Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.8% → 98.3%

Time: 9.2s

Alternatives: 5

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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.9%

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

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

   2. +-commutative N/A

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

   3. associate-+r+ N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac2 N/A

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

   6. sub-neg N/A

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

   7. associate--r- N/A

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

   8. metadata-eval N/A

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

   9. div-sub N/A

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

   10. metadata-eval N/A

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

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

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

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

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

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

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

   14. log-rec N/A

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

   15. associate-*r* N/A

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

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

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

6. Applied egg-rr 96.6%

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

   1. metadata-eval N/A

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

   2. distribute-neg-frac2 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. div-sub N/A

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

   6. remove-double-div N/A

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

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

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

   8. remove-double-div N/A

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

   9. --lowering--.f32 98.3%

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

8. Applied egg-rr 98.3%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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. distribute-neg-in N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. +-commutative 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. Final simplification 98.2%

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

Alternative 3: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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. distribute-neg-in N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. +-commutative 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%

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

Alternative 4: 94.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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%

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

Alternative 5: 29.6% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot 4\right) \end{array} \]

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(s * Float32(u * Float32(4.0)))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (u * single(4.0));
end

Derivation

1. Initial program 95.9%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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. distribute-neg-in N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. +-commutative 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 18.1%

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

9. Simplified 18.1%

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

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. associate-*l* N/A

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

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

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

    4. *-lowering-*.f32 29.5%

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

12. Simplified 29.5%

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

Reproduce

herbie shell --seed 2024164 
(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))))))