Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 19.9s

Alternatives: 17

Speedup: 1.0×

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 9.3%

   \[\leadsto \frac{0.25 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{r}{s}}{s}, \mathsf{fma}\left(-0.16666666666666666, \frac{r}{s}, 0.5\right), \frac{-1}{s}\right), r, 1\right)}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

6. Taylor expanded in s around 0

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

   1. mul-1-neg N/A

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

   2. rec-exp N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. lower-exp.f32 N/A

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

   7. lower-/.f32 99.4

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

8. Applied rewrites 99.4%

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

9. Taylor expanded in s around 0

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

    1. associate-*r/ N/A

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

    2. lower-/.f32 N/A

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

    3. mul-1-neg N/A

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

    4. exp-neg N/A

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

    5. associate-*r/ N/A

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

    6. metadata-eval N/A

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

    7. lower-/.f32 N/A

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

    8. lower-exp.f32 N/A

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

    9. lower-/.f32 N/A

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

    10. *-commutative N/A

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

    11. lower-*.f32 N/A

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

    12. *-commutative N/A

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

    13. lower-*.f32 N/A

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

    14. lower-PI.f32 99.4

        \[\leadsto \frac{\frac{0.125}{e^{\frac{r}{s}}}}{\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

11. Applied rewrites 99.4%

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

    1. lift-neg.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. neg-mul-1 N/A

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

    5. times-frac N/A

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

    6. lift-/.f32 N/A

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

    7. lower-*.f32 N/A

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

    8. metadata-eval 99.4

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

13. Applied rewrites 99.4%

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

14. Final simplification 99.4%

    \[\leadsto \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{0.125}{e^{\frac{r}{s}}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]
15. Add Preprocessing

Alternative 2: 12.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 4.999999980020986 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{s}, \frac{0.25}{\mathsf{PI}\left(\right)}, \left({s}^{-2} \cdot \mathsf{fma}\left(\frac{0.06944444444444445}{\mathsf{PI}\left(\right)}, \frac{r}{s}, \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\right)\right) \cdot r\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.06944444444444445}{{s}^{3} \cdot \mathsf{PI}\left(\right)} + \frac{\frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot r} - \frac{\frac{0.16666666666666666}{\mathsf{PI}\left(\right)}}{s}}{s}}{r}\right) \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<=
      (+
       (/ (* (exp (/ (- r) s)) 0.25) (* (* (* 2.0 (PI)) s) r))
       (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r)))
      4.999999980020986e-12)
   (/
    (fma
     (/ 1.0 s)
     (/ 0.25 (PI))
     (*
      (*
       (pow s -2.0)
       (fma
        (/ 0.06944444444444445 (PI))
        (/ r s)
        (/ -0.16666666666666666 (PI))))
      r))
    r)
   (*
    (+
     (/ 0.06944444444444445 (* (pow s 3.0) (PI)))
     (/ (/ (- (/ 0.25 (* (PI) r)) (/ (/ 0.16666666666666666 (PI)) s)) s) r))
    r)))

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))) < 4.99999998e-12

   1. Initial program 99.5%

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

   3. Taylor expanded in r around 0

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

   5. Applied rewrites 4.5%

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

   7. Applied rewrites 6.1%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{s}, \frac{0.25}{\mathsf{PI}\left(\right)}, \left(\mathsf{fma}\left(\frac{0.06944444444444445}{\mathsf{PI}\left(\right)}, \frac{r}{s}, \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\right) \cdot {s}^{-2}\right) \cdot r\right)}{r} \]

   if 4.99999998e-12 < (+.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 98.4%

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

   3. Taylor expanded in r around 0

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

   5. Applied rewrites 51.6%

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

   6. Taylor expanded in r around -inf

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

   8. Applied rewrites 70.1%

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

4. Final simplification 12.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 4.999999980020986 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{s}, \frac{0.25}{\mathsf{PI}\left(\right)}, \left({s}^{-2} \cdot \mathsf{fma}\left(\frac{0.06944444444444445}{\mathsf{PI}\left(\right)}, \frac{r}{s}, \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\right)\right) \cdot r\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.06944444444444445}{{s}^{3} \cdot \mathsf{PI}\left(\right)} + \frac{\frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot r} - \frac{\frac{0.16666666666666666}{\mathsf{PI}\left(\right)}}{s}}{s}}{r}\right) \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 14.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{{\left(\left(14.4 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right)}^{-1} \cdot r + \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}}{s}}{s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.06944444444444445}{{s}^{3} \cdot \mathsf{PI}\left(\right)} + \frac{\frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot r} - \frac{\frac{0.16666666666666666}{\mathsf{PI}\left(\right)}}{s}}{s}}{r}\right) \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<=
      (+
       (/ (* (exp (/ (- r) s)) 0.25) (* (* (* 2.0 (PI)) s) r))
       (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r)))
      1.9999999996399175e-23)
   (/
    (fma
     (/
      (/
       (+ (* (pow (* (* 14.4 (PI)) s) -1.0) r) (/ -0.16666666666666666 (PI)))
       s)
      s)
     r
     (/ 0.25 (* (PI) s)))
    r)
   (*
    (+
     (/ 0.06944444444444445 (* (pow s 3.0) (PI)))
     (/ (/ (- (/ 0.25 (* (PI) r)) (/ (/ 0.16666666666666666 (PI)) s)) s) r))
    r)))

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))) < 2e-23

   1. Initial program 99.5%

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

   3. Taylor expanded in r around 0

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

   5. Applied rewrites 4.4%

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

   7. Applied rewrites 12.3%

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

   9. Applied rewrites 8.5%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} + r \cdot {\left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 14.4\right)\right)}^{-1}}{s}}{s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} \]

   if 2e-23 < (+.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 98.0%

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

   3. Taylor expanded in r around 0

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

   5. Applied rewrites 48.8%

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

   6. Taylor expanded in r around -inf

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

   8. Applied rewrites 66.1%

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

4. Final simplification 15.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{{\left(\left(14.4 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right)}^{-1} \cdot r + \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}}{s}}{s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.06944444444444445}{{s}^{3} \cdot \mathsf{PI}\left(\right)} + \frac{\frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot r} - \frac{\frac{0.16666666666666666}{\mathsf{PI}\left(\right)}}{s}}{s}}{r}\right) \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 14.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;\frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{{\left(\left(14.4 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right)}^{-1} \cdot r + t\_0}{s}}{s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ -0.16666666666666666 (PI))))
   (if (<=
        (+
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* 2.0 (PI)) s) r))
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r)))
        1.9999999996399175e-23)
     (/
      (fma
       (/ (/ (+ (* (pow (* (* 14.4 (PI)) s) -1.0) r) t_0) s) s)
       r
       (/ 0.25 (* (PI) s)))
      r)
     (/
      (-
       (/
        (-
         t_0
         (/
          (*
           (-
            (/ -0.06944444444444445 (PI))
            (* (/ -0.021604938271604937 s) (/ r (PI))))
           r)
          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))) < 2e-23

   1. Initial program 99.5%

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

   3. Taylor expanded in r around 0

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

   5. Applied rewrites 4.4%

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

   7. Applied rewrites 12.8%

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

   9. Applied rewrites 8.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} + r \cdot {\left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 14.4\right)\right)}^{-1}}{s}}{s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} \]

   if 2e-23 < (+.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 98.0%

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

   3. Taylor expanded in s around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

   5. Applied rewrites 64.9%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{{\left(\left(14.4 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right)}^{-1} \cdot r + \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}}{s}}{s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 14.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;\frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 4.999999980020986 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(r, {\left(\left(14.4 \cdot s\right) \cdot \mathsf{PI}\left(\right)\right)}^{-1}, t\_0\right)}{s}}{s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ -0.16666666666666666 (PI))))
   (if (<=
        (+
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* 2.0 (PI)) s) r))
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r)))
        4.999999980020986e-12)
     (/
      (fma
       (/ (/ (fma r (pow (* (* 14.4 s) (PI)) -1.0) t_0) s) s)
       r
       (/ 0.25 (* (PI) s)))
      r)
     (/
      (-
       (/
        (-
         t_0
         (/
          (*
           (-
            (/ -0.06944444444444445 (PI))
            (* (/ -0.021604938271604937 s) (/ r (PI))))
           r)
          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))) < 4.99999998e-12

   1. Initial program 99.5%

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

   3. Taylor expanded in r around 0

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

   5. Applied rewrites 4.5%

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

   7. Applied rewrites 8.2%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(r, {\left(\mathsf{PI}\left(\right) \cdot \left(s \cdot 14.4\right)\right)}^{-1}, \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\right)}{s}}{s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} \]

   if 4.99999998e-12 < (+.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 98.4%

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

   3. Taylor expanded in s around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

   5. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 4.999999980020986 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(r, {\left(\left(14.4 \cdot s\right) \cdot \mathsf{PI}\left(\right)\right)}^{-1}, \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\right)}{s}}{s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 9.3%

   \[\leadsto \frac{0.25 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{r}{s}}{s}, \mathsf{fma}\left(-0.16666666666666666, \frac{r}{s}, 0.5\right), \frac{-1}{s}\right), r, 1\right)}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

6. Taylor expanded in s around 0

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

   1. mul-1-neg N/A

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

   2. rec-exp N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. lower-exp.f32 N/A

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

   7. lower-/.f32 99.4

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

8. Applied rewrites 99.4%

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

9. Taylor expanded in s around 0

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

    1. associate-*r/ N/A

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

    2. lower-/.f32 N/A

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

    3. mul-1-neg N/A

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

    4. exp-neg N/A

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

    5. associate-*r/ N/A

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

    6. metadata-eval N/A

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

    7. lower-/.f32 N/A

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

    8. lower-exp.f32 N/A

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

    9. lower-/.f32 N/A

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

    10. *-commutative N/A

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

    11. lower-*.f32 N/A

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

    12. *-commutative N/A

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

    13. lower-*.f32 N/A

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

    14. lower-PI.f32 99.4

        \[\leadsto \frac{\frac{0.125}{e^{\frac{r}{s}}}}{\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

11. Applied rewrites 99.4%

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

13. Applied rewrites 99.4%

    \[\leadsto \frac{e^{\frac{-r}{s}} \cdot 0.125}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot s\right)} \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

14. Final simplification 99.4%

    \[\leadsto \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.125}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]
15. Add Preprocessing

Alternative 7: 33.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 0.019999999552965164:\\ \;\;\;\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} - \frac{\frac{-1}{1 + \frac{\mathsf{fma}\left(\frac{r}{s} \cdot r, 0.5, r\right)}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= s 0.019999999552965164)
   (-
    (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))
    (/
     (* (/ -1.0 (+ 1.0 (/ (fma (* (/ r s) r) 0.5 r) s))) 0.25)
     (* (* (* 2.0 (PI)) s) r)))
   (/
    (-
     (/
      (-
       (/ -0.16666666666666666 (PI))
       (/
        (*
         (-
          (/ -0.06944444444444445 (PI))
          (* (/ -0.021604938271604937 s) (/ r (PI))))
         r)
        s))
      s)
     (/ -0.25 (* (PI) r)))
    s)))

Derivation

1. Split input into 2 regimes

2. if s < 0.0199999996

   1. Initial program 99.4%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in s around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

   7. Applied rewrites 12.7%

      \[\leadsto \frac{0.25 \cdot \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, 0.5, r\right)}{s} + 1}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   if 0.0199999996 < s

   1. Initial program 98.8%

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

   3. Taylor expanded in s around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

   5. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 0.019999999552965164:\\ \;\;\;\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} - \frac{\frac{-1}{1 + \frac{\mathsf{fma}\left(\frac{r}{s} \cdot r, 0.5, r\right)}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 15.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} - \frac{\frac{-1}{1 + \frac{r}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (-
  (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))
  (/ (* (/ -1.0 (+ 1.0 (/ r s))) 0.25) (* (* (* 2.0 (PI)) s) r))))

Derivation

1. Initial program 99.4%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 17.7

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

7. Applied rewrites 17.7%

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

8. Final simplification 17.7%

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

Alternative 9: 13.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 3.999999984016789 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{{\left(\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{fma}\left(0.06944444444444445 \cdot \frac{r}{s}, \mathsf{PI}\left(\right), -0.16666666666666666 \cdot \mathsf{PI}\left(\right)\right)}\right)}^{-1}}{s}}{s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.06944444444444445}{{s}^{3} \cdot \mathsf{PI}\left(\right)} + \frac{\frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot r} - \frac{\frac{0.16666666666666666}{\mathsf{PI}\left(\right)}}{s}}{s}}{r}\right) \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= s 3.999999984016789e-12)
   (/
    (fma
     (/
      (/
       (pow
        (/
         (* (PI) (PI))
         (fma
          (* 0.06944444444444445 (/ r s))
          (PI)
          (* -0.16666666666666666 (PI))))
        -1.0)
       s)
      s)
     r
     (/ 0.25 (* (PI) s)))
    r)
   (*
    (+
     (/ 0.06944444444444445 (* (pow s 3.0) (PI)))
     (/ (/ (- (/ 0.25 (* (PI) r)) (/ (/ 0.16666666666666666 (PI)) s)) s) r))
    r)))

Derivation

1. Split input into 2 regimes

2. if s < 3.99999998e-12

   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 r around 0

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

   5. Applied rewrites 4.1%

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

   7. Applied rewrites 12.7%

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

   9. Applied rewrites 12.4%

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

   11. Applied rewrites 8.2%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{{\left(\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{fma}\left(0.06944444444444445 \cdot \frac{r}{s}, \mathsf{PI}\left(\right), -0.16666666666666666 \cdot \mathsf{PI}\left(\right)\right)}\right)}^{-1}}{s}}{s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} \]

   if 3.99999998e-12 < s

   1. Initial program 97.8%

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

   3. Taylor expanded in r around 0

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

   5. Applied rewrites 23.2%

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

   6. Taylor expanded in r around -inf

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

   8. Applied rewrites 29.4%

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

4. Final simplification 16.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 3.999999984016789 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{{\left(\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{fma}\left(0.06944444444444445 \cdot \frac{r}{s}, \mathsf{PI}\left(\right), -0.16666666666666666 \cdot \mathsf{PI}\left(\right)\right)}\right)}^{-1}}{s}}{s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.06944444444444445}{{s}^{3} \cdot \mathsf{PI}\left(\right)} + \frac{\frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot r} - \frac{\frac{0.16666666666666666}{\mathsf{PI}\left(\right)}}{s}}{s}}{r}\right) \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 13.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 5.499999815583578 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{\frac{0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s} + \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}}{s}}{s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.06944444444444445}{{s}^{3} \cdot \mathsf{PI}\left(\right)} + \frac{\frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot r} - \frac{\frac{0.16666666666666666}{\mathsf{PI}\left(\right)}}{s}}{s}}{r}\right) \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= s 5.499999815583578e-7)
   (/
    (fma
     (/
      (/
       (+
        (* (/ 0.06944444444444445 (PI)) (/ r s))
        (/ -0.16666666666666666 (PI)))
       s)
      s)
     r
     (/ 0.25 (* (PI) s)))
    r)
   (*
    (+
     (/ 0.06944444444444445 (* (pow s 3.0) (PI)))
     (/ (/ (- (/ 0.25 (* (PI) r)) (/ (/ 0.16666666666666666 (PI)) s)) s) r))
    r)))

Derivation

1. Split input into 2 regimes

2. if s < 5.49999982e-7

   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 r around 0

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

   5. Applied rewrites 4.4%

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

   7. Applied rewrites 12.6%

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

   9. Applied rewrites 11.5%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} + \frac{0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}}{s}, r, \frac{0.25}{{\mathsf{PI}\left(\right)}^{1} \cdot s}\right)}{r} \]

   if 5.49999982e-7 < s

   1. Initial program 95.9%

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

   3. Taylor expanded in r around 0

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

   5. Applied rewrites 38.2%

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

   6. Taylor expanded in r around -inf

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

   8. Applied rewrites 50.9%

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

4. Final simplification 12.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 5.499999815583578 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{\frac{0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s} + \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}}{s}}{s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.06944444444444445}{{s}^{3} \cdot \mathsf{PI}\left(\right)} + \frac{\frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot r} - \frac{\frac{0.16666666666666666}{\mathsf{PI}\left(\right)}}{s}}{s}}{r}\right) \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 9.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (-
   (/
    (-
     (/ -0.16666666666666666 (PI))
     (/
      (*
       (-
        (/ -0.06944444444444445 (PI))
        (* (/ -0.021604938271604937 s) (/ r (PI))))
       r)
      s))
    s)
   (/ -0.25 (* (PI) r)))
  s))

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 10.5%

   \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]

6. Final simplification 10.5%

   \[\leadsto \frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s} \]
7. Add Preprocessing

Alternative 12: 6.5% accurate, 3.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\left(\frac{1}{144} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{16} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)\right) - \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 10.1%

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

Alternative 13: 9.2% accurate, 6.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 10.1%

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

Alternative 14: 9.0% 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 99.4%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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 9.6

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

5. Applied rewrites 9.6%

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

7. Applied rewrites 9.6%

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

9. Applied rewrites 9.6%

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

11. Applied rewrites 9.6%

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

12. Final simplification 9.6%

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

Alternative 15: 9.0% accurate, 8.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. associate-/l* N/A

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

   9. lower-/.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. lower-PI.f32 9.6

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

7. Applied rewrites 9.6%

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

Alternative 16: 9.0% 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 99.4%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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 9.6

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

5. Applied rewrites 9.6%

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

7. Applied rewrites 9.6%

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

Alternative 17: 9.0% 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 99.4%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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 9.6

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

5. Applied rewrites 9.6%

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

7. Applied rewrites 9.6%

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

Reproduce

herbie shell --seed 2024288 
(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))))