Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.4%

Time: 15.0s

Alternatives: 20

Speedup: N/A×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{0.16666666666666666}{r \cdot \pi}, e^{-0.3333333333333333 \cdot \frac{r}{s}} \cdot \frac{0.75}{s}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(s \cdot \pi\right) \cdot r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ 0.16666666666666666 (* r PI))
  (* (exp (* -0.3333333333333333 (/ r s))) (/ 0.75 s))
  (* 0.125 (/ (exp (/ (- r) s)) (* (* s PI) r)))))

C

float code(float s, float r) {
	return fmaf((0.16666666666666666f / (r * ((float) M_PI))), (expf((-0.3333333333333333f * (r / s))) * (0.75f / s)), (0.125f * (expf((-r / s)) / ((s * ((float) M_PI)) * r))));
}

Julia

function code(s, r)
	return fma(Float32(Float32(0.16666666666666666) / Float32(r * Float32(pi))), Float32(exp(Float32(Float32(-0.3333333333333333) * Float32(r / s))) * Float32(Float32(0.75) / s)), Float32(Float32(0.125) * Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(s * Float32(pi)) * r))))
end

Derivation

1. Initial program 99.8%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\color{blue}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\color{blue}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right)} \cdot r} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\color{blue}{\left(6 \cdot \mathsf{PI}\left(\right)\right)} \cdot s\right) \cdot r} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\color{blue}{\left(6 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)\right)} \cdot r} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\color{blue}{6 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r\right)}} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}} \]

4. Applied rewrites 99.8%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot 0.75\right) \cdot 0.16666666666666666}{\left(\pi \cdot s\right) \cdot r}} \]

5. Taylor expanded in r around inf

   \[\leadsto \color{blue}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{8}} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{8}} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   3. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   4. lower-exp.f32 N/A

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \frac{r}{s}}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{-1 \cdot r}{s}}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   6. lower-/.f32 N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{-1 \cdot r}{s}}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   7. mul-1-neg N/A

      \[\leadsto \frac{e^{\frac{\color{blue}{\mathsf{neg}\left(r\right)}}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   8. lower-neg.f32 N/A

      \[\leadsto \frac{e^{\frac{\color{blue}{\mathsf{neg}\left(r\right)}}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   9. associate-*r* N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\color{blue}{\left(r \cdot s\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\color{blue}{\left(r \cdot s\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   11. *-commutative N/A

       \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\color{blue}{\left(s \cdot r\right)} \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   12. lower-*.f32 N/A

       \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\color{blue}{\left(s \cdot r\right)} \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   13. lower-PI.f32 99.8

       \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(s \cdot r\right) \cdot \color{blue}{\pi}} \cdot 0.125 + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot 0.75\right) \cdot 0.16666666666666666}{\left(\pi \cdot s\right) \cdot r} \]

7. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{e^{\frac{-r}{s}}}{\left(s \cdot r\right) \cdot \pi} \cdot 0.125} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot 0.75\right) \cdot 0.16666666666666666}{\left(\pi \cdot s\right) \cdot r} \]

9. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{-0.3333333333333333 \cdot \frac{r}{s}} \cdot 0.75}{s}, \frac{0.16666666666666666}{\pi \cdot r}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r}\right)} \]
10. Step-by-step derivation

    1. lift-fma.f32 N/A

       \[\leadsto \color{blue}{\frac{e^{\frac{-1}{3} \cdot \frac{r}{s}} \cdot \frac{3}{4}}{s} \cdot \frac{\frac{1}{6}}{\mathsf{PI}\left(\right) \cdot r} + \frac{1}{8} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}} \]

    2. *-commutative N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{6}}{\mathsf{PI}\left(\right) \cdot r} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}} \cdot \frac{3}{4}}{s}} + \frac{1}{8} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

    3. lower-fma.f32 99.8

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.16666666666666666}{\pi \cdot r}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}} \cdot 0.75}{s}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r}\right)} \]

11. Applied rewrites 99.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.16666666666666666}{\pi \cdot r}, \frac{0.75}{s} \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}, \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125\right)} \]

12. Final simplification 99.8%

    \[\leadsto \mathsf{fma}\left(\frac{0.16666666666666666}{r \cdot \pi}, e^{-0.3333333333333333 \cdot \frac{r}{s}} \cdot \frac{0.75}{s}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(s \cdot \pi\right) \cdot r}\right) \]
13. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot 0.75\right) \cdot 0.16666666666666666}{\left(s \cdot \pi\right) \cdot r} + \frac{e^{\frac{-r}{s}}}{\left(s \cdot r\right) \cdot \pi} \cdot 0.125 \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* (* (exp (/ r (* -3.0 s))) 0.75) 0.16666666666666666) (* (* s PI) r))
  (* (/ (exp (/ (- r) s)) (* (* s r) PI)) 0.125)))

C

float code(float s, float r) {
	return (((expf((r / (-3.0f * s))) * 0.75f) * 0.16666666666666666f) / ((s * ((float) M_PI)) * r)) + ((expf((-r / s)) / ((s * r) * ((float) M_PI))) * 0.125f);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) * Float32(0.75)) * Float32(0.16666666666666666)) / Float32(Float32(s * Float32(pi)) * r)) + Float32(Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(s * r) * Float32(pi))) * Float32(0.125)))
end

MATLAB

function tmp = code(s, r)
	tmp = (((exp((r / (single(-3.0) * s))) * single(0.75)) * single(0.16666666666666666)) / ((s * single(pi)) * r)) + ((exp((-r / s)) / ((s * r) * single(pi))) * single(0.125));
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\color{blue}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\color{blue}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right)} \cdot r} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\color{blue}{\left(6 \cdot \mathsf{PI}\left(\right)\right)} \cdot s\right) \cdot r} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\color{blue}{\left(6 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)\right)} \cdot r} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\color{blue}{6 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r\right)}} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}} \]

4. Applied rewrites 99.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot 0.75\right) \cdot 0.16666666666666666}{\left(\pi \cdot s\right) \cdot r}} \]

5. Taylor expanded in r around inf

   \[\leadsto \color{blue}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{8}} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{8}} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   3. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   4. lower-exp.f32 N/A

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \frac{r}{s}}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{-1 \cdot r}{s}}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   6. lower-/.f32 N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{-1 \cdot r}{s}}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   7. mul-1-neg N/A

      \[\leadsto \frac{e^{\frac{\color{blue}{\mathsf{neg}\left(r\right)}}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   8. lower-neg.f32 N/A

      \[\leadsto \frac{e^{\frac{\color{blue}{\mathsf{neg}\left(r\right)}}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   9. associate-*r* N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\color{blue}{\left(r \cdot s\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\color{blue}{\left(r \cdot s\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   11. *-commutative N/A

       \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\color{blue}{\left(s \cdot r\right)} \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   12. lower-*.f32 N/A

       \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\color{blue}{\left(s \cdot r\right)} \cdot \mathsf{PI}\left(\right)} \cdot \frac{1}{8} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{3}{4}\right) \cdot \frac{1}{6}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   13. lower-PI.f32 99.6

       \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(s \cdot r\right) \cdot \color{blue}{\pi}} \cdot 0.125 + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot 0.75\right) \cdot 0.16666666666666666}{\left(\pi \cdot s\right) \cdot r} \]

7. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{e^{\frac{-r}{s}}}{\left(s \cdot r\right) \cdot \pi} \cdot 0.125} + \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot 0.75\right) \cdot 0.16666666666666666}{\left(\pi \cdot s\right) \cdot r} \]

8. Final simplification 99.6%

   \[\leadsto \frac{\left(e^{\frac{r}{-3 \cdot s}} \cdot 0.75\right) \cdot 0.16666666666666666}{\left(s \cdot \pi\right) \cdot r} + \frac{e^{\frac{-r}{s}}}{\left(s \cdot r\right) \cdot \pi} \cdot 0.125 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024229 
(FPCore (s r)
  :name "Disney BSSRDF, PDF of scattering profile"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (< 1e-6 r) (< r 1000000.0)))
  (+ (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r)) (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))