Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 96.0% → 96.7%

Time: 7.9s

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: 96.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = (single(-3.0) * log((single(1.0) - ((u - single(0.25)) / single(0.75))))) * 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 32.5%

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

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

   4. lift-fma.f32 N/A

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

   5. lower-log.f32 10.3

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

   6. lift-fma.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

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

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

   14. div-inv N/A

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

   15. lift--.f32 N/A

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

   16. sub-neg N/A

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

   17. lower--.f32 N/A

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

   18. div-sub N/A

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

   19. sub-neg N/A

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

   20. div-inv N/A

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

   21. lower-fma.f32 N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

   24. metadata-eval 8.8

       \[\leadsto \left(-3 \cdot \log \left(1 - \mathsf{fma}\left(u, 1.3333333333333333, \color{blue}{-0.3333333333333333}\right)\right)\right) \cdot s \]

6. Applied rewrites 8.8%

   \[\leadsto \left(-3 \cdot \color{blue}{\log \left(1 - \mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right)}\right) \cdot s \]
7. Step-by-step derivation

   1. lift-fma.f32 N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. div-sub N/A

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

   8. lift--.f32 N/A

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

   9. lower-/.f32 96.8

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

8. Applied rewrites 96.8%

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

Alternative 2: 96.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = (single(-3.0) * log((single(1.0) - ((u - single(0.25)) * single(1.3333333333333333))))) * 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 34.1%

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

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

   4. lift-fma.f32 N/A

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

   5. lower-log.f32 10.3

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

   6. lift-fma.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

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

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

   14. div-inv N/A

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

   15. lift--.f32 N/A

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

   16. sub-neg N/A

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

   17. lower--.f32 N/A

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

   18. div-sub N/A

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

   19. sub-neg N/A

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

   20. div-inv N/A

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

   21. lower-fma.f32 N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

   24. metadata-eval 8.8

       \[\leadsto \left(-3 \cdot \log \left(1 - \mathsf{fma}\left(u, 1.3333333333333333, \color{blue}{-0.3333333333333333}\right)\right)\right) \cdot s \]

6. Applied rewrites 8.8%

   \[\leadsto \left(-3 \cdot \color{blue}{\log \left(1 - \mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right)}\right) \cdot s \]
7. Step-by-step derivation

   1. lift-fma.f32 N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. div-sub N/A

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

   8. lift--.f32 N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f32 96.4

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

8. Applied rewrites 96.4%

   \[\leadsto \left(-3 \cdot \log \left(1 - \color{blue}{\left(u - 0.25\right) \cdot 1.3333333333333333}\right)\right) \cdot s \]
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 33.3%

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

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

   4. lift-fma.f32 N/A

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

   5. lower-log.f32 10.3

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

   6. lift-fma.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

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

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

   14. div-inv N/A

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

   15. lift--.f32 N/A

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

   16. sub-neg N/A

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

   17. lower--.f32 N/A

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

   18. div-sub N/A

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

   19. sub-neg N/A

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

   20. div-inv N/A

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

   21. lower-fma.f32 N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

   24. metadata-eval 8.8

       \[\leadsto \left(-3 \cdot \log \left(1 - \mathsf{fma}\left(u, 1.3333333333333333, \color{blue}{-0.3333333333333333}\right)\right)\right) \cdot s \]

6. Applied rewrites 8.7%

   \[\leadsto \left(-3 \cdot \color{blue}{\log \left(1 - \mathsf{fma}\left(u, 1.3333333333333333, -0.3333333333333333\right)\right)}\right) \cdot s \]
7. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. +-commutative N/A

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

   4. associate--r+ N/A

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

   5. metadata-eval N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 96.3

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

8. Applied rewrites 96.3%

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

Alternative 4: 30.3% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(3 \cdot u + \left(\left(1.5 + u\right) \cdot u\right) \cdot u\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (+ (* 3.0 u) (* (* (+ 1.5 u) u) u))))

C

float code(float s, float u) {
	return s * ((3.0f * u) + (((1.5f + u) * u) * u));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * ((3.0e0 * u) + (((1.5e0 + u) * u) * u))
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(Float32(3.0) * u) + Float32(Float32(Float32(Float32(1.5) + u) * u) * u)))
end

MATLAB

function tmp = code(s, u)
	tmp = s * ((single(3.0) * u) + (((single(1.5) + u) * u) * u));
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. 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 14.8%

   \[\leadsto \color{blue}{s \cdot \mathsf{fma}\left(\log 0.75 + u, 3, \left(1.5 + u\right) \cdot \left(u \cdot u\right)\right)} \]
6. Step-by-step derivation

7. Applied rewrites 36.5%

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

8. Taylor expanded in u around inf

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

10. Applied rewrites 30.4%

    \[\leadsto s \cdot \left(3 \cdot u + \color{blue}{\left(\left(1.5 + u\right) \cdot u\right)} \cdot u\right) \]
11. Add Preprocessing

Alternative 5: 23.3% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = s * ((u * u) * u);
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. 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 14.8%

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

6. Taylor expanded in u around inf

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

8. Applied rewrites 23.3%

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

10. Applied rewrites 23.3%

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

Reproduce

herbie shell --seed 2024325 
(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))))))