Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 13.3s

Alternatives: 12

Speedup: N/A×

Specification

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

Math

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

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))))

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

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))))

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 99.6

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{6 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(s \cdot r\right)}\right)} \]

4. Applied rewrites 99.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\color{blue}{6 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)}} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.6

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{r}{\color{blue}{-3} \cdot s}}}{6 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} \]

6. Applied rewrites 99.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{r}{-3 \cdot s}}}}{6 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)\right)} \]

7. Final simplification 99.6%

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

Alternative 2: 95.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{\frac{\frac{0.125}{s}}{\mathsf{PI}\left(\right)}}{r}, \frac{e^{\frac{r}{-3 \cdot s}}}{\left(s \cdot r\right) \cdot \mathsf{PI}\left(\right)} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{\frac{-0.06944444444444445 \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s}}{s} - \frac{0.25}{\mathsf{PI}\left(\right) \cdot r}\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<=
      (+
       (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))
       (/ (* (exp (/ (- r) s)) 0.25) (* (* (* (PI) 2.0) s) r)))
      5.000000058430487e-8)
   (fma
    1.0
    (/ (/ (/ 0.125 s) (PI)) r)
    (* (/ (exp (/ r (* -3.0 s))) (* (* s r) (PI))) 0.125))
   (/
    (-
     (/ (/ -0.16666666666666666 (PI)) s)
     (- (/ (/ (* -0.06944444444444445 (/ r (PI))) s) s) (/ 0.25 (* (PI) r))))
    s)))

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))) < 5.00000006e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 4.7%

      \[\leadsto \frac{0.25 \cdot \color{blue}{1}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   7. Applied rewrites 4.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{e^{\frac{-r}{s}}}}{6 \cdot \mathsf{PI}\left(\right)}, \frac{0.75}{s \cdot r}, \frac{1}{r} \cdot \frac{\frac{0.125}{s}}{\mathsf{PI}\left(\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-/.f32 N/A

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

   9. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{\frac{\frac{0.125}{s}}{\mathsf{PI}\left(\right)}}{r}, \frac{\sqrt[3]{e^{\frac{-r}{s}}}}{\left(s \cdot r\right) \cdot \mathsf{PI}\left(\right)} \cdot 0.125\right)} \]
   10. Step-by-step derivation

       1. lift-cbrt.f32 N/A

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

       2. pow1/3 N/A

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

       3. lift-exp.f32 N/A

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

       4. lift-neg.f32 N/A

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

       5. lift-/.f32 N/A

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

       6. div-inv N/A

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

       7. lift-/.f32 N/A

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

       8. exp-prod N/A

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

       9. pow-pow N/A

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

       10. lift-/.f32 N/A

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

       11. inv-pow N/A

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

       12. metadata-eval N/A

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

       13. unpow-prod-down N/A

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

       14. *-commutative N/A

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

       15. inv-pow N/A

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

       16. exp-prod N/A

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

       17. div-inv N/A

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

       18. metadata-eval N/A

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

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

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

       20. lift-*.f32 N/A

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

       21. frac-2neg N/A

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

       22. lift-/.f32 N/A

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

       23. lift-exp.f32 99.2

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

   11. Applied rewrites 99.2%

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

   if 5.00000006e-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 97.2%

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

   3. Taylor expanded in s around inf

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

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4} \cdot 1}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

      4. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4} \cdot 1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4}}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

      9. lower-*.f32 N/A

         \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

      10. lower-PI.f32 39.4

          \[\leadsto \frac{\frac{0.25}{\color{blue}{\mathsf{PI}\left(\right)} \cdot s}}{r} \]

   5. Applied rewrites 39.4%

      \[\leadsto \color{blue}{\frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot s}}{r}} \]

   6. Taylor expanded in s around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s}} \]

   8. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s}}{s} + \frac{0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
   9. Step-by-step derivation

   10. Applied rewrites 54.2%

       \[\leadsto \frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{\frac{\frac{r}{\mathsf{PI}\left(\right)} \cdot -0.06944444444444445}{s}}{s} - \frac{0.25}{\mathsf{PI}\left(\right) \cdot r}\right)}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{\frac{\frac{0.125}{s}}{\mathsf{PI}\left(\right)}}{r}, \frac{e^{\frac{r}{-3 \cdot s}}}{\left(s \cdot r\right) \cdot \mathsf{PI}\left(\right)} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{\frac{-0.06944444444444445 \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s}}{s} - \frac{0.25}{\mathsf{PI}\left(\right) \cdot r}\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 11.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(t\_0, 0.06944444444444445, \frac{s}{\mathsf{PI}\left(\right)} \cdot -0.16666666666666666\right)}{{s}^{3}}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{\frac{-0.06944444444444445 \cdot t\_0}{s}}{s} - \frac{0.25}{\mathsf{PI}\left(\right) \cdot r}\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ r (PI))))
   (if (<=
        (+
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* (PI) 2.0) s) r)))
        0.0)
     (/
      (fma
       (/
        (fma t_0 0.06944444444444445 (* (/ s (PI)) -0.16666666666666666))
        (pow s 3.0))
       r
       (/ 0.25 (* (PI) s)))
      r)
     (/
      (-
       (/ (/ -0.16666666666666666 (PI)) s)
       (- (/ (/ (* -0.06944444444444445 t_0) s) s) (/ 0.25 (* (PI) r))))
      s))))

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))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4} \cdot 1}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

      4. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4} \cdot 1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4}}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

      9. lower-*.f32 N/A

         \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

      10. lower-PI.f32 4.6

          \[\leadsto \frac{\frac{0.25}{\color{blue}{\mathsf{PI}\left(\right)} \cdot s}}{r} \]

   5. Applied rewrites 4.6%

      \[\leadsto \color{blue}{\frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot s}}{r}} \]

   6. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   8. Applied rewrites 5.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.06944444444444445}{{s}^{3} \cdot \mathsf{PI}\left(\right)}, r, \frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}}{s}}{s}\right), r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}} \]

   9. Taylor expanded in s around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{6} \cdot \frac{s}{\mathsf{PI}\left(\right)} + \frac{5}{72} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{{s}^{3}}, r, \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} \]
   10. Step-by-step derivation

   11. Applied rewrites 4.7%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r}{\mathsf{PI}\left(\right)}, 0.06944444444444445, \frac{s}{\mathsf{PI}\left(\right)} \cdot -0.16666666666666666\right)}{{s}^{3}}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} \]

   if 0.0 < (+.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 96.6%

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

   3. Taylor expanded in s around inf

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

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4} \cdot 1}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

      4. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4} \cdot 1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4}}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

      7. lower-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

      8. *-commutative N/A

         \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

      9. lower-*.f32 N/A

         \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

      10. lower-PI.f32 37.1

          \[\leadsto \frac{\frac{0.25}{\color{blue}{\mathsf{PI}\left(\right)} \cdot s}}{r} \]

   5. Applied rewrites 37.1%

      \[\leadsto \color{blue}{\frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot s}}{r}} \]

   6. Taylor expanded in s around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s}} \]

   8. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s}}{s} + \frac{0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.4%

       \[\leadsto \frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{\frac{\frac{r}{\mathsf{PI}\left(\right)} \cdot -0.06944444444444445}{s}}{s} - \frac{0.25}{\mathsf{PI}\left(\right) \cdot r}\right)}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r}{\mathsf{PI}\left(\right)}, 0.06944444444444445, \frac{s}{\mathsf{PI}\left(\right)} \cdot -0.16666666666666666\right)}{{s}^{3}}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{\frac{-0.06944444444444445 \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s}}{s} - \frac{0.25}{\mathsf{PI}\left(\right) \cdot r}\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-0.3333333333333333}{s} \cdot r} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* (exp (* (/ -0.3333333333333333 s) r)) 0.75) (* (* (* 6.0 (PI)) s) r))
  (/ (* (exp (/ (- r) s)) 0.25) (* (* (* (PI) 2.0) s) r))))

Derivation

1. Initial program 99.6%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. lift-neg.f32 N/A

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

   5. remove-double-neg N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

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

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 N/A

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

   14. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

   \[\leadsto \frac{e^{\frac{-0.3333333333333333}{s} \cdot r} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \]
6. Add Preprocessing

Alternative 5: 92.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, \frac{\frac{0.125}{s}}{\mathsf{PI}\left(\right) \cdot r}, \frac{\sqrt[3]{e^{\frac{-r}{s}}}}{\left(s \cdot r\right) \cdot \mathsf{PI}\left(\right)} \cdot 0.125\right) \end{array} \]

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 8.9%

   \[\leadsto \frac{0.25 \cdot \color{blue}{1}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

7. Applied rewrites 7.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{e^{\frac{-r}{s}}}}{6 \cdot \mathsf{PI}\left(\right)}, \frac{0.75}{s \cdot r}, \frac{1}{r} \cdot \frac{\frac{0.125}{s}}{\mathsf{PI}\left(\right)}\right)} \]
8. Step-by-step derivation

   1. lift-fma.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. frac-times N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/l* N/A

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

   10. lower-fma.f32 N/A

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

9. Applied rewrites 93.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{\frac{0.125}{s}}{\mathsf{PI}\left(\right) \cdot r}, \frac{\sqrt[3]{e^{\frac{-r}{s}}}}{\left(s \cdot r\right) \cdot \mathsf{PI}\left(\right)} \cdot 0.125\right)} \]
10. Add Preprocessing

Alternative 6: 9.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{\frac{-0.06944444444444445 \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s}}{s} - \frac{0.25}{\mathsf{PI}\left(\right) \cdot r}\right)}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (-
   (/ (/ -0.16666666666666666 (PI)) s)
   (- (/ (/ (* -0.06944444444444445 (/ r (PI))) s) s) (/ 0.25 (* (PI) r))))
  s))

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4} \cdot 1}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   4. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4} \cdot 1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4}}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

   10. lower-PI.f32 8.8

       \[\leadsto \frac{\frac{0.25}{\color{blue}{\mathsf{PI}\left(\right)} \cdot s}}{r} \]

5. Applied rewrites 8.8%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot s}}{r}} \]

6. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s}} \]

8. Applied rewrites 9.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s}}{s} + \frac{0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
9. Step-by-step derivation

10. Applied rewrites 9.6%

    \[\leadsto \frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{\frac{\frac{r}{\mathsf{PI}\left(\right)} \cdot -0.06944444444444445}{s}}{s} - \frac{0.25}{\mathsf{PI}\left(\right) \cdot r}\right)}{s} \]

11. Final simplification 9.6%

    \[\leadsto \frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}}{s} - \left(\frac{\frac{-0.06944444444444445 \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s}}{s} - \frac{0.25}{\mathsf{PI}\left(\right) \cdot r}\right)}{s} \]
12. Add Preprocessing

Alternative 7: 10.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot r} - \frac{\frac{-0.06944444444444445 \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}}{s}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (-
   (/ 0.25 (* (PI) r))
   (/
    (- (/ (* -0.06944444444444445 (/ r (PI))) s) (/ -0.16666666666666666 (PI)))
    s))
  s))

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4} \cdot 1}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   4. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4} \cdot 1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4}}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

   10. lower-PI.f32 8.8

       \[\leadsto \frac{\frac{0.25}{\color{blue}{\mathsf{PI}\left(\right)} \cdot s}}{r} \]

5. Applied rewrites 8.8%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot s}}{r}} \]

6. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s}} \]

8. Applied rewrites 9.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s}}{s} + \frac{0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]

9. Final simplification 9.6%

   \[\leadsto \frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot r} - \frac{\frac{-0.06944444444444445 \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}}{s}}{s} \]
10. Add Preprocessing

Alternative 8: 8.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \frac{0.25}{\left(\left(t\_0 \cdot r\right) \cdot t\_0\right) \cdot s} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (sqrt (PI)))) (/ 0.25 (* (* (* t_0 r) t_0) s))))

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4} \cdot 1}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   4. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4} \cdot 1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4}}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

   10. lower-PI.f32 8.8

       \[\leadsto \frac{\frac{0.25}{\color{blue}{\mathsf{PI}\left(\right)} \cdot s}}{r} \]

5. Applied rewrites 8.8%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot s}}{r}} \]
6. Step-by-step derivation

7. Applied rewrites 8.8%

   \[\leadsto \frac{0.25}{\color{blue}{\left(s \cdot r\right) \cdot \mathsf{PI}\left(\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 8.8%

   \[\leadsto \frac{0.25}{\left(\mathsf{PI}\left(\right) \cdot r\right) \cdot \color{blue}{s}} \]
10. Step-by-step derivation

11. Applied rewrites 8.8%

    \[\leadsto \frac{0.25}{\left(\left(r \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot s} \]

12. Final simplification 8.8%

    \[\leadsto \frac{0.25}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot r\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot s} \]
13. Add Preprocessing

Alternative 9: 8.9% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{\mathsf{PI}\left(\right)}}{s \cdot r} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ (/ 0.25 (PI)) (* s r)))

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4} \cdot 1}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   4. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4} \cdot 1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4}}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

   10. lower-PI.f32 8.8

       \[\leadsto \frac{\frac{0.25}{\color{blue}{\mathsf{PI}\left(\right)} \cdot s}}{r} \]

5. Applied rewrites 8.8%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot s}}{r}} \]
6. Step-by-step derivation

7. Applied rewrites 8.8%

   \[\leadsto \frac{\frac{0.25}{\mathsf{PI}\left(\right)}}{\color{blue}{s \cdot r}} \]
8. Add Preprocessing

Alternative 10: 8.9% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{r}}{\mathsf{PI}\left(\right) \cdot s} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ (/ 0.25 r) (* (PI) s)))

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4} \cdot 1}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   4. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4} \cdot 1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4}}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

   10. lower-PI.f32 8.8

       \[\leadsto \frac{\frac{0.25}{\color{blue}{\mathsf{PI}\left(\right)} \cdot s}}{r} \]

5. Applied rewrites 8.8%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot s}}{r}} \]
6. Step-by-step derivation

7. Applied rewrites 8.8%

   \[\leadsto \frac{\frac{0.25}{r}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}} \]
8. Add Preprocessing

Alternative 11: 8.9% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\left(\mathsf{PI}\left(\right) \cdot r\right) \cdot s} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 0.25 (* (* (PI) r) s)))

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4} \cdot 1}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   4. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4} \cdot 1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4}}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

   10. lower-PI.f32 8.8

       \[\leadsto \frac{\frac{0.25}{\color{blue}{\mathsf{PI}\left(\right)} \cdot s}}{r} \]

5. Applied rewrites 8.8%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot s}}{r}} \]
6. Step-by-step derivation

7. Applied rewrites 8.8%

   \[\leadsto \frac{0.25}{\color{blue}{\left(s \cdot r\right) \cdot \mathsf{PI}\left(\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 8.8%

   \[\leadsto \frac{0.25}{\left(\mathsf{PI}\left(\right) \cdot r\right) \cdot \color{blue}{s}} \]
10. Add Preprocessing

Alternative 12: 8.9% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\left(s \cdot r\right) \cdot \mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 0.25 (* (* s r) (PI))))

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4} \cdot 1}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   4. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4} \cdot 1}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{4}}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{4}}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\color{blue}{\mathsf{PI}\left(\right) \cdot s}}}{r} \]

   10. lower-PI.f32 8.8

       \[\leadsto \frac{\frac{0.25}{\color{blue}{\mathsf{PI}\left(\right)} \cdot s}}{r} \]

5. Applied rewrites 8.8%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot s}}{r}} \]
6. Step-by-step derivation

7. Applied rewrites 8.8%

   \[\leadsto \frac{0.25}{\color{blue}{\left(s \cdot r\right) \cdot \mathsf{PI}\left(\right)}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024254 
(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))))