Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 98.2%

Time: 8.9s

Alternatives: 6

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: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, u)
	return Float32(Float32(3.0) * Float32(log1p(Float32(Float32(u - Float32(0.25)) / Float32(-0.75))) * Float32(-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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

4. Applied rewrites 97.8%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval 98.3

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

6. Applied rewrites 98.3%

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

7. Final simplification 98.3%

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

Alternative 2: 97.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, u)
	return Float32(Float32(log1p(fma(u, Float32(-1.3333333333333333), Float32(0.3333333333333333))) * Float32(-s)) * Float32(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

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

   5. lower-*.f32 N/A

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

4. Applied rewrites 97.8%

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

6. Applied rewrites 97.8%

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

7. Final simplification 97.8%

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

Alternative 3: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, u)
	return Float32(Float32(Float32(-3.0) * log1p(fma(Float32(-1.3333333333333333), u, Float32(0.3333333333333333)))) * 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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

4. Applied rewrites 97.8%

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

6. Applied rewrites 97.8%

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

   1. lift-fma.f32 N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. div-sub N/A

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

   8. lift--.f32 N/A

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

   9. lift-/.f32 98.3

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

8. Applied rewrites 98.3%

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

   1. lift-*.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. distribute-lft-neg-out N/A

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

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

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift--.f32 N/A

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

   10. div-sub N/A

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

   11. div-inv N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. sub-neg N/A

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

   15. lower-fma.f32 N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval 97.8

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

10. Applied rewrites 97.8%

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

11. Final simplification 97.8%

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

Alternative 4: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   5. lower-*.f32 N/A

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

4. Applied rewrites 97.8%

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

6. Applied rewrites 97.8%

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

   1. lift-fma.f32 N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. div-inv N/A

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

   7. div-sub N/A

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

   8. lift--.f32 N/A

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

   9. lift-/.f32 98.3

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

8. Applied rewrites 98.3%

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

9. Taylor expanded in s around 0

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

    1. associate-*r* N/A

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

    2. lower-*.f32 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f32 N/A

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

    5. lower-log1p.f32 N/A

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

    6. sub-neg N/A

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

    7. distribute-rgt-in N/A

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

    8. metadata-eval N/A

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

    9. metadata-eval N/A

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

    10. lower-fma.f32 97.8

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

11. Applied rewrites 97.8%

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

12. Final simplification 97.8%

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

Alternative 5: 96.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, u)
	return Float32(log(fma(Float32(-1.3333333333333333), u, Float32(1.3333333333333333))) * Float32(Float32(-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 s around 0

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

5. Applied rewrites 96.4%

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

Alternative 6: 29.9% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

4. Applied rewrites 94.7%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{\frac{1}{\mathsf{fma}\left(1.3333333333333333, u, 0.6666666666666666\right)} \cdot \frac{1}{\mathsf{fma}\left(1.3333333333333333, u, 0.6666666666666666\right)} - \left(\left(\left(0.25 - u\right) \cdot \left(0.25 - u\right)\right) \cdot \frac{1.7777777777777777}{\mathsf{fma}\left(1.3333333333333333, u, 0.6666666666666666\right)}\right) \cdot \left(\left(\left(0.25 - u\right) \cdot \left(0.25 - u\right)\right) \cdot \frac{1.7777777777777777}{\mathsf{fma}\left(1.3333333333333333, u, 0.6666666666666666\right)}\right)}{\frac{1}{\mathsf{fma}\left(1.3333333333333333, u, 0.6666666666666666\right)} + \left(\left(0.25 - u\right) \cdot \left(0.25 - u\right)\right) \cdot \frac{1.7777777777777777}{\mathsf{fma}\left(1.3333333333333333, u, 0.6666666666666666\right)}}}}\right) \]

5. Taylor expanded in u around 0

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

7. Applied rewrites 7.5%

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

8. Taylor expanded in u around 0

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. distribute-lft-out N/A

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

   5. lower-*.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f32 N/A

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

   8. lower-log.f32 25.4

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

10. Applied rewrites 25.4%

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

11. Taylor expanded in u around inf

    \[\leadsto \left(s \cdot u\right) \cdot 3 \]
12. Step-by-step derivation

13. Applied rewrites 30.1%

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

Reproduce

herbie shell --seed 2024235 
(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))))))