Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 16.5s

Alternatives: 17

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: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{r}{-s}} \cdot 0.125}{\pi \cdot \left(r \cdot s\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{s \cdot \left({\left(\sqrt[3]{\pi}\right)}^{3} \cdot \left(r \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* (exp (/ r (- s))) 0.125) (* PI (* r s)))
  (/
   (* 0.75 (exp (* r (/ -0.3333333333333333 s))))
   (* s (* (pow (cbrt PI) 3.0) (* r 6.0))))))

C

float code(float s, float r) {
	return ((expf((r / -s)) * 0.125f) / (((float) M_PI) * (r * s))) + ((0.75f * expf((r * (-0.3333333333333333f / s)))) / (s * (powf(cbrtf(((float) M_PI)), 3.0f) * (r * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(r / Float32(-s))) * Float32(0.125)) / Float32(Float32(pi) * Float32(r * s))) + Float32(Float32(Float32(0.75) * exp(Float32(r * Float32(Float32(-0.3333333333333333) / s)))) / Float32(s * Float32((cbrt(Float32(pi)) ^ Float32(3.0)) * Float32(r * Float32(6.0))))))
end

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. lift-neg.f32 N/A

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

   5. remove-double-neg N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. distribute-lft-neg-in N/A

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 N/A

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

   14. metadata-eval 99.5

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{r \cdot \frac{\color{blue}{-0.3333333333333333}}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Applied rewrites 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{r \cdot \frac{-0.3333333333333333}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. associate-/r* N/A

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

   13. metadata-eval N/A

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

   14. times-frac N/A

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

   15. lift-*.f32 N/A

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

   16. associate-*r* N/A

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f32 N/A

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

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{e^{\frac{r}{-s}} \cdot 0.125}{\pi \cdot \left(r \cdot s\right)}} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

7. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 99.6

      \[\leadsto \frac{e^{\frac{r}{-s}} \cdot 0.125}{\pi \cdot \left(r \cdot s\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{s \cdot \left(\pi \cdot \color{blue}{\left(r \cdot 6\right)}\right)} \]

9. Applied rewrites 99.6%

   \[\leadsto \frac{e^{\frac{r}{-s}} \cdot 0.125}{\pi \cdot \left(r \cdot s\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{\color{blue}{s \cdot \left(\pi \cdot \left(r \cdot 6\right)\right)}} \]
10. Step-by-step derivation

11. Applied rewrites 99.6%

    \[\leadsto \frac{e^{\frac{r}{-s}} \cdot 0.125}{\pi \cdot \left(r \cdot s\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{s \cdot \left({\left(\sqrt[3]{\pi}\right)}^{3} \cdot \left(\color{blue}{r} \cdot 6\right)\right)} \]
12. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{r}{-s}} \cdot 0.125}{\pi \cdot \left(r \cdot s\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{s \cdot \left(\pi \cdot \left(r \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* (exp (/ r (- s))) 0.125) (* PI (* r s)))
  (/ (* 0.75 (exp (* r (/ -0.3333333333333333 s)))) (* s (* PI (* r 6.0))))))

C

float code(float s, float r) {
	return ((expf((r / -s)) * 0.125f) / (((float) M_PI) * (r * s))) + ((0.75f * expf((r * (-0.3333333333333333f / s)))) / (s * (((float) M_PI) * (r * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(r / Float32(-s))) * Float32(0.125)) / Float32(Float32(pi) * Float32(r * s))) + Float32(Float32(Float32(0.75) * exp(Float32(r * Float32(Float32(-0.3333333333333333) / s)))) / Float32(s * Float32(Float32(pi) * Float32(r * Float32(6.0))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp((r / -s)) * single(0.125)) / (single(pi) * (r * s))) + ((single(0.75) * exp((r * (single(-0.3333333333333333) / s)))) / (s * (single(pi) * (r * single(6.0)))));
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

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. lift-neg.f32 N/A

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

   5. remove-double-neg N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. distribute-lft-neg-in N/A

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 N/A

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

   14. metadata-eval 99.6

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{r \cdot \frac{\color{blue}{-0.3333333333333333}}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Applied rewrites 99.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{r \cdot \frac{-0.3333333333333333}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. associate-/r* N/A

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

   13. metadata-eval N/A

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

   14. times-frac N/A

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

   15. lift-*.f32 N/A

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

   16. associate-*r* N/A

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f32 N/A

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

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{e^{\frac{r}{-s}} \cdot 0.125}{\pi \cdot \left(r \cdot s\right)}} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

7. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 99.5

      \[\leadsto \frac{e^{\frac{r}{-s}} \cdot 0.125}{\pi \cdot \left(r \cdot s\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{s \cdot \left(\pi \cdot \color{blue}{\left(r \cdot 6\right)}\right)} \]

9. Applied rewrites 99.5%

   \[\leadsto \frac{e^{\frac{r}{-s}} \cdot 0.125}{\pi \cdot \left(r \cdot s\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{\color{blue}{s \cdot \left(\pi \cdot \left(r \cdot 6\right)\right)}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(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))))