Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 96.7%

Time: 7.2s

Alternatives: 4

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: 95.9% 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: 96.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   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 33.0%

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

   1. lift-log1p.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-+.f32 N/A

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

   4. lower-log.f32 96.2

      \[\leadsto \left(-3 \cdot \color{blue}{\log \left(-1.3333333333333333 \cdot \left(u - 0.25\right) + 1\right)}\right) \cdot s \]

   5. lift-+.f32 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

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

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

   11. metadata-eval N/A

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

   12. div-inv N/A

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

   13. lift--.f32 N/A

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

   14. sub-neg N/A

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

   15. div-sub N/A

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

   16. div-inv N/A

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

   17. metadata-eval N/A

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

   18. *-commutative N/A

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

   19. sub-neg N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

   22. +-commutative N/A

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

   23. associate--r+ N/A

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

   24. metadata-eval N/A

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

   25. lower--.f32 N/A

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

   26. lower-*.f32 95.9

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

6. Applied rewrites 95.9%

   \[\leadsto \left(-3 \cdot \color{blue}{\log \left(1.3333333333333333 - 1.3333333333333333 \cdot u\right)}\right) \cdot s \]
7. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. associate-+r+ N/A

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

   8. +-commutative N/A

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

   9. associate-+r+ N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

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

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

   13. lift-*.f32 N/A

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

   14. associate-+l- N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. metadata-eval N/A

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

   18. div-inv N/A

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

   19. metadata-eval N/A

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

   20. div-sub N/A

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

   21. sub-neg N/A

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

   22. +-commutative N/A

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

   23. lower-+.f32 N/A

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

8. Applied rewrites 96.2%

   \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\left(u - 0.25\right) \cdot -1.3333333333333333 + 1\right)}\right) \cdot s \]
9. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. metadata-eval N/A

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

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

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lower-/.f32 N/A

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

   9. lift--.f32 N/A

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

   10. metadata-eval 96.5

       \[\leadsto \left(-3 \cdot \log \left(\frac{u - 0.25}{\color{blue}{-0.75}} + 1\right)\right) \cdot s \]

10. Applied rewrites 96.5%

    \[\leadsto \left(-3 \cdot \log \left(\color{blue}{\frac{u - 0.25}{-0.75}} + 1\right)\right) \cdot s \]
11. Add Preprocessing

Alternative 2: 96.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(-3 \cdot \log \left(\left(u - 0.25\right) \cdot -1.3333333333333333 + 1\right)\right) \cdot s \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* -3.0 (log (+ (* (- u 0.25) -1.3333333333333333) 1.0))) s))

C

float code(float s, float u) {
	return (-3.0f * logf((((u - 0.25f) * -1.3333333333333333f) + 1.0f))) * s;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((-3.0e0) * log((((u - 0.25e0) * (-1.3333333333333333e0)) + 1.0e0))) * s
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(-3.0) * log(Float32(Float32(Float32(u - Float32(0.25)) * Float32(-1.3333333333333333)) + Float32(1.0)))) * s)
end

MATLAB

function tmp = code(s, u)
	tmp = (single(-3.0) * log((((u - single(0.25)) * single(-1.3333333333333333)) + single(1.0)))) * s;
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

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

   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 32.9%

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

   1. lift-log1p.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-+.f32 N/A

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

   4. lower-log.f32 96.2

      \[\leadsto \left(-3 \cdot \color{blue}{\log \left(-1.3333333333333333 \cdot \left(u - 0.25\right) + 1\right)}\right) \cdot s \]

   5. lift-+.f32 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

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

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

   11. metadata-eval N/A

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

   12. div-inv N/A

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

   13. lift--.f32 N/A

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

   14. sub-neg N/A

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

   15. div-sub N/A

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

   16. div-inv N/A

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

   17. metadata-eval N/A

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

   18. *-commutative N/A

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

   19. sub-neg N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

   22. +-commutative N/A

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

   23. associate--r+ N/A

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

   24. metadata-eval N/A

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

   25. lower--.f32 N/A

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

   26. lower-*.f32 95.9

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

6. Applied rewrites 95.9%

   \[\leadsto \left(-3 \cdot \color{blue}{\log \left(1.3333333333333333 - 1.3333333333333333 \cdot u\right)}\right) \cdot s \]
7. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. associate-+r+ N/A

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

   8. +-commutative N/A

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

   9. associate-+r+ N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

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

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

   13. lift-*.f32 N/A

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

   14. associate-+l- N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. metadata-eval N/A

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

   18. div-inv N/A

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

   19. metadata-eval N/A

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

   20. div-sub N/A

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

   21. sub-neg N/A

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

   22. +-commutative N/A

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

   23. lower-+.f32 N/A

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

8. Applied rewrites 96.2%

   \[\leadsto \left(-3 \cdot \log \color{blue}{\left(\left(u - 0.25\right) \cdot -1.3333333333333333 + 1\right)}\right) \cdot s \]
9. Add Preprocessing

Alternative 3: 96.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   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 34.3%

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

   1. lift-log1p.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-+.f32 N/A

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

   4. lower-log.f32 96.2

      \[\leadsto \left(-3 \cdot \color{blue}{\log \left(-1.3333333333333333 \cdot \left(u - 0.25\right) + 1\right)}\right) \cdot s \]

   5. lift-+.f32 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

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

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

   11. metadata-eval N/A

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

   12. div-inv N/A

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

   13. lift--.f32 N/A

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

   14. sub-neg N/A

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

   15. div-sub N/A

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

   16. div-inv N/A

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

   17. metadata-eval N/A

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

   18. *-commutative N/A

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

   19. sub-neg N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

   22. +-commutative N/A

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

   23. associate--r+ N/A

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

   24. metadata-eval N/A

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

   25. lower--.f32 N/A

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

   26. lower-*.f32 95.9

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

6. Applied rewrites 95.9%

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

Alternative 4: 29.9% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   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 33.5%

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

5. Taylor expanded in u around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. lower-*.f32 8.1

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

7. Applied rewrites 8.2%

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

8. Taylor expanded in u around inf

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

10. Applied rewrites 29.8%

    \[\leadsto \left(3 \cdot \color{blue}{u}\right) \cdot s \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024315 
(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))))))