Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.8% → 98.3%

Time: 9.1s

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: 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: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, u)
	return Float32(s * Float32(log1p(Float32(Float32(u - Float32(0.25)) / Float32(-0.75))) * Float32(-3.0)))
end

Derivation

1. Initial program 96.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

4. Applied rewrites 98.0%

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

   1. lift-*.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. distribute-rgt-neg-in N/A

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

   5. metadata-eval N/A

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

   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. distribute-neg-frac2 N/A

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

   8. lower-/.f32 N/A

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

   9. metadata-eval 98.4

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

6. Applied rewrites 98.4%

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

7. Final simplification 98.4%

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

Alternative 2: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, u)
	return Float32(Float32(log1p(fma(u, Float32(-1.3333333333333333), Float32(0.3333333333333333))) * Float32(-3.0)) * 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 97.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 97.9

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

   4. lift-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. sub-neg N/A

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

   7. distribute-rgt-in N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f32 N/A

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

   12. metadata-eval 97.9

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

6. Applied rewrites 97.9%

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

Reproduce

herbie shell --seed 2024230 
(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))))))