Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 21.1s

Alternatives: 17

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. times-frac N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 93.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := s \cdot \mathsf{PI}\left(\right)\\ t_1 := s \cdot t\_0\\ t_2 := e^{\frac{r}{-s}}\\ t_3 := \frac{0.25 \cdot t\_2}{r \cdot \left(s \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}\\ \mathbf{if}\;t\_3 + \frac{0.75 \cdot e^{-\frac{r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)} \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \frac{0.125}{\mathsf{PI}\left(\right)}, s \cdot \left(r \cdot -0.041666666666666664\right)\right), \frac{1}{s \cdot \left(r \cdot t\_1\right)}, \frac{t\_2}{r \cdot \left(t\_0 \cdot 8\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \frac{\frac{\frac{r \cdot 0.006944444444444444}{\mathsf{PI}\left(\right)} + \frac{-0.0007716049382716049 \cdot \left(r \cdot r\right)}{t\_0}}{s \cdot s} + \left(\frac{0.125}{r \cdot \mathsf{PI}\left(\right)} + \frac{-0.041666666666666664}{t\_0}\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* s (PI)))
        (t_1 (* s t_0))
        (t_2 (exp (/ r (- s))))
        (t_3 (/ (* 0.25 t_2) (* r (* s (* 2.0 (PI)))))))
   (if (<=
        (+ t_3 (/ (* 0.75 (exp (- (/ r (* s 3.0))))) (* r (* s (* (PI) 6.0)))))
        1.999999987845058e-8)
     (fma
      (fma t_1 (/ 0.125 (PI)) (* s (* r -0.041666666666666664)))
      (/ 1.0 (* s (* r t_1)))
      (/ t_2 (* r (* t_0 8.0))))
     (+
      t_3
      (/
       (+
        (/
         (+
          (/ (* r 0.006944444444444444) (PI))
          (/ (* -0.0007716049382716049 (* r r)) t_0))
         (* s s))
        (+ (/ 0.125 (* r (PI))) (/ -0.041666666666666664 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))) < 1.99999999e-8

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} - \frac{\frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{\frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}\right)\right)\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{24} \cdot \frac{\frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \color{blue}{\frac{1}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{\color{blue}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{\color{blue}{{s}^{2}} \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      7. lower-+.f32 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)} \]

   5. Applied rewrites 3.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   7. Applied rewrites 3.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{\color{blue}{\left(r \cdot s\right) \cdot \mathsf{PI}\left(\right)}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      5. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{\color{blue}{\left(r \cdot s\right)} \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \color{blue}{\frac{\frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}}{r \cdot s}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      8. lower-/.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \color{blue}{\frac{\frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}}{r \cdot s}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      9. lower-/.f32 3.7

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\color{blue}{\frac{0.125}{\mathsf{PI}\left(\right)}}}{r \cdot s} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      10. lift-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}}{\color{blue}{r \cdot s}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}}{\color{blue}{s \cdot r}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      12. lower-*.f32 3.7

          \[\leadsto \mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{0.125}{\mathsf{PI}\left(\right)}}{\color{blue}{s \cdot r}} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   9. Applied rewrites 3.7%

      \[\leadsto \mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \color{blue}{\frac{\frac{0.125}{\mathsf{PI}\left(\right)}}{s \cdot r}} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   11. Applied rewrites 98.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right), \frac{0.125}{\mathsf{PI}\left(\right)}, s \cdot \left(r \cdot -0.041666666666666664\right)\right), \frac{1}{s \cdot \left(r \cdot \left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, \frac{e^{\frac{r}{-s}}}{r \cdot \left(\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot 8\right)}\right)} \]

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

   1. Initial program 98.2%

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

   3. Taylor expanded in s around -inf

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{1296} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}}{s} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

   5. Applied rewrites 34.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.0007716049382716049, \frac{r \cdot r}{s \cdot \mathsf{PI}\left(\right)}, \frac{r \cdot 0.006944444444444444}{\mathsf{PI}\left(\right)}\right)}{s \cdot s} + \left(\frac{0.125}{r \cdot \mathsf{PI}\left(\right)} + \frac{-0.041666666666666664}{s \cdot \mathsf{PI}\left(\right)}\right)}{s}} \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{-1}{1296} \cdot \frac{\color{blue}{r \cdot r}}{s \cdot \mathsf{PI}\left(\right)} + \frac{r \cdot \frac{1}{144}}{\mathsf{PI}\left(\right)}}{s \cdot s} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      2. lift-PI.f32 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{-1}{1296} \cdot \frac{r \cdot r}{s \cdot \color{blue}{\mathsf{PI}\left(\right)}} + \frac{r \cdot \frac{1}{144}}{\mathsf{PI}\left(\right)}}{s \cdot s} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      3. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{-1}{1296} \cdot \frac{r \cdot r}{\color{blue}{s \cdot \mathsf{PI}\left(\right)}} + \frac{r \cdot \frac{1}{144}}{\mathsf{PI}\left(\right)}}{s \cdot s} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      4. lift-/.f32 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{-1}{1296} \cdot \color{blue}{\frac{r \cdot r}{s \cdot \mathsf{PI}\left(\right)}} + \frac{r \cdot \frac{1}{144}}{\mathsf{PI}\left(\right)}}{s \cdot s} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      5. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{-1}{1296} \cdot \frac{r \cdot r}{s \cdot \mathsf{PI}\left(\right)} + \frac{\color{blue}{r \cdot \frac{1}{144}}}{\mathsf{PI}\left(\right)}}{s \cdot s} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      6. lift-PI.f32 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{-1}{1296} \cdot \frac{r \cdot r}{s \cdot \mathsf{PI}\left(\right)} + \frac{r \cdot \frac{1}{144}}{\color{blue}{\mathsf{PI}\left(\right)}}}{s \cdot s} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      7. lift-/.f32 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{-1}{1296} \cdot \frac{r \cdot r}{s \cdot \mathsf{PI}\left(\right)} + \color{blue}{\frac{r \cdot \frac{1}{144}}{\mathsf{PI}\left(\right)}}}{s \cdot s} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\color{blue}{\frac{r \cdot \frac{1}{144}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{r \cdot r}{s \cdot \mathsf{PI}\left(\right)}}}{s \cdot s} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      9. lower-+.f32 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\color{blue}{\frac{r \cdot \frac{1}{144}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{r \cdot r}{s \cdot \mathsf{PI}\left(\right)}}}{s \cdot s} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      10. lift-/.f32 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{r \cdot \frac{1}{144}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \color{blue}{\frac{r \cdot r}{s \cdot \mathsf{PI}\left(\right)}}}{s \cdot s} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{r \cdot \frac{1}{144}}{\mathsf{PI}\left(\right)} + \color{blue}{\frac{\frac{-1}{1296} \cdot \left(r \cdot r\right)}{s \cdot \mathsf{PI}\left(\right)}}}{s \cdot s} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      12. lower-/.f32 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{r \cdot \frac{1}{144}}{\mathsf{PI}\left(\right)} + \color{blue}{\frac{\frac{-1}{1296} \cdot \left(r \cdot r\right)}{s \cdot \mathsf{PI}\left(\right)}}}{s \cdot s} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      13. lower-*.f32 66.3

          \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{r \cdot 0.006944444444444444}{\mathsf{PI}\left(\right)} + \frac{\color{blue}{-0.0007716049382716049 \cdot \left(r \cdot r\right)}}{s \cdot \mathsf{PI}\left(\right)}}{s \cdot s} + \left(\frac{0.125}{r \cdot \mathsf{PI}\left(\right)} + \frac{-0.041666666666666664}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

   7. Applied rewrites 66.3%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\color{blue}{\frac{r \cdot 0.006944444444444444}{\mathsf{PI}\left(\right)} + \frac{-0.0007716049382716049 \cdot \left(r \cdot r\right)}{s \cdot \mathsf{PI}\left(\right)}}}{s \cdot s} + \left(\frac{0.125}{r \cdot \mathsf{PI}\left(\right)} + \frac{-0.041666666666666664}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{0.75 \cdot e^{-\frac{r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)} \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right), \frac{0.125}{\mathsf{PI}\left(\right)}, s \cdot \left(r \cdot -0.041666666666666664\right)\right), \frac{1}{s \cdot \left(r \cdot \left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, \frac{e^{\frac{r}{-s}}}{r \cdot \left(\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot 8\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{\frac{\frac{r \cdot 0.006944444444444444}{\mathsf{PI}\left(\right)} + \frac{-0.0007716049382716049 \cdot \left(r \cdot r\right)}{s \cdot \mathsf{PI}\left(\right)}}{s \cdot s} + \left(\frac{0.125}{r \cdot \mathsf{PI}\left(\right)} + \frac{-0.041666666666666664}{s \cdot \mathsf{PI}\left(\right)}\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := s \cdot \mathsf{PI}\left(\right)\\ t_1 := s \cdot t\_0\\ t_2 := e^{\frac{r}{-s}}\\ t_3 := \frac{0.25 \cdot t\_2}{r \cdot \left(s \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}\\ \mathbf{if}\;t\_3 + \frac{0.75 \cdot e^{-\frac{r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \frac{0.125}{\mathsf{PI}\left(\right)}, s \cdot \left(r \cdot -0.041666666666666664\right)\right), \frac{1}{s \cdot \left(r \cdot t\_1\right)}, \frac{t\_2}{r \cdot \left(t\_0 \cdot 8\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 + \frac{\left(\frac{0.125}{r \cdot \mathsf{PI}\left(\right)} + \frac{-0.041666666666666664}{t\_0}\right) + \frac{r \cdot 0.006944444444444444}{t\_1}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* s (PI)))
        (t_1 (* s t_0))
        (t_2 (exp (/ r (- s))))
        (t_3 (/ (* 0.25 t_2) (* r (* s (* 2.0 (PI)))))))
   (if (<=
        (+ t_3 (/ (* 0.75 (exp (- (/ r (* s 3.0))))) (* r (* s (* (PI) 6.0)))))
        0.0)
     (fma
      (fma t_1 (/ 0.125 (PI)) (* s (* r -0.041666666666666664)))
      (/ 1.0 (* s (* r t_1)))
      (/ t_2 (* r (* t_0 8.0))))
     (+
      t_3
      (/
       (+
        (+ (/ 0.125 (* r (PI))) (/ -0.041666666666666664 t_0))
        (/ (* r 0.006944444444444444) 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))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} - \frac{\frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{\frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}\right)\right)\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{24} \cdot \frac{\frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \color{blue}{\frac{1}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{\color{blue}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{\color{blue}{{s}^{2}} \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      7. lower-+.f32 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)} \]

   5. Applied rewrites 3.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   7. Applied rewrites 3.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{\color{blue}{\left(r \cdot s\right) \cdot \mathsf{PI}\left(\right)}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      5. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{\color{blue}{\left(r \cdot s\right)} \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \color{blue}{\frac{\frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}}{r \cdot s}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      8. lower-/.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \color{blue}{\frac{\frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}}{r \cdot s}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      9. lower-/.f32 3.7

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\color{blue}{\frac{0.125}{\mathsf{PI}\left(\right)}}}{r \cdot s} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      10. lift-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}}{\color{blue}{r \cdot s}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}}{\color{blue}{s \cdot r}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      12. lower-*.f32 3.7

          \[\leadsto \mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{0.125}{\mathsf{PI}\left(\right)}}{\color{blue}{s \cdot r}} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   9. Applied rewrites 3.7%

      \[\leadsto \mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \color{blue}{\frac{\frac{0.125}{\mathsf{PI}\left(\right)}}{s \cdot r}} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right), \frac{0.125}{\mathsf{PI}\left(\right)}, s \cdot \left(r \cdot -0.041666666666666664\right)\right), \frac{1}{s \cdot \left(r \cdot \left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, \frac{e^{\frac{r}{-s}}}{r \cdot \left(\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot 8\right)}\right)} \]

   if 0.0 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 96.6%

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

   3. Taylor expanded in s around -inf

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{1296} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}}{s} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

   5. Applied rewrites 30.4%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.0007716049382716049, \frac{r \cdot r}{s \cdot \mathsf{PI}\left(\right)}, \frac{r \cdot 0.006944444444444444}{\mathsf{PI}\left(\right)}\right)}{s \cdot s} + \left(\frac{0.125}{r \cdot \mathsf{PI}\left(\right)} + \frac{-0.041666666666666664}{s \cdot \mathsf{PI}\left(\right)}\right)}{s}} \]

   6. Taylor expanded in r around 0

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{1}{144} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)}} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{\frac{1}{144} \cdot r}{{s}^{2} \cdot \mathsf{PI}\left(\right)}} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      2. lower-/.f32 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{\frac{1}{144} \cdot r}{{s}^{2} \cdot \mathsf{PI}\left(\right)}} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\color{blue}{r \cdot \frac{1}{144}}}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      4. lower-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\color{blue}{r \cdot \frac{1}{144}}}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{r \cdot \frac{1}{144}}{\color{blue}{\left(s \cdot s\right)} \cdot \mathsf{PI}\left(\right)} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{r \cdot \frac{1}{144}}{\color{blue}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      7. lower-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{r \cdot \frac{1}{144}}{\color{blue}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      8. lower-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{r \cdot \frac{1}{144}}{s \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} + \left(\frac{\frac{1}{8}}{r \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

      9. lower-PI.f32 53.6

         \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{r \cdot 0.006944444444444444}{s \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} + \left(\frac{0.125}{r \cdot \mathsf{PI}\left(\right)} + \frac{-0.041666666666666664}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

   8. Applied rewrites 53.6%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{r \cdot 0.006944444444444444}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} + \left(\frac{0.125}{r \cdot \mathsf{PI}\left(\right)} + \frac{-0.041666666666666664}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{0.75 \cdot e^{-\frac{r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right), \frac{0.125}{\mathsf{PI}\left(\right)}, s \cdot \left(r \cdot -0.041666666666666664\right)\right), \frac{1}{s \cdot \left(r \cdot \left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, \frac{e^{\frac{r}{-s}}}{r \cdot \left(\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot 8\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{\left(\frac{0.125}{r \cdot \mathsf{PI}\left(\right)} + \frac{-0.041666666666666664}{s \cdot \mathsf{PI}\left(\right)}\right) + \frac{r \cdot 0.006944444444444444}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := s \cdot \mathsf{PI}\left(\right)\\ t_1 := s \cdot t\_0\\ t_2 := e^{\frac{r}{-s}}\\ \mathbf{if}\;\frac{0.25 \cdot t\_2}{r \cdot \left(s \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{0.75 \cdot e^{-\frac{r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t\_1, \frac{0.125}{\mathsf{PI}\left(\right)}, s \cdot \left(r \cdot -0.041666666666666664\right)\right), \frac{1}{s \cdot \left(r \cdot t\_1\right)}, \frac{t\_2}{r \cdot \left(t\_0 \cdot 8\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.06944444444444445 \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s \cdot s} + \left(\frac{0.25}{r \cdot \mathsf{PI}\left(\right)} + \frac{-0.16666666666666666}{t\_0}\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* s (PI))) (t_1 (* s t_0)) (t_2 (exp (/ r (- s)))))
   (if (<=
        (+
         (/ (* 0.25 t_2) (* r (* s (* 2.0 (PI)))))
         (/ (* 0.75 (exp (- (/ r (* s 3.0))))) (* r (* s (* (PI) 6.0)))))
        0.0)
     (fma
      (fma t_1 (/ 0.125 (PI)) (* s (* r -0.041666666666666664)))
      (/ 1.0 (* s (* r t_1)))
      (/ t_2 (* r (* t_0 8.0))))
     (/
      (+
       (/ (* 0.06944444444444445 (/ r (PI))) (* s s))
       (+ (/ 0.25 (* r (PI))) (/ -0.16666666666666666 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))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} - \frac{\frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{\frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}\right)\right)\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{24} \cdot \frac{\frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \color{blue}{\frac{1}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{\color{blue}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{\color{blue}{{s}^{2}} \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      7. lower-+.f32 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)} \]

   5. Applied rewrites 3.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   7. Applied rewrites 3.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{\color{blue}{\left(r \cdot s\right) \cdot \mathsf{PI}\left(\right)}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      5. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{\color{blue}{\left(r \cdot s\right)} \cdot \mathsf{PI}\left(\right)} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \color{blue}{\frac{\frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}}{r \cdot s}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      8. lower-/.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \color{blue}{\frac{\frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}}{r \cdot s}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      9. lower-/.f32 3.7

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\color{blue}{\frac{0.125}{\mathsf{PI}\left(\right)}}}{r \cdot s} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      10. lift-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}}{\color{blue}{r \cdot s}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{\frac{1}{8}}{\mathsf{PI}\left(\right)}}{\color{blue}{s \cdot r}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      12. lower-*.f32 3.7

          \[\leadsto \mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{0.125}{\mathsf{PI}\left(\right)}}{\color{blue}{s \cdot r}} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   9. Applied rewrites 3.7%

      \[\leadsto \mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \color{blue}{\frac{\frac{0.125}{\mathsf{PI}\left(\right)}}{s \cdot r}} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right), \frac{0.125}{\mathsf{PI}\left(\right)}, s \cdot \left(r \cdot -0.041666666666666664\right)\right), \frac{1}{s \cdot \left(r \cdot \left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, \frac{e^{\frac{r}{-s}}}{r \cdot \left(\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot 8\right)}\right)} \]

   if 0.0 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 96.6%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. times-frac N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f32 N/A

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in s around -inf

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

   7. Applied rewrites 47.4%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{0.75 \cdot e^{-\frac{r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right), \frac{0.125}{\mathsf{PI}\left(\right)}, s \cdot \left(r \cdot -0.041666666666666664\right)\right), \frac{1}{s \cdot \left(r \cdot \left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, \frac{e^{\frac{r}{-s}}}{r \cdot \left(\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot 8\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.06944444444444445 \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s \cdot s} + \left(\frac{0.25}{r \cdot \mathsf{PI}\left(\right)} + \frac{-0.16666666666666666}{s \cdot \mathsf{PI}\left(\right)}\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := s \cdot \mathsf{PI}\left(\right)\\ t_1 := s \cdot t\_0\\ t_2 := e^{\frac{r}{-s}}\\ \mathbf{if}\;\frac{0.25 \cdot t\_2}{r \cdot \left(s \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{0.75 \cdot e^{-\frac{r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)} \leq 1.0000000168623835 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), -0.041666666666666664 \cdot \left(r \cdot s\right), 0.125 \cdot t\_1\right), \frac{1}{t\_0 \cdot \left(r \cdot t\_1\right)}, \frac{t\_2}{r \cdot \left(t\_0 \cdot 8\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.06944444444444445 \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s \cdot s} + \left(\frac{0.25}{r \cdot \mathsf{PI}\left(\right)} + \frac{-0.16666666666666666}{t\_0}\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* s (PI))) (t_1 (* s t_0)) (t_2 (exp (/ r (- s)))))
   (if (<=
        (+
         (/ (* 0.25 t_2) (* r (* s (* 2.0 (PI)))))
         (/ (* 0.75 (exp (- (/ r (* s 3.0))))) (* r (* s (* (PI) 6.0)))))
        1.0000000168623835e-16)
     (fma
      (fma (PI) (* -0.041666666666666664 (* r s)) (* 0.125 t_1))
      (/ 1.0 (* t_0 (* r t_1)))
      (/ t_2 (* r (* t_0 8.0))))
     (/
      (+
       (/ (* 0.06944444444444445 (/ r (PI))) (* s s))
       (+ (/ 0.25 (* r (PI))) (/ -0.16666666666666666 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))) < 1.00000002e-16

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} - \frac{\frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{\frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}\right)\right)\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{24} \cdot \frac{\frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \color{blue}{\frac{1}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{\color{blue}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{\color{blue}{{s}^{2}} \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      7. lower-+.f32 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)} \]

   5. Applied rewrites 3.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   7. Applied rewrites 3.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   9. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \left(s \cdot r\right) \cdot -0.041666666666666664, 0.125 \cdot \left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)\right), \frac{1}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(r \cdot \left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, \frac{e^{\frac{r}{-s}}}{r \cdot \left(\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot 8\right)}\right)} \]

   if 1.00000002e-16 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 97.5%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. times-frac N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f32 N/A

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in s around -inf

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

   7. Applied rewrites 50.5%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{0.75 \cdot e^{-\frac{r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)} \leq 1.0000000168623835 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), -0.041666666666666664 \cdot \left(r \cdot s\right), 0.125 \cdot \left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)\right), \frac{1}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(r \cdot \left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, \frac{e^{\frac{r}{-s}}}{r \cdot \left(\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot 8\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.06944444444444445 \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s \cdot s} + \left(\frac{0.25}{r \cdot \mathsf{PI}\left(\right)} + \frac{-0.16666666666666666}{s \cdot \mathsf{PI}\left(\right)}\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := s \cdot \mathsf{PI}\left(\right)\\ t_1 := s \cdot t\_0\\ t_2 := e^{\frac{r}{-s}}\\ \mathbf{if}\;\frac{0.25 \cdot t\_2}{r \cdot \left(s \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{0.75 \cdot e^{-\frac{r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(r, t\_0 \cdot -0.041666666666666664, 0.125 \cdot t\_1\right), \frac{1}{t\_0 \cdot \left(r \cdot t\_1\right)}, \frac{t\_2}{\left(r \cdot t\_0\right) \cdot 8}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.06944444444444445 \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s \cdot s} + \left(\frac{0.25}{r \cdot \mathsf{PI}\left(\right)} + \frac{-0.16666666666666666}{t\_0}\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* s (PI))) (t_1 (* s t_0)) (t_2 (exp (/ r (- s)))))
   (if (<=
        (+
         (/ (* 0.25 t_2) (* r (* s (* 2.0 (PI)))))
         (/ (* 0.75 (exp (- (/ r (* s 3.0))))) (* r (* s (* (PI) 6.0)))))
        0.0)
     (fma
      (fma r (* t_0 -0.041666666666666664) (* 0.125 t_1))
      (/ 1.0 (* t_0 (* r t_1)))
      (/ t_2 (* (* r t_0) 8.0)))
     (/
      (+
       (/ (* 0.06944444444444445 (/ r (PI))) (* s s))
       (+ (/ 0.25 (* r (PI))) (/ -0.16666666666666666 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))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} - \frac{\frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{\frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}\right)\right)\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{24} \cdot \frac{\frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \color{blue}{\frac{1}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{\color{blue}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{\color{blue}{{s}^{2}} \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      7. lower-+.f32 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)} \]

   5. Applied rewrites 3.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(r, \left(s \cdot \mathsf{PI}\left(\right)\right) \cdot -0.041666666666666664, 0.125 \cdot \left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)\right), \frac{1}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(r \cdot \left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, \frac{e^{\frac{r}{-s}}}{\left(r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 8}\right)} \]

   if 0.0 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 96.6%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. times-frac N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f32 N/A

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in s around -inf

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

   7. Applied rewrites 47.4%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{0.75 \cdot e^{-\frac{r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(r, \left(s \cdot \mathsf{PI}\left(\right)\right) \cdot -0.041666666666666664, 0.125 \cdot \left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)\right), \frac{1}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot \left(r \cdot \left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, \frac{e^{\frac{r}{-s}}}{\left(r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 8}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.06944444444444445 \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s \cdot s} + \left(\frac{0.25}{r \cdot \mathsf{PI}\left(\right)} + \frac{-0.16666666666666666}{s \cdot \mathsf{PI}\left(\right)}\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 13.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := s \cdot \mathsf{PI}\left(\right)\\ t_1 := e^{\frac{r}{-s}}\\ t_2 := r \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;\frac{0.25 \cdot t\_1}{r \cdot \left(s \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{0.75 \cdot e^{-\frac{r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)} \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \frac{0.125}{r \cdot t\_0}, \mathsf{fma}\left(-0.041666666666666664, \frac{1}{s \cdot t\_0}, \frac{0.125}{s \cdot t\_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.06944444444444445 \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s \cdot s} + \left(\frac{0.25}{t\_2} + \frac{-0.16666666666666666}{t\_0}\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* s (PI))) (t_1 (exp (/ r (- s)))) (t_2 (* r (PI))))
   (if (<=
        (+
         (/ (* 0.25 t_1) (* r (* s (* 2.0 (PI)))))
         (/ (* 0.75 (exp (- (/ r (* s 3.0))))) (* r (* s (* (PI) 6.0)))))
        3.999999999279835e-23)
     (fma
      t_1
      (/ 0.125 (* r t_0))
      (fma -0.041666666666666664 (/ 1.0 (* s t_0)) (/ 0.125 (* s t_2))))
     (/
      (+
       (/ (* 0.06944444444444445 (/ r (PI))) (* s s))
       (+ (/ 0.25 t_2) (/ -0.16666666666666666 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))) < 4e-23

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} - \frac{\frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{\frac{1}{24} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}\right)\right)\right)} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{24} \cdot \frac{\frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \color{blue}{\frac{1}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{\color{blue}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{\color{blue}{{s}^{2}} \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

      7. lower-+.f32 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{24} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)} \]

   5. Applied rewrites 3.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   7. Applied rewrites 3.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{-0.041666666666666664}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      4. lift-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right) \]

      5. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{\frac{-1}{24}}{s \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{\frac{-1}{24}}{\color{blue}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      7. lift-/.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \color{blue}{\frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      8. lift-/.f32 N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \color{blue}{\frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \color{blue}{\frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}\right) \]

      10. lift-/.f32 N/A

          \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \color{blue}{\frac{\frac{-1}{24}}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      11. div-inv N/A

          \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \color{blue}{\frac{-1}{24} \cdot \frac{1}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      12. lower-fma.f32 N/A

          \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \color{blue}{\mathsf{fma}\left(\frac{-1}{24}, \frac{1}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)}\right) \]

      13. lower-/.f32 7.1

          \[\leadsto \mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \mathsf{fma}\left(-0.041666666666666664, \color{blue}{\frac{1}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      14. lift-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \mathsf{fma}\left(\frac{-1}{24}, \frac{1}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \mathsf{fma}\left(\frac{-1}{24}, \frac{1}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{\color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot r}}\right)\right) \]

      16. lift-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \mathsf{fma}\left(\frac{-1}{24}, \frac{1}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{\color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot r}\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \mathsf{fma}\left(\frac{-1}{24}, \frac{1}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{\frac{1}{8}}{\color{blue}{s \cdot \left(\mathsf{PI}\left(\right) \cdot r\right)}}\right)\right) \]

   9. Applied rewrites 7.1%

      \[\leadsto \mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \color{blue}{\mathsf{fma}\left(-0.041666666666666664, \frac{1}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{0.125}{s \cdot \left(\mathsf{PI}\left(\right) \cdot r\right)}\right)}\right) \]

   if 4e-23 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 97.0%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. times-frac N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f32 N/A

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in s around -inf

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

   7. Applied rewrites 48.9%

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

4. Final simplification 10.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{0.75 \cdot e^{-\frac{r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)} \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(e^{\frac{r}{-s}}, \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \mathsf{fma}\left(-0.041666666666666664, \frac{1}{s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}, \frac{0.125}{s \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.06944444444444445 \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s \cdot s} + \left(\frac{0.25}{r \cdot \mathsf{PI}\left(\right)} + \frac{-0.16666666666666666}{s \cdot \mathsf{PI}\left(\right)}\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

   1. lift-neg.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. times-frac N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-*.f32 N/A

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

   2. distribute-frac-neg N/A

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

   3. distribute-frac-neg2 N/A

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

   4. lower-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

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

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 9: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

   1. lift-neg.f32 N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f32 N/A

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

   4. frac-2neg N/A

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

   5. lift-neg.f32 N/A

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

   6. remove-double-neg N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f32 N/A

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

   12. metadata-eval 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 10: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.6%

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

Alternative 11: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 96.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-exp.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-/.f32 N/A

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 12: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 96.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-exp.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-/.f32 N/A

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

6. Applied rewrites 96.9%

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

7. Final simplification 96.9%

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

Alternative 13: 9.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ r (- s)))) (* r (* s (* 2.0 (PI)))))
  (/ 0.125 (* r (* s (PI))))))

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.0

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.125}{r \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} \]

5. Applied rewrites 8.0%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

6. Final simplification 8.0%

   \[\leadsto \frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)} + \frac{0.125}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]
7. Add Preprocessing

Alternative 14: 10.0% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. times-frac N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around -inf

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

7. Applied rewrites 8.0%

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

Alternative 15: 4.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := s \cdot \mathsf{PI}\left(\right)\\ \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{s \cdot t\_0}, \frac{0.25}{t\_0}\right)}{r} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* s (PI))))
   (/ (fma -0.16666666666666666 (/ r (* s t_0)) (/ 0.25 t_0)) r)))

Derivation

1. Initial program 99.7%

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

   1. lift-*.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. times-frac N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-PI.f32 7.6

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

7. Applied rewrites 7.6%

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

Alternative 16: 9.0% accurate, 7.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 7.6

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

5. Applied rewrites 7.6%

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

   1. lift-PI.f32 N/A

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

   2. associate-*r* N/A

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

   3. lift-*.f32 N/A

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

   4. lower-*.f32 7.6

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

7. Applied rewrites 7.6%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 7.6

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 7.6

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

9. Applied rewrites 7.6%

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

10. Final simplification 7.6%

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

Alternative 17: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 7.6

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

5. Applied rewrites 7.6%

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

Reproduce

herbie shell --seed 2024216 
(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))))