Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.8% → 98.2%

Time: 13.6s

Alternatives: 15

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\frac{\left(u + -0.25\right) \cdot \left(u + -0.25\right)}{\frac{-0.421875}{u + -0.25}}\right) - s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{0.6666666666666666 + \frac{u}{0.75}}{\frac{0.75}{u + -0.25}}\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (-
  (*
   (* s -3.0)
   (log1p (/ (* (+ u -0.25) (+ u -0.25)) (/ -0.421875 (+ u -0.25)))))
  (*
   s
   (*
    -3.0
    (log1p (/ (+ 0.6666666666666666 (/ u 0.75)) (/ 0.75 (+ u -0.25))))))))

C

float code(float s, float u) {
	return ((s * -3.0f) * log1pf((((u + -0.25f) * (u + -0.25f)) / (-0.421875f / (u + -0.25f))))) - (s * (-3.0f * log1pf(((0.6666666666666666f + (u / 0.75f)) / (0.75f / (u + -0.25f))))));
}

Julia

function code(s, u)
	return Float32(Float32(Float32(s * Float32(-3.0)) * log1p(Float32(Float32(Float32(u + Float32(-0.25)) * Float32(u + Float32(-0.25))) / Float32(Float32(-0.421875) / Float32(u + Float32(-0.25)))))) - Float32(s * Float32(Float32(-3.0) * log1p(Float32(Float32(Float32(0.6666666666666666) + Float32(u / Float32(0.75))) / Float32(Float32(0.75) / Float32(u + Float32(-0.25))))))))
end

Derivation

1. Initial program 95.7%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.7%

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

3. Simplified 96.7%

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

6. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(\frac{\left(u + -0.25\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)}{-0.421875}\right) - \mathsf{log1p}\left(\frac{u + -0.25}{0.75} \cdot \left(0.6666666666666666 + \frac{u}{0.75}\right)\right)\right)} \cdot \left(s \cdot -3\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(s \cdot -3\right) \cdot \color{blue}{\left(\log \left(1 + \frac{\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)}{\frac{-27}{64}}\right) - \log \left(1 + \frac{u + \frac{-1}{4}}{\frac{3}{4}} \cdot \left(\frac{2}{3} + \frac{u}{\frac{3}{4}}\right)\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \left(s \cdot -3\right) \cdot \left(\log \left(1 + \frac{\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)}{\frac{-27}{64}}\right) + \color{blue}{\left(\mathsf{neg}\left(\log \left(1 + \frac{u + \frac{-1}{4}}{\frac{3}{4}} \cdot \left(\frac{2}{3} + \frac{u}{\frac{3}{4}}\right)\right)\right)\right)}\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \left(s \cdot -3\right) \cdot \log \left(1 + \frac{\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)}{\frac{-27}{64}}\right) + \color{blue}{\left(s \cdot -3\right) \cdot \left(\mathsf{neg}\left(\log \left(1 + \frac{u + \frac{-1}{4}}{\frac{3}{4}} \cdot \left(\frac{2}{3} + \frac{u}{\frac{3}{4}}\right)\right)\right)\right)} \]

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

      \[\leadsto \mathsf{+.f32}\left(\left(\left(s \cdot -3\right) \cdot \log \left(1 + \frac{\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)}{\frac{-27}{64}}\right)\right), \color{blue}{\left(\left(s \cdot -3\right) \cdot \left(\mathsf{neg}\left(\log \left(1 + \frac{u + \frac{-1}{4}}{\frac{3}{4}} \cdot \left(\frac{2}{3} + \frac{u}{\frac{3}{4}}\right)\right)\right)\right)\right)}\right) \]

8. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\left(u + -0.25\right) \cdot \left(\left(u + -0.25\right) \cdot \frac{u + -0.25}{-0.421875}\right)\right) + \left(s \cdot -3\right) \cdot \left(-\mathsf{log1p}\left(\frac{0.6666666666666666 + \frac{u}{0.75}}{\frac{0.75}{u + -0.25}}\right)\right)} \]
9. Step-by-step derivation

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

      \[\leadsto \left(s \cdot -3\right) \cdot \log \left(1 + \left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \frac{u + \frac{-1}{4}}{\frac{-27}{64}}\right)\right) + \left(\mathsf{neg}\left(\left(s \cdot -3\right) \cdot \log \left(1 + \frac{\frac{2}{3} + \frac{u}{\frac{3}{4}}}{\frac{\frac{3}{4}}{u + \frac{-1}{4}}}\right)\right)\right) \]

   2. unsub-neg N/A

      \[\leadsto \left(s \cdot -3\right) \cdot \log \left(1 + \left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \frac{u + \frac{-1}{4}}{\frac{-27}{64}}\right)\right) - \color{blue}{\left(s \cdot -3\right) \cdot \log \left(1 + \frac{\frac{2}{3} + \frac{u}{\frac{3}{4}}}{\frac{\frac{3}{4}}{u + \frac{-1}{4}}}\right)} \]

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

      \[\leadsto \mathsf{\_.f32}\left(\left(\left(s \cdot -3\right) \cdot \log \left(1 + \left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \frac{u + \frac{-1}{4}}{\frac{-27}{64}}\right)\right)\right), \color{blue}{\left(\left(s \cdot -3\right) \cdot \log \left(1 + \frac{\frac{2}{3} + \frac{u}{\frac{3}{4}}}{\frac{\frac{3}{4}}{u + \frac{-1}{4}}}\right)\right)}\right) \]

10. Applied egg-rr 98.4%

    \[\leadsto \color{blue}{\left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\frac{\left(u + -0.25\right) \cdot \left(u + -0.25\right)}{\frac{-0.421875}{u + -0.25}}\right) - s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{0.6666666666666666 + \frac{u}{0.75}}{\frac{0.75}{u + -0.25}}\right)\right)} \]
11. Add Preprocessing

Alternative 2: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-3 \cdot \mathsf{log1p}\left(\left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right) \cdot \left(\left(u + -0.25\right) \cdot -2.3703703703703702\right)\right) + 3 \cdot \mathsf{log1p}\left(\left(u + -0.25\right) \cdot \left(0.8888888888888888 + 1.3333333333333333 \cdot \left(u \cdot 1.3333333333333333\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  s
  (+
   (*
    -3.0
    (log1p
     (* (* (+ u -0.25) (+ u -0.25)) (* (+ u -0.25) -2.3703703703703702))))
   (*
    3.0
    (log1p
     (*
      (+ u -0.25)
      (+
       0.8888888888888888
       (* 1.3333333333333333 (* u 1.3333333333333333)))))))))

C

float code(float s, float u) {
	return s * ((-3.0f * log1pf((((u + -0.25f) * (u + -0.25f)) * ((u + -0.25f) * -2.3703703703703702f)))) + (3.0f * log1pf(((u + -0.25f) * (0.8888888888888888f + (1.3333333333333333f * (u * 1.3333333333333333f)))))));
}

Julia

function code(s, u)
	return Float32(s * Float32(Float32(Float32(-3.0) * log1p(Float32(Float32(Float32(u + Float32(-0.25)) * Float32(u + Float32(-0.25))) * Float32(Float32(u + Float32(-0.25)) * Float32(-2.3703703703703702))))) + Float32(Float32(3.0) * log1p(Float32(Float32(u + Float32(-0.25)) * Float32(Float32(0.8888888888888888) + Float32(Float32(1.3333333333333333) * Float32(u * Float32(1.3333333333333333)))))))))
end

Derivation

1. Initial program 95.7%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.7%

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

3. Simplified 96.7%

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

6. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(\frac{\left(u + -0.25\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)}{-0.421875}\right) - \mathsf{log1p}\left(\frac{u + -0.25}{0.75} \cdot \left(0.6666666666666666 + \frac{u}{0.75}\right)\right)\right)} \cdot \left(s \cdot -3\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(s \cdot -3\right) \cdot \color{blue}{\left(\log \left(1 + \frac{\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)}{\frac{-27}{64}}\right) - \log \left(1 + \frac{u + \frac{-1}{4}}{\frac{3}{4}} \cdot \left(\frac{2}{3} + \frac{u}{\frac{3}{4}}\right)\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \left(s \cdot -3\right) \cdot \left(\log \left(1 + \frac{\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)}{\frac{-27}{64}}\right) + \color{blue}{\left(\mathsf{neg}\left(\log \left(1 + \frac{u + \frac{-1}{4}}{\frac{3}{4}} \cdot \left(\frac{2}{3} + \frac{u}{\frac{3}{4}}\right)\right)\right)\right)}\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \left(s \cdot -3\right) \cdot \log \left(1 + \frac{\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)}{\frac{-27}{64}}\right) + \color{blue}{\left(s \cdot -3\right) \cdot \left(\mathsf{neg}\left(\log \left(1 + \frac{u + \frac{-1}{4}}{\frac{3}{4}} \cdot \left(\frac{2}{3} + \frac{u}{\frac{3}{4}}\right)\right)\right)\right)} \]

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

      \[\leadsto \mathsf{+.f32}\left(\left(\left(s \cdot -3\right) \cdot \log \left(1 + \frac{\left(u + \frac{-1}{4}\right) \cdot \left(\left(u + \frac{-1}{4}\right) \cdot \left(u + \frac{-1}{4}\right)\right)}{\frac{-27}{64}}\right)\right), \color{blue}{\left(\left(s \cdot -3\right) \cdot \left(\mathsf{neg}\left(\log \left(1 + \frac{u + \frac{-1}{4}}{\frac{3}{4}} \cdot \left(\frac{2}{3} + \frac{u}{\frac{3}{4}}\right)\right)\right)\right)\right)}\right) \]

8. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\left(u + -0.25\right) \cdot \left(\left(u + -0.25\right) \cdot \frac{u + -0.25}{-0.421875}\right)\right) + \left(s \cdot -3\right) \cdot \left(-\mathsf{log1p}\left(\frac{0.6666666666666666 + \frac{u}{0.75}}{\frac{0.75}{u + -0.25}}\right)\right)} \]

9. Taylor expanded in s around 0

   \[\leadsto \color{blue}{s \cdot \left(-3 \cdot \log \left(1 + \frac{-64}{27} \cdot {\left(u - \frac{1}{4}\right)}^{3}\right) + 3 \cdot \log \left(1 + \frac{4}{3} \cdot \left(\left(\frac{2}{3} + \frac{4}{3} \cdot u\right) \cdot \left(u - \frac{1}{4}\right)\right)\right)\right)} \]
10. Step-by-step derivation

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

       \[\leadsto \mathsf{*.f32}\left(s, \color{blue}{\left(-3 \cdot \log \left(1 + \frac{-64}{27} \cdot {\left(u - \frac{1}{4}\right)}^{3}\right) + 3 \cdot \log \left(1 + \frac{4}{3} \cdot \left(\left(\frac{2}{3} + \frac{4}{3} \cdot u\right) \cdot \left(u - \frac{1}{4}\right)\right)\right)\right)}\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(-3 \cdot \log \left(1 + \frac{-64}{27} \cdot {\left(u - \frac{1}{4}\right)}^{3}\right)\right), \color{blue}{\left(3 \cdot \log \left(1 + \frac{4}{3} \cdot \left(\left(\frac{2}{3} + \frac{4}{3} \cdot u\right) \cdot \left(u - \frac{1}{4}\right)\right)\right)\right)}\right)\right) \]

11. Simplified 98.3%

    \[\leadsto \color{blue}{s \cdot \left(-3 \cdot \mathsf{log1p}\left(\left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right) \cdot \left(\left(u + -0.25\right) \cdot -2.3703703703703702\right)\right) + 3 \cdot \mathsf{log1p}\left(\left(u + -0.25\right) \cdot \left(0.8888888888888888 + 1.3333333333333333 \cdot \left(u \cdot 1.3333333333333333\right)\right)\right)\right)} \]
12. Add Preprocessing

Alternative 3: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot -3\right) \cdot \left(\mathsf{log1p}\left(\frac{\left(u + -0.25\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)}{-0.421875}\right) - \mathsf{log1p}\left(\left(0.6666666666666666 + \frac{u}{0.75}\right) \cdot \frac{u + -0.25}{0.75}\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  (* s -3.0)
  (-
   (log1p (/ (* (+ u -0.25) (* (+ u -0.25) (+ u -0.25))) -0.421875))
   (log1p (* (+ 0.6666666666666666 (/ u 0.75)) (/ (+ u -0.25) 0.75))))))

C

float code(float s, float u) {
	return (s * -3.0f) * (log1pf((((u + -0.25f) * ((u + -0.25f) * (u + -0.25f))) / -0.421875f)) - log1pf(((0.6666666666666666f + (u / 0.75f)) * ((u + -0.25f) / 0.75f))));
}

Julia

function code(s, u)
	return Float32(Float32(s * Float32(-3.0)) * Float32(log1p(Float32(Float32(Float32(u + Float32(-0.25)) * Float32(Float32(u + Float32(-0.25)) * Float32(u + Float32(-0.25)))) / Float32(-0.421875))) - log1p(Float32(Float32(Float32(0.6666666666666666) + Float32(u / Float32(0.75))) * Float32(Float32(u + Float32(-0.25)) / Float32(0.75))))))
end

Derivation

1. Initial program 95.7%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.7%

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

3. Simplified 96.7%

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

6. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(\frac{\left(u + -0.25\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)}{-0.421875}\right) - \mathsf{log1p}\left(\frac{u + -0.25}{0.75} \cdot \left(0.6666666666666666 + \frac{u}{0.75}\right)\right)\right)} \cdot \left(s \cdot -3\right) \]

7. Final simplification 98.3%

   \[\leadsto \left(s \cdot -3\right) \cdot \left(\mathsf{log1p}\left(\frac{\left(u + -0.25\right) \cdot \left(\left(u + -0.25\right) \cdot \left(u + -0.25\right)\right)}{-0.421875}\right) - \mathsf{log1p}\left(\left(0.6666666666666666 + \frac{u}{0.75}\right) \cdot \frac{u + -0.25}{0.75}\right)\right) \]
8. Add Preprocessing

Alternative 4: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot -3\right) \cdot \left(\mathsf{log1p}\left(\frac{\frac{u}{-0.75} - -0.3333333333333333}{\frac{0.75}{u + -0.25}}\right) - \mathsf{log1p}\left(\frac{u + -0.25}{0.75}\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  (* s -3.0)
  (-
   (log1p (/ (- (/ u -0.75) -0.3333333333333333) (/ 0.75 (+ u -0.25))))
   (log1p (/ (+ u -0.25) 0.75)))))

C

float code(float s, float u) {
	return (s * -3.0f) * (log1pf((((u / -0.75f) - -0.3333333333333333f) / (0.75f / (u + -0.25f)))) - log1pf(((u + -0.25f) / 0.75f)));
}

Julia

function code(s, u)
	return Float32(Float32(s * Float32(-3.0)) * Float32(log1p(Float32(Float32(Float32(u / Float32(-0.75)) - Float32(-0.3333333333333333)) / Float32(Float32(0.75) / Float32(u + Float32(-0.25))))) - log1p(Float32(Float32(u + Float32(-0.25)) / Float32(0.75)))))
end

Derivation

1. Initial program 95.7%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.7%

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

3. Simplified 96.7%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
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. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-frac-neg N/A

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

   6. sub-neg N/A

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

   7. associate--r- N/A

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

   8. metadata-eval N/A

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

   9. div-sub N/A

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

   10. flip-- N/A

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

   11. log-div N/A

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

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

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

6. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(\frac{\frac{u}{-0.75} - -0.3333333333333333}{\frac{0.75}{u + -0.25}}\right) - \mathsf{log1p}\left(\frac{u + -0.25}{0.75}\right)\right)} \cdot \left(s \cdot -3\right) \]

7. Final simplification 98.0%

   \[\leadsto \left(s \cdot -3\right) \cdot \left(\mathsf{log1p}\left(\frac{\frac{u}{-0.75} - -0.3333333333333333}{\frac{0.75}{u + -0.25}}\right) - \mathsf{log1p}\left(\frac{u + -0.25}{0.75}\right)\right) \]
8. Add Preprocessing

Alternative 5: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* s -3.0) (log1p (fma u -1.3333333333333333 0.3333333333333333))))

C

float code(float s, float u) {
	return (s * -3.0f) * log1pf(fmaf(u, -1.3333333333333333f, 0.3333333333333333f));
}

Julia

function code(s, u)
	return Float32(Float32(s * Float32(-3.0)) * log1p(fma(u, Float32(-1.3333333333333333), Float32(0.3333333333333333))))
end

Derivation

1. Initial program 95.7%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.7%

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

3. Simplified 96.7%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
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. div-inv N/A

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

   3. fma-define N/A

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

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

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

   5. metadata-eval 98.0%

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

6. Applied egg-rr 98.0%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)}\right) \cdot \left(s \cdot -3\right) \]

7. Final simplification 98.0%

   \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, -1.3333333333333333, 0.3333333333333333\right)\right) \]
8. Add Preprocessing

Alternative 6: 96.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\frac{u}{-1.3333333333333333 + \frac{-0.3333333333333333}{u}} \cdot \left(1.7777777777777777 + \frac{-0.1111111111111111}{u \cdot u}\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  (* s -3.0)
  (log1p
   (*
    (/ u (+ -1.3333333333333333 (/ -0.3333333333333333 u)))
    (+ 1.7777777777777777 (/ -0.1111111111111111 (* u u)))))))

C

float code(float s, float u) {
	return (s * -3.0f) * log1pf(((u / (-1.3333333333333333f + (-0.3333333333333333f / u))) * (1.7777777777777777f + (-0.1111111111111111f / (u * u)))));
}

Julia

function code(s, u)
	return Float32(Float32(s * Float32(-3.0)) * log1p(Float32(Float32(u / Float32(Float32(-1.3333333333333333) + Float32(Float32(-0.3333333333333333) / u))) * Float32(Float32(1.7777777777777777) + Float32(Float32(-0.1111111111111111) / Float32(u * u))))))
end

Derivation

1. Initial program 95.7%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.7%

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

3. Simplified 96.7%

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

5. Taylor expanded in u around inf

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

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

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. /-lowering-/.f32 96.9%

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

7. Simplified 96.9%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{u \cdot \left(-1.3333333333333333 + \frac{0.3333333333333333}{u}\right)}\right) \cdot \left(s \cdot -3\right) \]
8. Step-by-step derivation

   1. *-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*l/ N/A

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

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

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

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

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

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\frac{16}{9} - \frac{\frac{1}{3}}{u} \cdot \frac{\frac{1}{3}}{u}\right), u\right), \left(\frac{-4}{3} - \frac{\frac{1}{3}}{u}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(\frac{16}{9}, \left(\frac{\frac{1}{3}}{u} \cdot \frac{\frac{1}{3}}{u}\right)\right), u\right), \left(\frac{-4}{3} - \frac{\frac{1}{3}}{u}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   8. frac-times N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(\frac{16}{9}, \left(\frac{\frac{1}{3} \cdot \frac{1}{3}}{u \cdot u}\right)\right), u\right), \left(\frac{-4}{3} - \frac{\frac{1}{3}}{u}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(\frac{16}{9}, \left(\frac{\frac{1}{9}}{u \cdot u}\right)\right), u\right), \left(\frac{-4}{3} - \frac{\frac{1}{3}}{u}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(\frac{16}{9}, \mathsf{/.f32}\left(\frac{1}{9}, \left(u \cdot u\right)\right)\right), u\right), \left(\frac{-4}{3} - \frac{\frac{1}{3}}{u}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(\frac{16}{9}, \mathsf{/.f32}\left(\frac{1}{9}, \mathsf{*.f32}\left(u, u\right)\right)\right), u\right), \left(\frac{-4}{3} - \frac{\frac{1}{3}}{u}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(\frac{16}{9}, \mathsf{/.f32}\left(\frac{1}{9}, \mathsf{*.f32}\left(u, u\right)\right)\right), u\right), \mathsf{\_.f32}\left(\frac{-4}{3}, \left(\frac{\frac{1}{3}}{u}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   13. /-lowering-/.f32 97.0%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(\frac{16}{9}, \mathsf{/.f32}\left(\frac{1}{9}, \mathsf{*.f32}\left(u, u\right)\right)\right), u\right), \mathsf{\_.f32}\left(\frac{-4}{3}, \mathsf{/.f32}\left(\frac{1}{3}, u\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

9. Applied egg-rr 97.0%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{\left(1.7777777777777777 - \frac{0.1111111111111111}{u \cdot u}\right) \cdot u}{-1.3333333333333333 - \frac{0.3333333333333333}{u}}}\right) \cdot \left(s \cdot -3\right) \]
10. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\left(\frac{16}{9} - \frac{\frac{1}{9}}{u \cdot u}\right) \cdot \frac{u}{\frac{-4}{3} - \frac{\frac{1}{3}}{u}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-4}{3} - \frac{\frac{1}{3}}{u}} \cdot \left(\frac{16}{9} - \frac{\frac{1}{9}}{u \cdot u}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\left(\frac{u}{\frac{-4}{3} - \frac{\frac{1}{3}}{u}} \cdot \left(\frac{-4}{3} \cdot \frac{-4}{3} - \frac{\frac{1}{9}}{u \cdot u}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    4. metadata-eval N/A

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

    5. frac-times N/A

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

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

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

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

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

    8. sub-neg N/A

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

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

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

    10. distribute-neg-frac N/A

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

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

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

    12. metadata-eval N/A

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

    13. metadata-eval N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{/.f32}\left(\frac{-1}{3}, u\right)\right)\right), \left(\frac{16}{9} - \frac{\frac{1}{3}}{u} \cdot \frac{\frac{1}{3}}{u}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    14. frac-times N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{/.f32}\left(\frac{-1}{3}, u\right)\right)\right), \left(\frac{16}{9} - \frac{\frac{1}{3} \cdot \frac{1}{3}}{u \cdot u}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    15. metadata-eval N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{/.f32}\left(\frac{-1}{3}, u\right)\right)\right), \left(\frac{16}{9} - \frac{\frac{1}{9}}{u \cdot u}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    16. sub-neg N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{/.f32}\left(\frac{-1}{3}, u\right)\right)\right), \left(\frac{16}{9} + \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{u \cdot u}\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{/.f32}\left(\frac{-1}{3}, u\right)\right)\right), \mathsf{+.f32}\left(\frac{16}{9}, \left(\mathsf{neg}\left(\frac{\frac{1}{9}}{u \cdot u}\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    18. distribute-neg-frac N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{/.f32}\left(\frac{-1}{3}, u\right)\right)\right), \mathsf{+.f32}\left(\frac{16}{9}, \left(\frac{\mathsf{neg}\left(\frac{1}{9}\right)}{u \cdot u}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    19. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{/.f32}\left(\frac{-1}{3}, u\right)\right)\right), \mathsf{+.f32}\left(\frac{16}{9}, \mathsf{/.f32}\left(\left(\mathsf{neg}\left(\frac{1}{9}\right)\right), \left(u \cdot u\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    20. metadata-eval N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{/.f32}\left(\frac{-1}{3}, u\right)\right)\right), \mathsf{+.f32}\left(\frac{16}{9}, \mathsf{/.f32}\left(\frac{-1}{9}, \left(u \cdot u\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    21. *-lowering-*.f32 97.2%

        \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(\frac{-4}{3}, \mathsf{/.f32}\left(\frac{-1}{3}, u\right)\right)\right), \mathsf{+.f32}\left(\frac{16}{9}, \mathsf{/.f32}\left(\frac{-1}{9}, \mathsf{*.f32}\left(u, u\right)\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

11. Applied egg-rr 97.2%

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{u}{-1.3333333333333333 + \frac{-0.3333333333333333}{u}} \cdot \left(1.7777777777777777 + \frac{-0.1111111111111111}{u \cdot u}\right)}\right) \cdot \left(s \cdot -3\right) \]

12. Final simplification 97.2%

    \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\frac{u}{-1.3333333333333333 + \frac{-0.3333333333333333}{u}} \cdot \left(1.7777777777777777 + \frac{-0.1111111111111111}{u \cdot u}\right)\right) \]
13. Add Preprocessing

Alternative 7: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -3 \cdot \left(s \cdot \mathsf{log1p}\left(0.3333333333333333 + u \cdot -1.3333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* -3.0 (* s (log1p (+ 0.3333333333333333 (* u -1.3333333333333333))))))

C

float code(float s, float u) {
	return -3.0f * (s * log1pf((0.3333333333333333f + (u * -1.3333333333333333f))));
}

Julia

function code(s, u)
	return Float32(Float32(-3.0) * Float32(s * log1p(Float32(Float32(0.3333333333333333) + Float32(u * Float32(-1.3333333333333333))))))
end

Derivation

1. Initial program 95.7%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.7%

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

3. Simplified 96.7%

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

5. Taylor expanded in u around inf

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

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

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. /-lowering-/.f32 96.9%

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

7. Simplified 96.9%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{u \cdot \left(-1.3333333333333333 + \frac{0.3333333333333333}{u}\right)}\right) \cdot \left(s \cdot -3\right) \]
8. Step-by-step derivation

   1. associate-*r* N/A

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

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

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

   3. *-commutative N/A

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

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

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

   5. log1p-define N/A

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

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

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

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

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

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

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

   9. /-lowering-/.f32 96.9%

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

9. Applied egg-rr 96.9%

   \[\leadsto \color{blue}{\left(s \cdot \mathsf{log1p}\left(u \cdot \left(-1.3333333333333333 + \frac{0.3333333333333333}{u}\right)\right)\right) \cdot -3} \]
10. Step-by-step derivation

    1. *-commutative N/A

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

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

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

    3. log1p-define N/A

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

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

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

    5. distribute-rgt-in N/A

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

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

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

    7. *-commutative N/A

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

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

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

    9. div-inv N/A

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

    10. associate-*l* N/A

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

    11. inv-pow N/A

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

    12. pow-plus N/A

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

    13. metadata-eval N/A

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

    14. metadata-eval N/A

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

    15. metadata-eval 97.2%

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

11. Applied egg-rr 97.2%

    \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(u \cdot -1.3333333333333333 + 0.3333333333333333\right) \cdot s\right)} \cdot -3 \]

12. Final simplification 97.2%

    \[\leadsto -3 \cdot \left(s \cdot \mathsf{log1p}\left(0.3333333333333333 + u \cdot -1.3333333333333333\right)\right) \]
13. Add Preprocessing

Alternative 8: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-3 \cdot \mathsf{log1p}\left(0.3333333333333333 + u \cdot -1.3333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (* -3.0 (log1p (+ 0.3333333333333333 (* u -1.3333333333333333))))))

C

float code(float s, float u) {
	return s * (-3.0f * log1pf((0.3333333333333333f + (u * -1.3333333333333333f))));
}

Julia

function code(s, u)
	return Float32(s * Float32(Float32(-3.0) * log1p(Float32(Float32(0.3333333333333333) + Float32(u * Float32(-1.3333333333333333))))))
end

Derivation

1. Initial program 95.7%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.7%

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

3. Simplified 96.7%

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

5. Taylor expanded in u around inf

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

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

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. /-lowering-/.f32 96.9%

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

7. Simplified 96.9%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{u \cdot \left(-1.3333333333333333 + \frac{0.3333333333333333}{u}\right)}\right) \cdot \left(s \cdot -3\right) \]
8. Step-by-step derivation

   1. associate-*r* N/A

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

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

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

   3. *-commutative N/A

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

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

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

   5. log1p-define N/A

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

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

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

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

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

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

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

   9. /-lowering-/.f32 96.9%

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

9. Applied egg-rr 96.9%

   \[\leadsto \color{blue}{\left(s \cdot \mathsf{log1p}\left(u \cdot \left(-1.3333333333333333 + \frac{0.3333333333333333}{u}\right)\right)\right) \cdot -3} \]
10. Step-by-step derivation

    1. associate-*l* N/A

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

    2. *-commutative N/A

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

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

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

    4. *-commutative N/A

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

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

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

    6. log1p-define N/A

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

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

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

    8. distribute-rgt-in N/A

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

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

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

    10. *-commutative N/A

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

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

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

    12. div-inv N/A

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

    13. associate-*l* N/A

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

    14. inv-pow N/A

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

    15. pow-plus N/A

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

    16. metadata-eval N/A

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

    17. metadata-eval N/A

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

    18. metadata-eval 97.1%

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

11. Applied egg-rr 97.1%

    \[\leadsto \color{blue}{\left(-3 \cdot \mathsf{log1p}\left(u \cdot -1.3333333333333333 + 0.3333333333333333\right)\right) \cdot s} \]

12. Final simplification 97.1%

    \[\leadsto s \cdot \left(-3 \cdot \mathsf{log1p}\left(0.3333333333333333 + u \cdot -1.3333333333333333\right)\right) \]
13. Add Preprocessing

Alternative 9: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(0.3333333333333333 + u \cdot -1.3333333333333333\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* s -3.0) (log1p (+ 0.3333333333333333 (* u -1.3333333333333333)))))

C

float code(float s, float u) {
	return (s * -3.0f) * log1pf((0.3333333333333333f + (u * -1.3333333333333333f)));
}

Julia

function code(s, u)
	return Float32(Float32(s * Float32(-3.0)) * log1p(Float32(Float32(0.3333333333333333) + Float32(u * Float32(-1.3333333333333333)))))
end

Derivation

1. Initial program 95.7%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.7%

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

3. Simplified 96.7%

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

5. Taylor expanded in s around 0

   \[\leadsto \color{blue}{-3 \cdot \left(s \cdot \log \left(\frac{4}{3} + \frac{-4}{3} \cdot u\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

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

   2. metadata-eval N/A

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

   3. associate-+r+ N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. distribute-lft-in N/A

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

   8. metadata-eval N/A

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

   9. sub-neg N/A

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

   10. cancel-sign-sub-inv N/A

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

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

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

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

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

   13. sub-neg N/A

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

   14. log1p-define N/A

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

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

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

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

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

   17. metadata-eval N/A

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

   18. sub-neg N/A

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

   19. distribute-lft-in N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

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

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

   23. *-commutative N/A

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

   24. *-lowering-*.f32 97.1%

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

7. Simplified 97.1%

   \[\leadsto \color{blue}{\left(-3 \cdot s\right) \cdot \mathsf{log1p}\left(u \cdot -1.3333333333333333 + 0.3333333333333333\right)} \]

8. Final simplification 97.1%

   \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(0.3333333333333333 + u \cdot -1.3333333333333333\right) \]
9. Add Preprocessing

Alternative 10: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\frac{u}{-0.75} + 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* s -3.0) (log1p (+ (/ u -0.75) 0.3333333333333333))))

C

float code(float s, float u) {
	return (s * -3.0f) * log1pf(((u / -0.75f) + 0.3333333333333333f));
}

Julia

function code(s, u)
	return Float32(Float32(s * Float32(-3.0)) * log1p(Float32(Float32(u / Float32(-0.75)) + Float32(0.3333333333333333))))
end

Derivation

1. Initial program 95.7%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.7%

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

3. Simplified 96.7%

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

5. Final simplification 96.7%

   \[\leadsto \left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\frac{u}{-0.75} + 0.3333333333333333\right) \]
6. Add Preprocessing

Alternative 11: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * (-1.3333333333333333e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 95.7%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 96.7%

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

3. Simplified 96.7%

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

5. Taylor expanded in u around inf

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

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

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. /-lowering-/.f32 96.9%

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

7. Simplified 96.9%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{u \cdot \left(-1.3333333333333333 + \frac{0.3333333333333333}{u}\right)}\right) \cdot \left(s \cdot -3\right) \]
8. Step-by-step derivation

   1. *-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*l/ N/A

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

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

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

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

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

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\frac{16}{9} - \frac{\frac{1}{3}}{u} \cdot \frac{\frac{1}{3}}{u}\right), u\right), \left(\frac{-4}{3} - \frac{\frac{1}{3}}{u}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(\frac{16}{9}, \left(\frac{\frac{1}{3}}{u} \cdot \frac{\frac{1}{3}}{u}\right)\right), u\right), \left(\frac{-4}{3} - \frac{\frac{1}{3}}{u}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   8. frac-times N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(\frac{16}{9}, \left(\frac{\frac{1}{3} \cdot \frac{1}{3}}{u \cdot u}\right)\right), u\right), \left(\frac{-4}{3} - \frac{\frac{1}{3}}{u}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(\frac{16}{9}, \left(\frac{\frac{1}{9}}{u \cdot u}\right)\right), u\right), \left(\frac{-4}{3} - \frac{\frac{1}{3}}{u}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(\frac{16}{9}, \mathsf{/.f32}\left(\frac{1}{9}, \left(u \cdot u\right)\right)\right), u\right), \left(\frac{-4}{3} - \frac{\frac{1}{3}}{u}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(\frac{16}{9}, \mathsf{/.f32}\left(\frac{1}{9}, \mathsf{*.f32}\left(u, u\right)\right)\right), u\right), \left(\frac{-4}{3} - \frac{\frac{1}{3}}{u}\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(\frac{16}{9}, \mathsf{/.f32}\left(\frac{1}{9}, \mathsf{*.f32}\left(u, u\right)\right)\right), u\right), \mathsf{\_.f32}\left(\frac{-4}{3}, \left(\frac{\frac{1}{3}}{u}\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

   13. /-lowering-/.f32 97.0%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{\_.f32}\left(\frac{16}{9}, \mathsf{/.f32}\left(\frac{1}{9}, \mathsf{*.f32}\left(u, u\right)\right)\right), u\right), \mathsf{\_.f32}\left(\frac{-4}{3}, \mathsf{/.f32}\left(\frac{1}{3}, u\right)\right)\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

9. Applied egg-rr 97.0%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{\left(1.7777777777777777 - \frac{0.1111111111111111}{u \cdot u}\right) \cdot u}{-1.3333333333333333 - \frac{0.3333333333333333}{u}}}\right) \cdot \left(s \cdot -3\right) \]
10. Step-by-step derivation

    1. log1p-define N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\frac{\left(\frac{16}{9} - \frac{\frac{1}{9}}{u \cdot u}\right) \cdot u}{\frac{-4}{3} - \frac{\frac{1}{3}}{u}}\right)\right), \mathsf{*.f32}\left(\color{blue}{s}, -3\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\frac{u \cdot \left(\frac{16}{9} - \frac{\frac{1}{9}}{u \cdot u}\right)}{\frac{-4}{3} - \frac{\frac{1}{3}}{u}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    3. associate-/l* N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(u \cdot \frac{\frac{16}{9} - \frac{\frac{1}{9}}{u \cdot u}}{\frac{-4}{3} - \frac{\frac{1}{3}}{u}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    4. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{log1p}\left(u \cdot \frac{\frac{-4}{3} \cdot \frac{-4}{3} - \frac{\frac{1}{9}}{u \cdot u}}{\frac{-4}{3} - \frac{\frac{1}{3}}{u}}\right)\right), \mathsf{*.f32}\left(s, -3\right)\right) \]

    5. metadata-eval N/A

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

    6. frac-times N/A

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

    7. flip-+ N/A

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

    8. log1p-define N/A

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

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

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

    10. +-commutative N/A

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

    11. distribute-rgt-in N/A

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

    12. associate-+l+ N/A

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

    13. div-inv N/A

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

    14. associate-*l* N/A

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

11. Applied egg-rr 96.3%

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

12. Final simplification 96.3%

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

Alternative 12: 25.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot u\right) + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. distribute-lft-out N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

   9. log-lowering-log.f32 25.3%

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

5. Simplified 25.3%

   \[\leadsto \color{blue}{s \cdot \left(3 \cdot \left(u + \log 0.75\right)\right)} \]
6. Add Preprocessing

Alternative 13: 25.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.7%

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

   1. div-sub N/A

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

   2. metadata-eval N/A

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

   3. associate--r- N/A

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

   4. sub-neg N/A

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

   5. distribute-frac-neg N/A

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

   6. metadata-eval N/A

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

   7. frac-2neg N/A

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

   8. associate-+r+ N/A

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

   9. +-commutative N/A

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

   10. associate-+r+ N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. flip-+ N/A

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

   14. div-inv N/A

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

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

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

4. Applied egg-rr 95.1%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(1.7777777777777777 - \frac{u \cdot u}{0.5625}\right) \cdot \frac{1}{1.3333333333333333 + \frac{u}{0.75}}}}\right) \]

5. Taylor expanded in u around 0

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot u\right) + 3 \cdot \left(s \cdot \log \frac{3}{4}\right)} \]
6. Step-by-step derivation

   1. distribute-lft-out N/A

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

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

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

   3. distribute-lft-out N/A

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

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

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

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

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

   6. log-lowering-log.f32 25.3%

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

7. Simplified 25.3%

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

Alternative 14: 7.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (s u) :precision binary32 (* (log 0.75) (* s 3.0)))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in u around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(3, s\right), \color{blue}{\log \frac{3}{4}}\right) \]
4. Step-by-step derivation

   1. log-lowering-log.f32 7.6%

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

5. Simplified 7.6%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log 0.75} \]

6. Final simplification 7.6%

   \[\leadsto \log 0.75 \cdot \left(s \cdot 3\right) \]
7. Add Preprocessing

Alternative 15: 7.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (s u) :precision binary32 (* s (* 3.0 (log 0.75))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. log-lowering-log.f32 7.6%

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

5. Simplified 7.6%

   \[\leadsto \color{blue}{s \cdot \left(3 \cdot \log 0.75\right)} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(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))))))