Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.8% → 98.3%

Time: 11.7s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 95.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{log1p}\left(\frac{\left(u + -0.25\right) \cdot \left(u + -0.25\right)}{-0.5625}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right)\right) \cdot \left(s \cdot -3\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  (-
   (log1p (/ (* (+ u -0.25) (+ u -0.25)) (- 0.5625)))
   (log1p (fma u 1.3333333333333333 -0.3333333333333333)))
  (* s -3.0)))

C

float code(float s, float u) {
	return (log1pf((((u + -0.25f) * (u + -0.25f)) / -0.5625f)) - log1pf(fmaf(u, 1.3333333333333333f, -0.3333333333333333f))) * (s * -3.0f);
}

Julia

function code(s, u)
	return Float32(Float32(log1p(Float32(Float32(Float32(u + Float32(-0.25)) * Float32(u + Float32(-0.25))) / Float32(-Float32(0.5625)))) - log1p(fma(u, Float32(1.3333333333333333), Float32(-0.3333333333333333)))) * Float32(s * Float32(-3.0)))
end

Derivation

1. Initial program 95.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. neg-sub0 N/A

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

   10. div-sub N/A

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

   11. associate--r- N/A

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

   12. neg-sub0 N/A

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

   13. +-commutative N/A

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

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

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac2 N/A

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

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

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

   18. metadata-eval N/A

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

   19. *-commutative N/A

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

   20. associate-*l* N/A

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

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

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

   22. metadata-eval 96.6%

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

3. Simplified 96.6%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
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 97.4%

   \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(-\frac{\left(u + -0.25\right) \cdot \left(u + -0.25\right)}{0.5625}\right) - \mathsf{log1p}\left(\frac{u}{0.75} + -0.3333333333333333\right)\right)} \cdot \left(s \cdot -3\right) \]
7. Step-by-step derivation

   1. div-inv N/A

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

   2. fma-define N/A

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

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

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

   4. metadata-eval 98.5%

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

8. Applied egg-rr 98.5%

   \[\leadsto \left(\mathsf{log1p}\left(-\frac{\left(u + -0.25\right) \cdot \left(u + -0.25\right)}{0.5625}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)}\right)\right) \cdot \left(s \cdot -3\right) \]

9. Final simplification 98.5%

   \[\leadsto \left(\mathsf{log1p}\left(\frac{\left(u + -0.25\right) \cdot \left(u + -0.25\right)}{-0.5625}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right)\right) \cdot \left(s \cdot -3\right) \]
10. Add Preprocessing

Alternative 2: 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 95.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. neg-sub0 N/A

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

   10. div-sub N/A

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

   11. associate--r- N/A

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

   12. neg-sub0 N/A

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

   13. +-commutative N/A

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

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

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac2 N/A

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

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

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

   18. metadata-eval N/A

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

   19. *-commutative N/A

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

   20. associate-*l* N/A

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

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

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

   22. metadata-eval 96.6%

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

3. Simplified 96.6%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
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 98.1%

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

6. Applied egg-rr 98.1%

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

7. Final simplification 98.1%

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

Alternative 3: 96.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. neg-sub0 N/A

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

   10. div-sub N/A

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

   11. associate--r- N/A

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

   12. neg-sub0 N/A

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

   13. +-commutative N/A

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

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

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac2 N/A

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

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

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

   18. metadata-eval N/A

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

   19. *-commutative N/A

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

   20. associate-*l* N/A

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

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

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

   22. metadata-eval 96.6%

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

3. Simplified 96.6%

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

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

   2. flip-+ N/A

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

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

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

   4. metadata-eval N/A

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

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

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

   6. frac-times N/A

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

   7. metadata-eval N/A

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

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

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

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

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

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

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

   11. /-lowering-/.f32 97.0%

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

9. Applied egg-rr 97.0%

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

10. Final simplification 97.0%

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

Alternative 4: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. neg-sub0 N/A

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

   10. div-sub N/A

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

   11. associate--r- N/A

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

   12. neg-sub0 N/A

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

   13. +-commutative N/A

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

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

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac2 N/A

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

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

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

   18. metadata-eval N/A

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

   19. *-commutative N/A

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

   20. associate-*l* N/A

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

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

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

   22. metadata-eval 96.6%

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

3. Simplified 96.6%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

6. Applied egg-rr 96.5%

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

7. Taylor expanded in u around inf

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

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

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. /-lowering-/.f32 97.0%

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

9. Simplified 97.0%

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

10. Final simplification 97.0%

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

Alternative 5: 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 95.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. neg-sub0 N/A

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

   10. div-sub N/A

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

   11. associate--r- N/A

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

   12. neg-sub0 N/A

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

   13. +-commutative N/A

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

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

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac2 N/A

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

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

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

   18. metadata-eval N/A

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

   19. *-commutative N/A

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

   20. associate-*l* N/A

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

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

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

   22. metadata-eval 96.6%

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

3. Simplified 96.6%

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

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

7. Simplified 96.9%

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

8. Final simplification 96.9%

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

Alternative 6: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. neg-sub0 N/A

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

   10. div-sub N/A

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

   11. associate--r- N/A

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

   12. neg-sub0 N/A

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

   13. +-commutative N/A

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

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

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac2 N/A

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

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

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

   18. metadata-eval N/A

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

   19. *-commutative N/A

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

   20. associate-*l* N/A

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

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

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

   22. metadata-eval 96.6%

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

3. Simplified 96.6%

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

5. Final simplification 96.6%

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

Alternative 7: 94.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. neg-sub0 N/A

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

   10. div-sub N/A

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

   11. associate--r- N/A

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

   12. neg-sub0 N/A

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

   13. +-commutative N/A

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

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

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac2 N/A

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

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

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

   18. metadata-eval N/A

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

   19. *-commutative N/A

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

   20. associate-*l* N/A

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

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

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

   22. metadata-eval 96.6%

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

3. Simplified 96.6%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(0.3333333333333333 + \frac{u}{-0.75}\right) \cdot \left(s \cdot -3\right)} \]
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. log-lowering-log.f32 N/A

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

   11. div-sub N/A

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

   12. metadata-eval N/A

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

   13. associate--r- N/A

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

   14. sub-neg N/A

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

   15. distribute-frac-neg N/A

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

   16. metadata-eval N/A

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

   17. frac-2neg N/A

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

   18. associate-+r+ N/A

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

   19. +-commutative N/A

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

   20. associate-+r+ N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. +-commutative N/A

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

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

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

6. Applied egg-rr 94.7%

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

7. Final simplification 94.7%

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

Alternative 8: 25.7% accurate, 1.1× speedup

Math

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

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(Float32(s * 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 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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. *-lowering-*.f32 N/A

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

   5. *-commutative N/A

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

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

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

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

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

   8. log-lowering-log.f32 25.5%

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

5. Simplified 25.5%

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

Alternative 9: 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 95.8%

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

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

   14. flip-+ N/A

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

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

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

4. Applied egg-rr 95.0%

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

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. 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 25.5%

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

7. Simplified 25.5%

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

Alternative 10: 7.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. 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 7.7%

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

5. Simplified 7.7%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log 0.75} \]
6. Step-by-step derivation

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. log-lowering-log.f32 7.7%

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

7. Applied egg-rr 7.7%

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

8. Final simplification 7.7%

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

Alternative 11: 7.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.8%

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

   1. log-rec N/A

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

   2. neg-mul-1 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. sub-neg N/A

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

   7. log1p-define N/A

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

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

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

   9. neg-sub0 N/A

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

   10. div-sub N/A

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

   11. associate--r- N/A

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

   12. neg-sub0 N/A

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

   13. +-commutative N/A

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

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

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac2 N/A

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

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

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

   18. metadata-eval N/A

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

   19. *-commutative N/A

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

   20. associate-*l* N/A

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

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

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

   22. metadata-eval 96.6%

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

3. Simplified 96.6%

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

5. Taylor expanded in u around 0

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

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

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

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

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

   3. log-lowering-log.f32 7.7%

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

7. Simplified 7.7%

   \[\leadsto \color{blue}{-3 \cdot \left(s \cdot \log 1.3333333333333333\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(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))))))