Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 14.9s

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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.125}{s \cdot \pi}\\ \frac{1}{r} \cdot \mathsf{fma}\left(e^{\frac{r}{-s}}, t\_0, t\_0 \cdot e^{\frac{r}{s \cdot -3}}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ 0.125 (* s PI))))
   (* (/ 1.0 r) (fma (exp (/ r (- s))) t_0 (* t_0 (exp (/ r (* s -3.0))))))))

C

float code(float s, float r) {
	float t_0 = 0.125f / (s * ((float) M_PI));
	return (1.0f / r) * fmaf(expf((r / -s)), t_0, (t_0 * expf((r / (s * -3.0f)))));
}

Julia

function code(s, r)
	t_0 = Float32(Float32(0.125) / Float32(s * Float32(pi)))
	return Float32(Float32(Float32(1.0) / r) * fma(exp(Float32(r / Float32(-s))), t_0, Float32(t_0 * exp(Float32(r / Float32(s * Float32(-3.0)))))))
end

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{1}{r} \cdot \mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{s \cdot \pi}, e^{\frac{r}{s \cdot -3}} \cdot \frac{0.125}{s \cdot \pi}\right)} \]

5. Final simplification 99.8%

   \[\leadsto \frac{1}{r} \cdot \mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{s \cdot \pi}, \frac{0.125}{s \cdot \pi} \cdot e^{\frac{r}{s \cdot -3}}\right) \]
6. Add Preprocessing

Alternative 2: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{r}{-s}}\\ t_1 := \frac{0.125}{s \cdot \pi}\\ \mathbf{if}\;\frac{t\_0 \cdot 0.25}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{\left(-s\right) \cdot 3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{0.125 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{s \cdot \pi}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{r} \cdot \mathsf{fma}\left(t\_0, t\_1, t\_1 \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot \left(s \cdot s\right)}, -0.006172839506172839, \frac{0.05555555555555555}{s \cdot s}\right), \frac{-0.3333333333333333}{s}\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (exp (/ r (- s)))) (t_1 (/ 0.125 (* s PI))))
   (if (<=
        (+
         (/ (* t_0 0.25) (* r (* s (* PI 2.0))))
         (/ (* 0.75 (exp (/ r (* (- s) 3.0)))) (* r (* s (* PI 6.0)))))
        1.999999987845058e-8)
     (/ (/ (* 0.125 (exp (* (/ r s) -0.3333333333333333))) (* s PI)) r)
     (*
      (/ 1.0 r)
      (fma
       t_0
       t_1
       (*
        t_1
        (fma
         r
         (fma
          r
          (fma
           (/ r (* s (* s s)))
           -0.006172839506172839
           (/ 0.05555555555555555 (* s s)))
          (/ -0.3333333333333333 s))
         1.0)))))))

C

float code(float s, float r) {
	float t_0 = expf((r / -s));
	float t_1 = 0.125f / (s * ((float) M_PI));
	float tmp;
	if ((((t_0 * 0.25f) / (r * (s * (((float) M_PI) * 2.0f)))) + ((0.75f * expf((r / (-s * 3.0f)))) / (r * (s * (((float) M_PI) * 6.0f))))) <= 1.999999987845058e-8f) {
		tmp = ((0.125f * expf(((r / s) * -0.3333333333333333f))) / (s * ((float) M_PI))) / r;
	} else {
		tmp = (1.0f / r) * fmaf(t_0, t_1, (t_1 * fmaf(r, fmaf(r, fmaf((r / (s * (s * s))), -0.006172839506172839f, (0.05555555555555555f / (s * s))), (-0.3333333333333333f / s)), 1.0f)));
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = exp(Float32(r / Float32(-s)))
	t_1 = Float32(Float32(0.125) / Float32(s * Float32(pi)))
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(t_0 * Float32(0.25)) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(Float32(-s) * Float32(3.0))))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0)))))) <= Float32(1.999999987845058e-8))
		tmp = Float32(Float32(Float32(Float32(0.125) * exp(Float32(Float32(r / s) * Float32(-0.3333333333333333)))) / Float32(s * Float32(pi))) / r);
	else
		tmp = Float32(Float32(Float32(1.0) / r) * fma(t_0, t_1, Float32(t_1 * fma(r, fma(r, fma(Float32(r / Float32(s * Float32(s * s))), Float32(-0.006172839506172839), Float32(Float32(0.05555555555555555) / Float32(s * s))), Float32(Float32(-0.3333333333333333) / s)), Float32(1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 1.99999999e-8

   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. Taylor expanded in r around inf

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

      1. distribute-lft-out N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-out N/A

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

      5. lower-/.f32 N/A

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

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{0.125 \cdot \left(\frac{e^{\frac{r}{-s}}}{s \cdot \pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{s \cdot \pi}\right)}{r}} \]

   7. Applied egg-rr 99.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{\pi}, \frac{0.125}{s}, \frac{0.125}{\left(s \cdot \pi\right) \cdot e^{\frac{r}{s}}}\right)}}{r} \]

   8. Taylor expanded in s around -inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{\mathsf{PI}\left(\right)}, \frac{\frac{1}{8}}{s}, \frac{\frac{1}{8}}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot r + \frac{-1}{2} \cdot \frac{{r}^{2}}{s}}{s}\right)}}\right)}{r} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{\mathsf{PI}\left(\right)}, \frac{\frac{1}{8}}{s}, \frac{\frac{1}{8}}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot r + \frac{-1}{2} \cdot \frac{{r}^{2}}{s}}{s}\right)\right)}\right)}\right)}{r} \]

      2. unsub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{\mathsf{PI}\left(\right)}, \frac{\frac{1}{8}}{s}, \frac{\frac{1}{8}}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(1 - \frac{-1 \cdot r + \frac{-1}{2} \cdot \frac{{r}^{2}}{s}}{s}\right)}}\right)}{r} \]

      3. lower--.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{\mathsf{PI}\left(\right)}, \frac{\frac{1}{8}}{s}, \frac{\frac{1}{8}}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(1 - \frac{-1 \cdot r + \frac{-1}{2} \cdot \frac{{r}^{2}}{s}}{s}\right)}}\right)}{r} \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{\mathsf{PI}\left(\right)}, \frac{\frac{1}{8}}{s}, \frac{\frac{1}{8}}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - \color{blue}{\frac{-1 \cdot r + \frac{-1}{2} \cdot \frac{{r}^{2}}{s}}{s}}\right)}\right)}{r} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{\mathsf{PI}\left(\right)}, \frac{\frac{1}{8}}{s}, \frac{\frac{1}{8}}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - \frac{\color{blue}{\frac{-1}{2} \cdot \frac{{r}^{2}}{s} + -1 \cdot r}}{s}\right)}\right)}{r} \]

      6. lower-fma.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{\mathsf{PI}\left(\right)}, \frac{\frac{1}{8}}{s}, \frac{\frac{1}{8}}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{{r}^{2}}{s}, -1 \cdot r\right)}}{s}\right)}\right)}{r} \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{\mathsf{PI}\left(\right)}, \frac{\frac{1}{8}}{s}, \frac{\frac{1}{8}}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{{r}^{2}}{s}}, -1 \cdot r\right)}{s}\right)}\right)}{r} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{\mathsf{PI}\left(\right)}, \frac{\frac{1}{8}}{s}, \frac{\frac{1}{8}}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{r \cdot r}}{s}, -1 \cdot r\right)}{s}\right)}\right)}{r} \]

      9. lower-*.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{\mathsf{PI}\left(\right)}, \frac{\frac{1}{8}}{s}, \frac{\frac{1}{8}}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{r \cdot r}}{s}, -1 \cdot r\right)}{s}\right)}\right)}{r} \]

      10. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{\mathsf{PI}\left(\right)}, \frac{\frac{1}{8}}{s}, \frac{\frac{1}{8}}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 - \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{r \cdot r}{s}, \color{blue}{\mathsf{neg}\left(r\right)}\right)}{s}\right)}\right)}{r} \]

      11. lower-neg.f32 55.7

          \[\leadsto \frac{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{\pi}, \frac{0.125}{s}, \frac{0.125}{\left(s \cdot \pi\right) \cdot \left(1 - \frac{\mathsf{fma}\left(-0.5, \frac{r \cdot r}{s}, \color{blue}{-r}\right)}{s}\right)}\right)}{r} \]

   10. Simplified 55.7%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{\pi}, \frac{0.125}{s}, \frac{0.125}{\left(s \cdot \pi\right) \cdot \color{blue}{\left(1 - \frac{\mathsf{fma}\left(-0.5, \frac{r \cdot r}{s}, -r\right)}{s}\right)}}\right)}{r} \]

   11. Taylor expanded in r around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower-exp.f32 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f32 N/A

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

       7. lower-/.f32 N/A

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

       8. lower-*.f32 N/A

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

       9. lower-PI.f32 99.7

          \[\leadsto \frac{\frac{0.125 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{s \cdot \color{blue}{\pi}}}{r} \]

   13. Simplified 99.7%

       \[\leadsto \frac{\color{blue}{\frac{0.125 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{s \cdot \pi}}}{r} \]

   if 1.99999999e-8 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 97.9%

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

   4. Applied egg-rr 97.5%

      \[\leadsto \color{blue}{\frac{1}{r} \cdot \mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{s \cdot \pi}, e^{\frac{r}{s \cdot -3}} \cdot \frac{0.125}{s \cdot \pi}\right)} \]

   5. Taylor expanded in r around 0

      \[\leadsto \frac{1}{r} \cdot \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}, \color{blue}{\left(1 + r \cdot \left(r \cdot \left(\frac{-1}{162} \cdot \frac{r}{{s}^{3}} + \frac{1}{18} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{3} \cdot \frac{1}{s}\right)\right)} \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{r} \cdot \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}, \color{blue}{\left(r \cdot \left(r \cdot \left(\frac{-1}{162} \cdot \frac{r}{{s}^{3}} + \frac{1}{18} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{3} \cdot \frac{1}{s}\right) + 1\right)} \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}\right) \]

      2. lower-fma.f32 N/A

         \[\leadsto \frac{1}{r} \cdot \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}, \color{blue}{\mathsf{fma}\left(r, r \cdot \left(\frac{-1}{162} \cdot \frac{r}{{s}^{3}} + \frac{1}{18} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{3} \cdot \frac{1}{s}, 1\right)} \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}\right) \]

   7. Simplified 71.3%

      \[\leadsto \frac{1}{r} \cdot \mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{s \cdot \pi}, \color{blue}{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot \left(s \cdot s\right)}, -0.006172839506172839, \frac{0.05555555555555555}{s \cdot s}\right), \frac{-0.3333333333333333}{s}\right), 1\right)} \cdot \frac{0.125}{s \cdot \pi}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{r}{-s}} \cdot 0.25}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{\left(-s\right) \cdot 3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{0.125 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{s \cdot \pi}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{r} \cdot \mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{s \cdot \pi}, \frac{0.125}{s \cdot \pi} \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot \left(s \cdot s\right)}, -0.006172839506172839, \frac{0.05555555555555555}{s \cdot s}\right), \frac{-0.3333333333333333}{s}\right), 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024218 
(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))))