Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 96.0% → 98.3%

Time: 10.4s

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: 96.0% 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

\[\begin{array}{l} \\ \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\frac{0.25 - u}{-0.75}\right) - \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \left(1.7777777777777777 \cdot \left(u + -0.25\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  (* 3.0 s)
  (-
   (log1p (/ (- 0.25 u) -0.75))
   (log1p (* (- 0.25 u) (* 1.7777777777777777 (+ u -0.25)))))))

C

float code(float s, float u) {
	return (3.0f * s) * (log1pf(((0.25f - u) / -0.75f)) - log1pf(((0.25f - u) * (1.7777777777777777f * (u + -0.25f)))));
}

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * Float32(log1p(Float32(Float32(Float32(0.25) - u) / Float32(-0.75))) - log1p(Float32(Float32(Float32(0.25) - u) * Float32(Float32(1.7777777777777777) * Float32(u + Float32(-0.25)))))))
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 98.3%

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

   1. lift-fma.f32 N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. div-sub N/A

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

   8. frac-2neg N/A

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

   9. lower-/.f32 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

   13. distribute-neg-in N/A

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

   14. metadata-eval N/A

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

   15. sub-neg N/A

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

   16. lift--.f32 N/A

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

   17. metadata-eval 98.6

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\frac{0.25 - u}{\color{blue}{-0.75}}\right) - \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \left(1.7777777777777777 \cdot \left(u + -0.25\right)\right)\right)\right) \]

6. Applied rewrites 98.6%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(\mathsf{log1p}\left(\color{blue}{\frac{0.25 - u}{-0.75}}\right) - \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \left(1.7777777777777777 \cdot \left(u + -0.25\right)\right)\right)\right) \]
7. Add Preprocessing

Alternative 2: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

4. Applied rewrites 98.1%

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

   1. lift-neg.f32 N/A

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

   2. neg-sub0 N/A

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

   3. lift-fma.f32 N/A

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

   4. +-commutative N/A

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

   5. associate--r+ N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. div-inv N/A

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

   10. div-sub N/A

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

   11. lift--.f32 N/A

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

   12. lower-/.f32 98.5

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

6. Applied rewrites 98.5%

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

7. Final simplification 98.5%

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

Alternative 3: 97.9% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (s u)
 :precision binary32
 (* s (* -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(s * 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-*.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

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

4. Applied rewrites 98.1%

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

   1. lift-neg.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. distribute-neg-in N/A

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

   4. metadata-eval N/A

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

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

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

   6. metadata-eval N/A

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

   7. lower-fma.f32 98.1

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

6. Applied rewrites 98.1%

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

7. Final simplification 98.1%

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

Alternative 4: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, u)
	return Float32(Float32(s * Float32(-3.0)) * log1p(fma(Float32(-1.3333333333333333), u, 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-lft-in N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lower-fma.f32 98.1

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

5. Applied rewrites 98.1%

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

Alternative 5: 30.0% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

4. Applied rewrites 98.1%

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

5. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-log.f32 25.8

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

7. Applied rewrites 25.8%

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

8. Taylor expanded in u around inf

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

10. Applied rewrites 29.9%

    \[\leadsto \left(-3 \cdot \left(-u\right)\right) \cdot s \]

11. Final simplification 29.9%

    \[\leadsto s \cdot \left(-3 \cdot \left(-u\right)\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(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))))))