Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.8% → 98.3%

Time: 10.2s

Alternatives: 5

Speedup: 1.2×

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, 0.6× speedup

Math

\[\begin{array}{l} \\ s \cdot \mathsf{fma}\left(-3, \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.7777777777777777, -0.4444444444444444\right) \cdot \left(0.25 - u\right)\right), 3 \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  s
  (fma
   -3.0
   (log1p (* (fma u 1.7777777777777777 -0.4444444444444444) (- 0.25 u)))
   (* 3.0 (log1p (fma u 1.3333333333333333 -0.3333333333333333))))))

C

float code(float s, float u) {
	return s * fmaf(-3.0f, log1pf((fmaf(u, 1.7777777777777777f, -0.4444444444444444f) * (0.25f - u))), (3.0f * log1pf(fmaf(u, 1.3333333333333333f, -0.3333333333333333f))));
}

Julia

function code(s, u)
	return Float32(s * fma(Float32(-3.0), log1p(Float32(fma(u, Float32(1.7777777777777777), Float32(-0.4444444444444444)) * Float32(Float32(0.25) - u))), Float32(Float32(3.0) * log1p(fma(u, Float32(1.3333333333333333), Float32(-0.3333333333333333))))))
end

Derivation

1. Initial program 95.9%

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

   1. lift-log.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift--.f32 N/A

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

   4. flip-- N/A

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

   5. clear-num N/A

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

   6. log-div N/A

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

   7. lower--.f32 N/A

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

   8. lower-log1p.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\mathsf{log1p}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)} - \log \left(1 \cdot 1 - \frac{u - \frac{1}{4}}{\frac{3}{4}} \cdot \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) \]

   9. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}\right) - \log \left(1 \cdot 1 - \frac{u - \frac{1}{4}}{\frac{3}{4}} \cdot \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) \]

   10. lift--.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{u - \frac{1}{4}}}{\frac{3}{4}}\right) - \log \left(1 \cdot 1 - \frac{u - \frac{1}{4}}{\frac{3}{4}} \cdot \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) \]

   11. div-sub N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{\frac{u}{\frac{3}{4}} - \frac{\frac{1}{4}}{\frac{3}{4}}}\right) - \log \left(1 \cdot 1 - \frac{u - \frac{1}{4}}{\frac{3}{4}} \cdot \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) \]

   12. sub-neg N/A

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

   13. div-inv N/A

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

   14. lower-fma.f32 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(u, \frac{4}{3}, \color{blue}{\frac{-1}{3}}\right)\right) - \log \left(1 \cdot 1 - \frac{u - \frac{1}{4}}{\frac{3}{4}} \cdot \frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) \]

4. Applied rewrites 98.2%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right) - \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \left(1.7777777777777777 \cdot \left(u + -0.25\right)\right)\right)\right)} \]

6. Applied rewrites 95.4%

   \[\leadsto \color{blue}{\left(s \cdot \log \left(\frac{\mathsf{fma}\left(u, 1.3333333333333333, 0.6666666666666666\right)}{\mathsf{fma}\left(\mathsf{fma}\left(1.7777777777777777, u, -0.4444444444444444\right), 0.25 - u, 1\right)}\right)\right) \cdot 3} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(s \cdot \log \left(\frac{\mathsf{fma}\left(u, \frac{4}{3}, \frac{2}{3}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{16}{9}, u, \frac{-4}{9}\right), \frac{1}{4} - u, 1\right)}\right)\right) \cdot 3} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{\mathsf{fma}\left(u, \frac{4}{3}, \frac{2}{3}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{16}{9}, u, \frac{-4}{9}\right), \frac{1}{4} - u, 1\right)}\right)\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto 3 \cdot \color{blue}{\left(s \cdot \log \left(\frac{\mathsf{fma}\left(u, \frac{4}{3}, \frac{2}{3}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{16}{9}, u, \frac{-4}{9}\right), \frac{1}{4} - u, 1\right)}\right)\right)} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{\mathsf{fma}\left(u, \frac{4}{3}, \frac{2}{3}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{16}{9}, u, \frac{-4}{9}\right), \frac{1}{4} - u, 1\right)}\right)} \]

   5. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \left(\frac{\mathsf{fma}\left(u, \frac{4}{3}, \frac{2}{3}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{16}{9}, u, \frac{-4}{9}\right), \frac{1}{4} - u, 1\right)}\right) \]

   6. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\mathsf{fma}\left(u, \frac{4}{3}, \frac{2}{3}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{16}{9}, u, \frac{-4}{9}\right), \frac{1}{4} - u, 1\right)}\right)} \]

   7. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\mathsf{fma}\left(u, \frac{4}{3}, \frac{2}{3}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{16}{9}, u, \frac{-4}{9}\right), \frac{1}{4} - u, 1\right)}\right)} \]

   8. log-div N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{2}{3}\right)\right) - \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{16}{9}, u, \frac{-4}{9}\right), \frac{1}{4} - u, 1\right)\right)\right)} \]

   9. lift-fma.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{2}{3}\right)\right) - \log \color{blue}{\left(\mathsf{fma}\left(\frac{16}{9}, u, \frac{-4}{9}\right) \cdot \left(\frac{1}{4} - u\right) + 1\right)}\right) \]

   10. lift-*.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{2}{3}\right)\right) - \log \left(\color{blue}{\mathsf{fma}\left(\frac{16}{9}, u, \frac{-4}{9}\right) \cdot \left(\frac{1}{4} - u\right)} + 1\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{2}{3}\right)\right) - \log \color{blue}{\left(1 + \mathsf{fma}\left(\frac{16}{9}, u, \frac{-4}{9}\right) \cdot \left(\frac{1}{4} - u\right)\right)}\right) \]

   12. lift-log1p.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{2}{3}\right)\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{16}{9}, u, \frac{-4}{9}\right) \cdot \left(\frac{1}{4} - u\right)\right)}\right) \]

   13. sub-neg N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \left(\mathsf{fma}\left(u, \frac{4}{3}, \frac{2}{3}\right)\right) + \left(\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{16}{9}, u, \frac{-4}{9}\right) \cdot \left(\frac{1}{4} - u\right)\right)\right)\right)\right)} \]

8. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(s \cdot 3, \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right), \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\left(0.25 - u\right) \cdot \mathsf{fma}\left(u, 1.7777777777777777, -0.4444444444444444\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. lift-fma.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-out N/A

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

   9. lower-*.f32 N/A

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

   10. lift-neg.f32 N/A

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

   11. neg-mul-1 N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f32 N/A

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

10. Applied rewrites 98.3%

    \[\leadsto \color{blue}{s \cdot \mathsf{fma}\left(-3, \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.7777777777777777, -0.4444444444444444\right) \cdot \left(0.25 - u\right)\right), 3 \cdot \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right)\right)} \]
11. Add Preprocessing

Alternative 2: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

4. Applied rewrites 97.7%

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

   1. lift-fma.f32 N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. div-sub N/A

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

   8. lower-neg.f32 N/A

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

   9. distribute-neg-frac N/A

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

   10. lower-/.f32 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. +-commutative N/A

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

   14. distribute-neg-in N/A

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

   15. metadata-eval N/A

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

   16. sub-neg N/A

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

   17. lift--.f32 98.3

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

6. Applied rewrites 98.3%

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

7. Final simplification 98.3%

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

Alternative 3: 97.9% accurate, 1.2× 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 95.9%

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

3. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. log-rec N/A

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

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

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

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

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

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

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

   6. metadata-eval N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. sub-neg N/A

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

   11. lower-log1p.f32 N/A

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

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

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

   13. metadata-eval N/A

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

   14. sub-neg N/A

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

   15. distribute-rgt-in N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lower-fma.f32 97.8

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

5. Applied rewrites 97.8%

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

Alternative 4: 25.7% accurate, 1.2× 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 95.9%

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

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

   2. lower-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. lower-*.f32 N/A

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

   5. lower-+.f32 N/A

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

   6. lower-log.f32 25.4

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

5. Applied rewrites 25.4%

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

Alternative 5: 7.4% accurate, 1.3× 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 95.9%

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

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

   2. lower-*.f32 N/A

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

   3. lower-log.f32 7.1

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

5. Applied rewrites 7.1%

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

Reproduce

herbie shell --seed 2024234 
(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))))))