Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 18.6s

Alternatives: 16

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

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

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

   6. associate-*l* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 99.2

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

4. Applied rewrites 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 10.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot r\\ \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 0.002199999988079071:\\ \;\;\;\;\frac{e^{\frac{r}{-3 \cdot s}} \cdot 0.75}{\left(6 \cdot s\right) \cdot t\_0} + \frac{\mathsf{fma}\left(-0.125, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{0.125}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{t\_0}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (PI) r)))
   (if (<=
        (+
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* (PI) 2.0) s) r)))
        0.002199999988079071)
     (+
      (/ (* (exp (/ r (* -3.0 s))) 0.75) (* (* 6.0 s) t_0))
      (/ (fma -0.125 (/ r (* (* s s) (PI))) (/ 0.125 (* (PI) s))) r))
     (/
      (-
       (/
        (-
         (/ -0.16666666666666666 (PI))
         (/ (* (/ -0.06944444444444445 (PI)) r) s))
        s)
       (/ -0.25 t_0))
      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))) < 0.0022

   1. Initial program 99.3%

      \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
   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. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f32 N/A

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

      6. lift-*.f32 N/A

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

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

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

      8. lower-*.f32 N/A

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

      9. metadata-eval 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-*.f32 N/A

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

      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

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

      5. *-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-*.f32 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} + \frac{\frac{3}{4} \cdot e^{\frac{r}{-3 \cdot s}}}{\left(6 \cdot s\right) \cdot \left(\mathsf{PI}\left(\right) \cdot r\right)} \]
   8. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} + \frac{\frac{3}{4} \cdot e^{\frac{r}{-3 \cdot s}}}{\left(6 \cdot s\right) \cdot \left(\mathsf{PI}\left(\right) \cdot r\right)} \]

      2. lower-fma.f32 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{8}, \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)}, \frac{1}{8} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right)}}{r} + \frac{\frac{3}{4} \cdot e^{\frac{r}{-3 \cdot s}}}{\left(6 \cdot s\right) \cdot \left(\mathsf{PI}\left(\right) \cdot r\right)} \]

      3. lower-/.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \color{blue}{\frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)}}, \frac{1}{8} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right)}{r} + \frac{\frac{3}{4} \cdot e^{\frac{r}{-3 \cdot s}}}{\left(6 \cdot s\right) \cdot \left(\mathsf{PI}\left(\right) \cdot r\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{r}{\color{blue}{{s}^{2} \cdot \mathsf{PI}\left(\right)}}, \frac{1}{8} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right)}{r} + \frac{\frac{3}{4} \cdot e^{\frac{r}{-3 \cdot s}}}{\left(6 \cdot s\right) \cdot \left(\mathsf{PI}\left(\right) \cdot r\right)} \]

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-PI.f32 5.2

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

   9. Applied rewrites 5.2%

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

   if 0.0022 < (+.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 64.4%

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

   6. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{\frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)} - \frac{r \cdot \frac{\frac{-5}{72}}{\mathsf{PI}\left(\right)}}{s}}{s} - \frac{\frac{-1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.9%

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

4. Final simplification 10.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 0.002199999988079071:\\ \;\;\;\;\frac{e^{\frac{r}{-3 \cdot s}} \cdot 0.75}{\left(6 \cdot s\right) \cdot \left(\mathsf{PI}\left(\right) \cdot r\right)} + \frac{\mathsf{fma}\left(-0.125, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{0.125}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 13.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot r\\ \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left({\left(-4 \cdot t\_0\right)}^{-1}, \frac{-1}{s}, \frac{\frac{-0.16666666666666666 - \mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right) \cdot \frac{r}{s}}{\mathsf{PI}\left(\right)}}{s \cdot s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{t\_0}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (PI) r)))
   (if (<=
        (+
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* (PI) 2.0) s) r)))
        1.9999999593223797e-31)
     (fma
      (pow (* -4.0 t_0) -1.0)
      (/ -1.0 s)
      (/
       (/
        (-
         -0.16666666666666666
         (* (fma (/ 0.021604938271604937 s) r -0.06944444444444445) (/ r s)))
        (PI))
       (* s s)))
     (/
      (-
       (/
        (-
         (/ -0.16666666666666666 (PI))
         (/ (* (/ -0.06944444444444445 (PI)) r) s))
        s)
       (/ -0.25 t_0))
      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))) < 1.99999996e-31

   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

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

   5. Applied rewrites 3.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 5.1%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \color{blue}{\frac{1}{s}}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 15.0%

      \[\leadsto \mathsf{fma}\left(\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}, \color{blue}{\frac{-1}{s}}, \frac{\frac{-0.16666666666666666 - \frac{r}{s} \cdot \mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right)}{\mathsf{PI}\left(\right)}}{s \cdot s}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 8.1%

       \[\leadsto \mathsf{fma}\left({\left(\left(\mathsf{PI}\left(\right) \cdot r\right) \cdot -4\right)}^{-1}, \frac{\color{blue}{-1}}{s}, \frac{\frac{-0.16666666666666666 - \frac{r}{s} \cdot \mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right)}{\mathsf{PI}\left(\right)}}{s \cdot s}\right) \]

   if 1.99999996e-31 < (+.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 96.2%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 43.6%

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

   6. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{\frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)} - \frac{r \cdot \frac{\frac{-5}{72}}{\mathsf{PI}\left(\right)}}{s}}{s} - \frac{\frac{-1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.5%

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

4. Final simplification 12.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left({\left(-4 \cdot \left(\mathsf{PI}\left(\right) \cdot r\right)\right)}^{-1}, \frac{-1}{s}, \frac{\frac{-0.16666666666666666 - \mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right) \cdot \frac{r}{s}}{\mathsf{PI}\left(\right)}}{s \cdot s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 10.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\\ t_1 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 4.0000000126843074 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0 - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \frac{1}{s}, \frac{\frac{--0.25}{\left(t\_1 \cdot r\right) \cdot t\_1}}{s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ -0.16666666666666666 (PI))) (t_1 (sqrt (PI))))
   (if (<=
        (+
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* (PI) 2.0) s) r)))
        4.0000000126843074e-29)
     (fma
      (/
       (-
        t_0
        (*
         (/ (- -0.06944444444444445 (* (/ -0.021604938271604937 s) r)) (PI))
         (/ r s)))
       s)
      (/ 1.0 s)
      (/ (/ (- -0.25) (* (* t_1 r) t_1)) s))
     (/
      (-
       (/ (- t_0 (/ (* (/ -0.06944444444444445 (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))) < 4.00000001e-29

   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

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

   5. Applied rewrites 3.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 7.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \color{blue}{\frac{1}{s}}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 4.7%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \frac{1}{s}, -\frac{\frac{-0.25}{\left(r \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}{s}\right) \]

   if 4.00000001e-29 < (+.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 96.4%

      \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
   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 44.9%

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

   6. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{\frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)} - \frac{r \cdot \frac{\frac{-5}{72}}{\mathsf{PI}\left(\right)}}{s}}{s} - \frac{\frac{-1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.8%

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

4. Final simplification 10.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 4.0000000126843074 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \frac{1}{s}, \frac{\frac{--0.25}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot r\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}{s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 10.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\\ t_1 := \mathsf{PI}\left(\right) \cdot r\\ \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 4.0000000126843074 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0 - \left(\left(\frac{\frac{0.021604938271604937}{s}}{\mathsf{PI}\left(\right)} - \frac{\frac{0.06944444444444445}{r}}{\mathsf{PI}\left(\right)}\right) \cdot r\right) \cdot \frac{r}{s}}{s}, \frac{1}{s}, \frac{\frac{0.25}{t\_1}}{s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{t\_1}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ -0.16666666666666666 (PI))) (t_1 (* (PI) r)))
   (if (<=
        (+
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* (PI) 2.0) s) r)))
        4.0000000126843074e-29)
     (fma
      (/
       (-
        t_0
        (*
         (*
          (-
           (/ (/ 0.021604938271604937 s) (PI))
           (/ (/ 0.06944444444444445 r) (PI)))
          r)
         (/ r s)))
       s)
      (/ 1.0 s)
      (/ (/ 0.25 t_1) s))
     (/
      (- (/ (- t_0 (/ (* (/ -0.06944444444444445 (PI)) r) s)) s) (/ -0.25 t_1))
      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))) < 4.00000001e-29

   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

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

   5. Applied rewrites 3.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 5.6%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \color{blue}{\frac{1}{s}}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 5.6%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \frac{1}{s}, \frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot r}}{-\left(-s\right)}\right) \]

   10. Taylor expanded in r around inf

       \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)} - \left(r \cdot \left(\frac{7}{324} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)} - \frac{5}{72} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot \frac{r}{s}}{s}, \frac{1}{s}, \frac{\frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{-\left(-s\right)}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 5.2%

       \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \left(\left(\frac{\frac{0.021604938271604937}{s}}{\mathsf{PI}\left(\right)} - \frac{\frac{0.06944444444444445}{r}}{\mathsf{PI}\left(\right)}\right) \cdot r\right) \cdot \frac{r}{s}}{s}, \frac{1}{s}, \frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot r}}{-\left(-s\right)}\right) \]

   if 4.00000001e-29 < (+.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 96.4%

      \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
   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 44.9%

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

   6. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{\frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)} - \frac{r \cdot \frac{\frac{-5}{72}}{\mathsf{PI}\left(\right)}}{s}}{s} - \frac{\frac{-1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.8%

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

4. Final simplification 10.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 4.0000000126843074 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \left(\left(\frac{\frac{0.021604938271604937}{s}}{\mathsf{PI}\left(\right)} - \frac{\frac{0.06944444444444445}{r}}{\mathsf{PI}\left(\right)}\right) \cdot r\right) \cdot \frac{r}{s}}{s}, \frac{1}{s}, \frac{\frac{0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 99.2

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

4. Applied rewrites 99.2%

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

5. Final simplification 99.2%

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

Alternative 7: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
2. Add Preprocessing
3. 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. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.2

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

4. Applied rewrites 99.2%

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

   1. lift-*.f32 N/A

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

   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. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*r* N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-PI.f32 N/A

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

   11. add-sqr-sqrt N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f32 N/A

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

   18. lift-PI.f32 N/A

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

   19. lower-sqrt.f32 N/A

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

   20. lift-PI.f32 N/A

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

   21. lower-sqrt.f32 99.2

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

6. Applied rewrites 99.2%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-sqrt.f32 N/A

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

   8. lift-sqrt.f32 N/A

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

   9. rem-square-sqrt N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 99.2

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

8. Applied rewrites 99.2%

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

9. Final simplification 99.2%

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

Alternative 8: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

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

4. Applied rewrites 97.7%

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

   1. lift-cbrt.f32 N/A

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

   2. pow1/3 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. exp-prod N/A

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

   7. pow-pow N/A

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

   8. inv-pow N/A

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

   9. metadata-eval N/A

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

   10. unpow-prod-down N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. inv-pow N/A

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

   14. exp-prod N/A

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

   15. div-inv N/A

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

   16. lift-/.f32 N/A

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

   17. lift-exp.f32 99.1

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

   18. lift-/.f32 N/A

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

   19. lift-neg.f32 N/A

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

   20. lift-*.f32 N/A

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

   21. neg-mul-1 N/A

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

   22. times-frac N/A

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

   23. lower-*.f32 N/A

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

   24. metadata-eval N/A

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

   25. lower-/.f32 99.0

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

6. Applied rewrites 99.0%

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

7. Final simplification 99.0%

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

Alternative 9: 6.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.0000000168623835 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \frac{1}{s}, \frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, \frac{r}{s}, 1\right)}{-s}, r, 1\right) \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= s 1.0000000168623835e-16)
   (fma
    (/
     (-
      (/ -0.16666666666666666 (PI))
      (*
       (/ (- -0.06944444444444445 (* (/ -0.021604938271604937 s) r)) (PI))
       (/ r s)))
     s)
    (/ 1.0 s)
    (/ (/ -0.25 (* (PI) r)) (- s)))
   (+
    (/
     (* (fma (/ (fma -0.5 (/ r s) 1.0) (- s)) r 1.0) 0.25)
     (* (* (* (PI) 2.0) s) r))
    (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r)))))

Derivation

1. Split input into 2 regimes

2. if s < 1.00000002e-16

   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

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

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 5.5%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \color{blue}{\frac{1}{s}}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 3.9%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{{\mathsf{PI}\left(\right)}^{1}} \cdot \frac{r}{s}}{s}, \frac{1}{s}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]

   if 1.00000002e-16 < s

   1. Initial program 98.3%

      \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
   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 13.3%

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

   6. Taylor expanded in s around -inf

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

   8. Applied rewrites 12.8%

      \[\leadsto \frac{0.25 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, \frac{r}{s}, 1\right)}{-s}, r, 1\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 8.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.0000000168623835 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \frac{1}{s}, \frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5, \frac{r}{s}, 1\right)}{-s}, r, 1\right) \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 10.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}\\ t_1 := \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;s \leq 4.600000068355481 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_1 - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \frac{1}{s}, \frac{t\_0}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1 - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - t\_0}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ -0.25 (* (PI) r))) (t_1 (/ -0.16666666666666666 (PI))))
   (if (<= s 4.600000068355481e-12)
     (fma
      (/
       (-
        t_1
        (*
         (/ (- -0.06944444444444445 (* (/ -0.021604938271604937 s) r)) (PI))
         (/ r s)))
       s)
      (/ 1.0 s)
      (/ t_0 (- s)))
     (/ (- (/ (- t_1 (/ (* (/ -0.06944444444444445 (PI)) r) s)) s) t_0) s))))

Derivation

1. Split input into 2 regimes

2. if s < 4.60000007e-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 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 3.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 5.4%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \color{blue}{\frac{1}{s}}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 4.1%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{{\mathsf{PI}\left(\right)}^{1}} \cdot \frac{r}{s}}{s}, \frac{1}{s}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]

   if 4.60000007e-12 < s

   1. Initial program 97.9%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 17.3%

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

   6. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{\frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)} - \frac{r \cdot \frac{\frac{-5}{72}}{\mathsf{PI}\left(\right)}}{s}}{s} - \frac{\frac{-1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]
   7. Step-by-step derivation

   8. Applied rewrites 18.9%

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

4. Final simplification 10.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 4.600000068355481 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \frac{1}{s}, \frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 9.8% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 8.4%

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

6. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{\frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)} - \frac{r \cdot \frac{\frac{-5}{72}}{\mathsf{PI}\left(\right)}}{s}}{s} - \frac{\frac{-1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]
7. Step-by-step derivation

8. Applied rewrites 9.1%

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

9. Final simplification 9.1%

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

Alternative 12: 8.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r} \cdot \frac{0.25}{s} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (* (/ (/ 1.0 (PI)) r) (/ 0.25 s)))

Derivation

1. Initial program 99.2%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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 8.4

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

5. Applied rewrites 8.4%

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

7. Applied rewrites 8.4%

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

9. Applied rewrites 8.4%

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

11. Applied rewrites 8.4%

    \[\leadsto \frac{0.25}{s} \cdot \color{blue}{\frac{\frac{1}{\mathsf{PI}\left(\right)}}{r}} \]

12. Final simplification 8.4%

    \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r} \cdot \frac{0.25}{s} \]
13. Add Preprocessing

Alternative 13: 8.9% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\mathsf{PI}\left(\right) \cdot r} \cdot \frac{1}{s} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (* (/ 0.25 (* (PI) r)) (/ 1.0 s)))

Derivation

1. Initial program 99.2%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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 8.4

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

5. Applied rewrites 8.4%

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

7. Applied rewrites 8.4%

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

9. Applied rewrites 8.4%

   \[\leadsto \frac{0.25}{\mathsf{PI}\left(\right) \cdot r} \cdot \color{blue}{\frac{1}{s}} \]
10. Add Preprocessing

Alternative 14: 8.9% accurate, 10.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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 8.4

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

5. Applied rewrites 8.4%

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

7. Applied rewrites 8.4%

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

9. Applied rewrites 8.4%

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

Alternative 15: 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 99.2%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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 8.4

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

5. Applied rewrites 8.4%

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

7. Applied rewrites 8.4%

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

Alternative 16: 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 99.2%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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 8.4

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

5. Applied rewrites 8.4%

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

7. Applied rewrites 8.4%

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

9. Applied rewrites 8.4%

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

Reproduce

herbie shell --seed 2024276 
(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))))