Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.8% → 98.4%

Time: 10.6s

Alternatives: 7

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: 98.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 96.1%

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

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-neg N/A

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

   4. distribute-lft-in N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-neg.f32 98.3

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lift--.f32 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-in N/A

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

   14. +-commutative N/A

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

   15. lift-+.f32 N/A

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

   16. lower-*.f32 N/A

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in s around 0

   \[\leadsto \color{blue}{s \cdot \left(-3 \cdot \log \left(1 + \left(\frac{1}{4} - u\right) \cdot \left(\frac{16}{9} \cdot u - \frac{4}{9}\right)\right) + 3 \cdot \log \left(\frac{2}{3} + \frac{4}{3} \cdot u\right)\right)} \]
8. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-log1p.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. sub-neg N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. metadata-eval 96.0

       \[\leadsto s \cdot \mathsf{fma}\left(3, \log \left(\mathsf{fma}\left(u, 1.3333333333333333, 0.6666666666666666\right)\right), -3 \cdot \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \mathsf{fma}\left(u, 1.7777777777777777, \color{blue}{-0.4444444444444444}\right)\right)\right) \]

9. Applied rewrites 96.0%

   \[\leadsto \color{blue}{s \cdot \mathsf{fma}\left(3, \log \left(\mathsf{fma}\left(u, 1.3333333333333333, 0.6666666666666666\right)\right), -3 \cdot \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \mathsf{fma}\left(u, 1.7777777777777777, -0.4444444444444444\right)\right)\right)} \]
10. Step-by-step derivation

11. Applied rewrites 98.5%

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

12. Final simplification 98.5%

    \[\leadsto s \cdot \mathsf{fma}\left(3, \mathsf{log1p}\left(\mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right), -3 \cdot \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \mathsf{fma}\left(u, 1.7777777777777777, -0.4444444444444444\right)\right)\right) \]
13. 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 95.8%

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

   1. 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 97.9%

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

   1. lift-neg.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. div-sub N/A

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

   9. clear-num N/A

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

   10. clear-num N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lift-+.f32 N/A

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

   14. distribute-neg-frac N/A

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

   15. lower-/.f32 N/A

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

   16. lift-+.f32 N/A

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

   17. +-commutative N/A

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

   18. distribute-neg-in N/A

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

   19. metadata-eval N/A

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

   20. sub-neg N/A

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

   21. 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

Reproduce

herbie shell --seed 2024230 
(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))))))