3.0))))
end

MATLAB

function tmp = code(s, u)
	tmp = u * (single(1.3333333333333333) * (s * single(3.0)));
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \]

   2. neg-mul-1 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-1 \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(3 \cdot s\right) \cdot -1\right) \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)} \]

   4. *-commutative N/A

      \[\leadsto \log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right) \cdot \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)} \]

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

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right), \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   7. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   8. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   10. div-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   11. associate--r- N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(0 - \frac{u}{\frac{3}{4}}\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}} + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right), \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   16. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(s \cdot 3\right) \cdot -1\right)\right) \]

   20. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(s \cdot \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

   22. metadata-eval 96.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

3. Simplified 96.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{-1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{\frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u - \frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   8. frac-2neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\mathsf{neg}\left(\frac{-3}{4}\right)}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\frac{3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) + u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} - u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   17. --lowering--.f32 98.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{4}, u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

6. Applied egg-rr 98.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{0.25 - u}{0.75}}\right) \cdot \left(s \cdot -3\right) \]

7. Taylor expanded in u around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot u\right)}, \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(u\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. neg-lowering-neg.f32 19.3%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

9. Simplified 19.3%

   \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{-u}}{0.75}\right) \cdot \left(s \cdot -3\right) \]

10. Taylor expanded in u around 0

    \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right)} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(s \cdot u\right) \cdot \color{blue}{4} \]

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

       \[\leadsto \mathsf{*.f32}\left(\left(s \cdot u\right), \color{blue}{4}\right) \]

    3. *-lowering-*.f32 29.7%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, u\right), 4\right) \]

12. Simplified 29.7%

    \[\leadsto \color{blue}{\left(s \cdot u\right) \cdot 4} \]
13. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \left(s \cdot u\right) \cdot \frac{-3}{\color{blue}{\frac{-3}{4}}} \]

    2. associate-/l* N/A

       \[\leadsto \frac{\left(s \cdot u\right) \cdot -3}{\color{blue}{\frac{-3}{4}}} \]

    3. *-commutative N/A

       \[\leadsto \frac{\left(u \cdot s\right) \cdot -3}{\frac{-3}{4}} \]

    4. associate-*r* N/A

       \[\leadsto \frac{u \cdot \left(s \cdot -3\right)}{\frac{-3}{4}} \]

    5. *-commutative N/A

       \[\leadsto \frac{\left(s \cdot -3\right) \cdot u}{\frac{-3}{4}} \]

    6. frac-2neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(s \cdot -3\right) \cdot u\right)}{\color{blue}{\mathsf{neg}\left(\frac{-3}{4}\right)}} \]

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

       \[\leadsto \frac{\left(s \cdot -3\right) \cdot \left(\mathsf{neg}\left(u\right)\right)}{\mathsf{neg}\left(\color{blue}{\frac{-3}{4}}\right)} \]

    8. metadata-eval N/A

       \[\leadsto \frac{\left(s \cdot -3\right) \cdot \left(\mathsf{neg}\left(u\right)\right)}{\frac{3}{4}} \]

    9. neg-mul-1 N/A

       \[\leadsto \frac{\left(s \cdot -3\right) \cdot \left(-1 \cdot u\right)}{\frac{3}{4}} \]

    10. associate-*r* N/A

        \[\leadsto \frac{\left(\left(s \cdot -3\right) \cdot -1\right) \cdot u}{\frac{3}{4}} \]

    11. associate-*r/ N/A

        \[\leadsto \left(\left(s \cdot -3\right) \cdot -1\right) \cdot \color{blue}{\frac{u}{\frac{3}{4}}} \]

    12. div-inv N/A

        \[\leadsto \left(\left(s \cdot -3\right) \cdot -1\right) \cdot \left(u \cdot \color{blue}{\frac{1}{\frac{3}{4}}}\right) \]

    13. metadata-eval N/A

        \[\leadsto \left(\left(s \cdot -3\right) \cdot -1\right) \cdot \left(u \cdot \frac{4}{3}\right) \]

    14. *-commutative N/A

        \[\leadsto \left(\left(s \cdot -3\right) \cdot -1\right) \cdot \left(\frac{4}{3} \cdot \color{blue}{u}\right) \]

    15. associate-*r* N/A

        \[\leadsto \left(\left(\left(s \cdot -3\right) \cdot -1\right) \cdot \frac{4}{3}\right) \cdot \color{blue}{u} \]

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

        \[\leadsto \mathsf{*.f32}\left(\left(\left(\left(s \cdot -3\right) \cdot -1\right) \cdot \frac{4}{3}\right), \color{blue}{u}\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(\left(s \cdot -3\right) \cdot -1\right), \frac{4}{3}\right), u\right) \]

    18. associate-*l* N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(s \cdot \left(-3 \cdot -1\right)\right), \frac{4}{3}\right), u\right) \]

    19. metadata-eval N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(s \cdot 3\right), \frac{4}{3}\right), u\right) \]

    20. *-lowering-*.f32 29.7%

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, 3\right), \frac{4}{3}\right), u\right) \]

14. Applied egg-rr 29.7%

    \[\leadsto \color{blue}{\left(\left(s \cdot 3\right) \cdot 1.3333333333333333\right) \cdot u} \]

15. Final simplification 29.7%

    \[\leadsto u \cdot \left(1.3333333333333333 \cdot \left(s \cdot 3\right)\right) \]
16. Add Preprocessing

Alternative 12: 29.5% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ 1.3333333333333333 \cdot \left(u \cdot \left(s \cdot 3\right)\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* 1.3333333333333333 (* u (* s 3.0))))

C

float code(float s, float u) {
	return 1.3333333333333333f * (u * (s * 3.0f));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = 1.3333333333333333e0 * (u * (s * 3.0e0))
end function

Julia

function code(s, u)
	return Float32(Float32(1.3333333333333333) * Float32(u * Float32(s * Float32(3.0))))
end

MATLAB

function tmp = code(s, u)
	tmp = single(1.3333333333333333) * (u * (s * single(3.0)));
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \]

   2. neg-mul-1 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-1 \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(3 \cdot s\right) \cdot -1\right) \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)} \]

   4. *-commutative N/A

      \[\leadsto \log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right) \cdot \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)} \]

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

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right), \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   7. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   8. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   10. div-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   11. associate--r- N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(0 - \frac{u}{\frac{3}{4}}\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}} + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right), \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   16. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(s \cdot 3\right) \cdot -1\right)\right) \]

   20. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(s \cdot \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

   22. metadata-eval 96.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

3. Simplified 96.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{-1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{\frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u - \frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   8. frac-2neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\mathsf{neg}\left(\frac{-3}{4}\right)}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\frac{3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) + u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} - u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   17. --lowering--.f32 98.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{4}, u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

6. Applied egg-rr 98.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{0.25 - u}{0.75}}\right) \cdot \left(s \cdot -3\right) \]

7. Taylor expanded in u around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot u\right)}, \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(u\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. neg-lowering-neg.f32 19.3%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

9. Simplified 19.3%

   \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{-u}}{0.75}\right) \cdot \left(s \cdot -3\right) \]

10. Taylor expanded in u around 0

    \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right)} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(s \cdot u\right) \cdot \color{blue}{4} \]

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

       \[\leadsto \mathsf{*.f32}\left(\left(s \cdot u\right), \color{blue}{4}\right) \]

    3. *-lowering-*.f32 29.7%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, u\right), 4\right) \]

12. Simplified 29.7%

    \[\leadsto \color{blue}{\left(s \cdot u\right) \cdot 4} \]
13. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \left(s \cdot u\right) \cdot \frac{-3}{\color{blue}{\frac{-3}{4}}} \]

    2. associate-/l* N/A

       \[\leadsto \frac{\left(s \cdot u\right) \cdot -3}{\color{blue}{\frac{-3}{4}}} \]

    3. *-commutative N/A

       \[\leadsto \frac{\left(u \cdot s\right) \cdot -3}{\frac{-3}{4}} \]

    4. associate-*r* N/A

       \[\leadsto \frac{u \cdot \left(s \cdot -3\right)}{\frac{-3}{4}} \]

    5. *-commutative N/A

       \[\leadsto \frac{\left(s \cdot -3\right) \cdot u}{\frac{-3}{4}} \]

    6. frac-2neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(s \cdot -3\right) \cdot u\right)}{\color{blue}{\mathsf{neg}\left(\frac{-3}{4}\right)}} \]

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

       \[\leadsto \frac{\left(s \cdot -3\right) \cdot \left(\mathsf{neg}\left(u\right)\right)}{\mathsf{neg}\left(\color{blue}{\frac{-3}{4}}\right)} \]

    8. metadata-eval N/A

       \[\leadsto \frac{\left(s \cdot -3\right) \cdot \left(\mathsf{neg}\left(u\right)\right)}{\frac{3}{4}} \]

    9. div-inv N/A

       \[\leadsto \left(\left(s \cdot -3\right) \cdot \left(\mathsf{neg}\left(u\right)\right)\right) \cdot \color{blue}{\frac{1}{\frac{3}{4}}} \]

    10. metadata-eval N/A

        \[\leadsto \left(\left(s \cdot -3\right) \cdot \left(\mathsf{neg}\left(u\right)\right)\right) \cdot \frac{4}{3} \]

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

        \[\leadsto \mathsf{*.f32}\left(\left(\left(s \cdot -3\right) \cdot \left(\mathsf{neg}\left(u\right)\right)\right), \color{blue}{\frac{4}{3}}\right) \]

    12. *-commutative N/A

        \[\leadsto \mathsf{*.f32}\left(\left(\left(\mathsf{neg}\left(u\right)\right) \cdot \left(s \cdot -3\right)\right), \frac{4}{3}\right) \]

    13. neg-mul-1 N/A

        \[\leadsto \mathsf{*.f32}\left(\left(\left(-1 \cdot u\right) \cdot \left(s \cdot -3\right)\right), \frac{4}{3}\right) \]

    14. *-commutative N/A

        \[\leadsto \mathsf{*.f32}\left(\left(\left(u \cdot -1\right) \cdot \left(s \cdot -3\right)\right), \frac{4}{3}\right) \]

    15. associate-*l* N/A

        \[\leadsto \mathsf{*.f32}\left(\left(u \cdot \left(-1 \cdot \left(s \cdot -3\right)\right)\right), \frac{4}{3}\right) \]

    16. *-commutative N/A

        \[\leadsto \mathsf{*.f32}\left(\left(u \cdot \left(\left(s \cdot -3\right) \cdot -1\right)\right), \frac{4}{3}\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \left(\left(s \cdot -3\right) \cdot -1\right)\right), \frac{4}{3}\right) \]

    18. associate-*l* N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \left(s \cdot \left(-3 \cdot -1\right)\right)\right), \frac{4}{3}\right) \]

    19. metadata-eval N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \left(s \cdot 3\right)\right), \frac{4}{3}\right) \]

    20. *-lowering-*.f32 29.7%

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \mathsf{*.f32}\left(s, 3\right)\right), \frac{4}{3}\right) \]

14. Applied egg-rr 29.7%

    \[\leadsto \color{blue}{\left(u \cdot \left(s \cdot 3\right)\right) \cdot 1.3333333333333333} \]

15. Final simplification 29.7%

    \[\leadsto 1.3333333333333333 \cdot \left(u \cdot \left(s \cdot 3\right)\right) \]
16. Add Preprocessing

Alternative 13: 29.5% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot -3\right) \cdot \left(u \cdot -1.3333333333333333\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* s -3.0) (* u -1.3333333333333333)))

C

float code(float s, float u) {
	return (s * -3.0f) * (u * -1.3333333333333333f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (s * (-3.0e0)) * (u * (-1.3333333333333333e0))
end function

Julia

function code(s, u)
	return Float32(Float32(s * Float32(-3.0)) * Float32(u * Float32(-1.3333333333333333)))
end

MATLAB

function tmp = code(s, u)
	tmp = (s * single(-3.0)) * (u * single(-1.3333333333333333));
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \]

   2. neg-mul-1 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-1 \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(3 \cdot s\right) \cdot -1\right) \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)} \]

   4. *-commutative N/A

      \[\leadsto \log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right) \cdot \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)} \]

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

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right), \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   7. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   8. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   10. div-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   11. associate--r- N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(0 - \frac{u}{\frac{3}{4}}\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}} + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right), \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   16. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(s \cdot 3\right) \cdot -1\right)\right) \]

   20. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(s \cdot \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

   22. metadata-eval 96.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

3. Simplified 96.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{-1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{\frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u - \frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   8. frac-2neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\mathsf{neg}\left(\frac{-3}{4}\right)}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\frac{3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) + u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} - u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   17. --lowering--.f32 98.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{4}, u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

6. Applied egg-rr 98.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{0.25 - u}{0.75}}\right) \cdot \left(s \cdot -3\right) \]

7. Taylor expanded in u around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot u\right)}, \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(u\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. neg-lowering-neg.f32 19.3%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

9. Simplified 19.3%

   \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{-u}}{0.75}\right) \cdot \left(s \cdot -3\right) \]

10. Taylor expanded in u around 0

    \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(\frac{-4}{3} \cdot u\right)}, \mathsf{*.f32}\left(s, -3\right)\right) \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\left(u \cdot \frac{-4}{3}\right), \mathsf{*.f32}\left(\color{blue}{s}, -3\right)\right) \]

    2. *-lowering-*.f32 29.7%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, \frac{-4}{3}\right), \mathsf{*.f32}\left(\color{blue}{s}, -3\right)\right) \]

12. Simplified 29.7%

    \[\leadsto \color{blue}{\left(u \cdot -1.3333333333333333\right)} \cdot \left(s \cdot -3\right) \]

13. Final simplification 29.7%

    \[\leadsto \left(s \cdot -3\right) \cdot \left(u \cdot -1.3333333333333333\right) \]
14. Add Preprocessing

Alternative 14: 29.5% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \left(u \cdot -3\right) \cdot \left(s \cdot -1.3333333333333333\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* u -3.0) (* s -1.3333333333333333)))

C

float code(float s, float u) {
	return (u * -3.0f) * (s * -1.3333333333333333f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (u * (-3.0e0)) * (s * (-1.3333333333333333e0))
end function

Julia

function code(s, u)
	return Float32(Float32(u * Float32(-3.0)) * Float32(s * Float32(-1.3333333333333333)))
end

MATLAB

function tmp = code(s, u)
	tmp = (u * single(-3.0)) * (s * single(-1.3333333333333333));
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \]

   2. neg-mul-1 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-1 \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(3 \cdot s\right) \cdot -1\right) \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)} \]

   4. *-commutative N/A

      \[\leadsto \log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right) \cdot \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)} \]

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

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right), \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   7. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   8. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   10. div-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   11. associate--r- N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(0 - \frac{u}{\frac{3}{4}}\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}} + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right), \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   16. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(s \cdot 3\right) \cdot -1\right)\right) \]

   20. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(s \cdot \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

   22. metadata-eval 96.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

3. Simplified 96.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{-1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{\frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u - \frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   8. frac-2neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\mathsf{neg}\left(\frac{-3}{4}\right)}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\frac{3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) + u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} - u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   17. --lowering--.f32 98.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{4}, u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

6. Applied egg-rr 98.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{0.25 - u}{0.75}}\right) \cdot \left(s \cdot -3\right) \]

7. Taylor expanded in u around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot u\right)}, \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(u\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. neg-lowering-neg.f32 19.3%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

9. Simplified 19.3%

   \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{-u}}{0.75}\right) \cdot \left(s \cdot -3\right) \]

10. Taylor expanded in u around 0

    \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right)} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(s \cdot u\right) \cdot \color{blue}{4} \]

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

       \[\leadsto \mathsf{*.f32}\left(\left(s \cdot u\right), \color{blue}{4}\right) \]

    3. *-lowering-*.f32 29.7%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, u\right), 4\right) \]

12. Simplified 29.7%

    \[\leadsto \color{blue}{\left(s \cdot u\right) \cdot 4} \]
13. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \left(s \cdot u\right) \cdot \left(-3 \cdot \color{blue}{\frac{-4}{3}}\right) \]

    2. associate-*l* N/A

       \[\leadsto \left(\left(s \cdot u\right) \cdot -3\right) \cdot \color{blue}{\frac{-4}{3}} \]

    3. *-commutative N/A

       \[\leadsto \left(\left(u \cdot s\right) \cdot -3\right) \cdot \frac{-4}{3} \]

    4. associate-*r* N/A

       \[\leadsto \left(u \cdot \left(s \cdot -3\right)\right) \cdot \frac{-4}{3} \]

    5. *-commutative N/A

       \[\leadsto \left(u \cdot \left(-3 \cdot s\right)\right) \cdot \frac{-4}{3} \]

    6. associate-*r* N/A

       \[\leadsto \left(\left(u \cdot -3\right) \cdot s\right) \cdot \frac{-4}{3} \]

    7. associate-*l* N/A

       \[\leadsto \left(u \cdot -3\right) \cdot \color{blue}{\left(s \cdot \frac{-4}{3}\right)} \]

    8. *-commutative N/A

       \[\leadsto \left(-3 \cdot u\right) \cdot \left(\color{blue}{s} \cdot \frac{-4}{3}\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\left(-3 \cdot u\right), \color{blue}{\left(s \cdot \frac{-4}{3}\right)}\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{*.f32}\left(\left(u \cdot -3\right), \left(\color{blue}{s} \cdot \frac{-4}{3}\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, -3\right), \left(\color{blue}{s} \cdot \frac{-4}{3}\right)\right) \]

    12. *-lowering-*.f32 29.7%

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u, -3\right), \mathsf{*.f32}\left(s, \color{blue}{\frac{-4}{3}}\right)\right) \]

14. Applied egg-rr 29.7%

    \[\leadsto \color{blue}{\left(u \cdot -3\right) \cdot \left(s \cdot -1.3333333333333333\right)} \]
15. Add Preprocessing

Alternative 15: 29.5% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \frac{s}{\frac{0.25}{u}} \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (/ s (/ 0.25 u)))

C

float code(float s, float u) {
	return s / (0.25f / u);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s / (0.25e0 / u)
end function

Julia

function code(s, u)
	return Float32(s / Float32(Float32(0.25) / u))
end

MATLAB

function tmp = code(s, u)
	tmp = s / (single(0.25) / u);
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \]

   2. neg-mul-1 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-1 \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(3 \cdot s\right) \cdot -1\right) \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)} \]

   4. *-commutative N/A

      \[\leadsto \log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right) \cdot \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)} \]

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

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right), \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   7. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   8. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   10. div-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   11. associate--r- N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(0 - \frac{u}{\frac{3}{4}}\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}} + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right), \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   16. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(s \cdot 3\right) \cdot -1\right)\right) \]

   20. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(s \cdot \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

   22. metadata-eval 96.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

3. Simplified 96.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{-1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{\frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u - \frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   8. frac-2neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\mathsf{neg}\left(\frac{-3}{4}\right)}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\frac{3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) + u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} - u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   17. --lowering--.f32 98.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{4}, u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

6. Applied egg-rr 98.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{0.25 - u}{0.75}}\right) \cdot \left(s \cdot -3\right) \]

7. Taylor expanded in u around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot u\right)}, \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(u\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. neg-lowering-neg.f32 19.3%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

9. Simplified 19.3%

   \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{-u}}{0.75}\right) \cdot \left(s \cdot -3\right) \]

10. Taylor expanded in u around 0

    \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right)} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(s \cdot u\right) \cdot \color{blue}{4} \]

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

       \[\leadsto \mathsf{*.f32}\left(\left(s \cdot u\right), \color{blue}{4}\right) \]

    3. *-lowering-*.f32 29.7%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, u\right), 4\right) \]

12. Simplified 29.7%

    \[\leadsto \color{blue}{\left(s \cdot u\right) \cdot 4} \]
13. Step-by-step derivation

    1. associate-*l* N/A

       \[\leadsto s \cdot \color{blue}{\left(u \cdot 4\right)} \]

    2. metadata-eval N/A

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

    3. div-inv N/A

       \[\leadsto s \cdot \frac{u}{\color{blue}{\frac{1}{4}}} \]

    4. clear-num N/A

       \[\leadsto s \cdot \frac{1}{\color{blue}{\frac{\frac{1}{4}}{u}}} \]

    5. un-div-inv N/A

       \[\leadsto \frac{s}{\color{blue}{\frac{\frac{1}{4}}{u}}} \]

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

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

    7. /-lowering-/.f32 29.7%

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

14. Applied egg-rr 29.7%

    \[\leadsto \color{blue}{\frac{s}{\frac{0.25}{u}}} \]
15. Add Preprocessing

Alternative 16: 29.5% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot u\right) \cdot 4 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* s u) 4.0))

C

float code(float s, float u) {
	return (s * u) * 4.0f;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (s * u) * 4.0e0
end function

Julia

function code(s, u)
	return Float32(Float32(s * u) * Float32(4.0))
end

MATLAB

function tmp = code(s, u)
	tmp = (s * u) * single(4.0);
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. log-rec N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right) \]

   2. neg-mul-1 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(-1 \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(3 \cdot s\right) \cdot -1\right) \cdot \color{blue}{\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)} \]

   4. *-commutative N/A

      \[\leadsto \log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right) \cdot \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)} \]

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

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right), \color{blue}{\left(\left(3 \cdot s\right) \cdot -1\right)}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\log \left(1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   7. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   8. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\color{blue}{\left(3 \cdot s\right)} \cdot -1\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   10. div-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(0 - \left(\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   11. associate--r- N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(0 - \frac{u}{\frac{3}{4}}\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right) + \frac{\frac{1}{4}}{\frac{3}{4}}\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}} + \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\left(\frac{\frac{1}{4}}{\frac{3}{4}}\right), \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(\color{blue}{3} \cdot s\right) \cdot -1\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{u}{\frac{3}{4}}\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   16. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \left(\frac{u}{\mathsf{neg}\left(\frac{3}{4}\right)}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   17. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \left(\mathsf{neg}\left(\frac{3}{4}\right)\right)\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(3 \cdot s\right) \cdot -1\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(\left(s \cdot 3\right) \cdot -1\right)\right) \]

   20. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \left(s \cdot \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, \color{blue}{\left(3 \cdot -1\right)}\right)\right) \]

   22. metadata-eval 96.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{+.f32}\left(\frac{1}{3}, \mathsf{/.f32}\left(u, \frac{-3}{4}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

3. Simplified 96.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \frac{1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{-1}{3}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-3}{4}} - \frac{\frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u - \frac{1}{4}}{\frac{-3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   8. frac-2neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\mathsf{neg}\left(\frac{-3}{4}\right)}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)}{\frac{3}{4}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u - \frac{1}{4}\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) + u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} + \left(\mathsf{neg}\left(u\right)\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\frac{1}{4} - u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   17. --lowering--.f32 98.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{4}, u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

6. Applied egg-rr 98.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{0.25 - u}{0.75}}\right) \cdot \left(s \cdot -3\right) \]

7. Taylor expanded in u around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\color{blue}{\left(-1 \cdot u\right)}, \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(u\right)\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   2. neg-lowering-neg.f32 19.3%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(u\right), \frac{3}{4}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

9. Simplified 19.3%

   \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{-u}}{0.75}\right) \cdot \left(s \cdot -3\right) \]

10. Taylor expanded in u around 0

    \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right)} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(s \cdot u\right) \cdot \color{blue}{4} \]

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

       \[\leadsto \mathsf{*.f32}\left(\left(s \cdot u\right), \color{blue}{4}\right) \]

    3. *-lowering-*.f32 29.7%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, u\right), 4\right) \]

12. Simplified 29.7%

    \[\leadsto \color{blue}{\left(s \cdot u\right) \cdot 4} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024159 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, upper"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))