Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.8% → 96.1%

Time: 8.4s

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: 95.8% 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: 96.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.5%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 34.7%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

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

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

   6. div-inv N/A

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

   7. frac-2neg N/A

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

   8. distribute-neg-frac N/A

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

   9. frac-2neg N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. lower-neg.f32 N/A

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

   14. lower-neg.f32 N/A

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

   15. lift--.f32 N/A

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

   16. sub-neg N/A

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

   17. distribute-neg-in N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. lower-+.f32 N/A

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

   21. lower-neg.f32 34.7

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

6. Applied rewrites 34.7%

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

   1. lift-/.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-neg-frac2 N/A

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

   5. lower-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. distribute-neg-in N/A

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

   9. lift-neg.f32 N/A

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

   10. remove-double-neg N/A

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

   11. sub-neg N/A

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

   12. lift--.f32 N/A

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

   13. metadata-eval 34.7

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

8. Applied rewrites 34.6%

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

   1. lift-log1p.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

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

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

   8. metadata-eval N/A

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

   9. div-inv N/A

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

   10. lift--.f32 N/A

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

   11. lower-log.f32 N/A

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

   12. lift--.f32 N/A

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

   13. div-inv N/A

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

   14. metadata-eval N/A

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

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

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

   16. metadata-eval N/A

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

   17. div-inv N/A

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

   18. metadata-eval N/A

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

   19. frac-2neg N/A

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

   20. lift-/.f32 N/A

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

   21. +-commutative N/A

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

10. Applied rewrites 96.4%

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

11. Final simplification 96.4%

    \[\leadsto s \cdot \left(\log \left(u \cdot -1.3333333333333333 - -1.3333333333333333\right) \cdot -3\right) \]
12. Add Preprocessing

Alternative 2: 28.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\log 0.6666666666666666 \cdot s\right) \cdot -3 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (* (log 0.6666666666666666) s) -3.0))

C

float code(float s, float u) {
	return (logf(0.6666666666666666f) * s) * -3.0f;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (log(0.6666666666666666e0) * s) * (-3.0e0)
end function

Julia

function code(s, u)
	return Float32(Float32(log(Float32(0.6666666666666666)) * s) * Float32(-3.0))
end

MATLAB

function tmp = code(s, u)
	tmp = (log(single(0.6666666666666666)) * s) * single(-3.0);
end

Derivation

1. Initial program 95.5%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
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 34.7%

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

6. Applied rewrites 28.5%

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

7. Taylor expanded in u around 0

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

   1. lower-log.f32 28.4

      \[\leadsto \left(\color{blue}{\log 0.6666666666666666} \cdot s\right) \cdot -3 \]

9. Applied rewrites 28.4%

   \[\leadsto \left(\color{blue}{\log 0.6666666666666666} \cdot s\right) \cdot -3 \]
10. Add Preprocessing

Alternative 3: 23.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ {u}^{3} \cdot s \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 95.5%

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

3. Taylor expanded in u around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-lft-out N/A

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

   6. *-commutative N/A

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

   7. distribute-lft-out N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-out N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. distribute-lft-out N/A

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

5. Applied rewrites 14.6%

   \[\leadsto \color{blue}{s \cdot \mathsf{fma}\left(\log 0.75 + u, 3, \left(1.5 + u\right) \cdot \left(u \cdot u\right)\right)} \]

6. Taylor expanded in u around inf

   \[\leadsto s \cdot {u}^{\color{blue}{3}} \]
7. Step-by-step derivation

8. Applied rewrites 23.2%

   \[\leadsto s \cdot {u}^{\color{blue}{3}} \]

9. Final simplification 23.2%

   \[\leadsto {u}^{3} \cdot s \]
10. Add Preprocessing

Alternative 4: 23.3% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = ((u * u) * u) * s;
end

Derivation

1. Initial program 95.5%

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

3. Taylor expanded in u around 0

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

   1. distribute-rgt-in N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-lft-out N/A

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

   6. *-commutative N/A

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

   7. distribute-lft-out N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-out N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. distribute-lft-out N/A

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

5. Applied rewrites 14.7%

   \[\leadsto \color{blue}{s \cdot \mathsf{fma}\left(\log 0.75 + u, 3, \left(1.5 + u\right) \cdot \left(u \cdot u\right)\right)} \]

6. Taylor expanded in u around inf

   \[\leadsto s \cdot {u}^{\color{blue}{3}} \]
7. Step-by-step derivation

8. Applied rewrites 23.2%

   \[\leadsto s \cdot {u}^{\color{blue}{3}} \]
9. Step-by-step derivation

10. Applied rewrites 23.2%

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

11. Final simplification 23.2%

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

Reproduce

herbie shell --seed 2024276 
(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))))))