Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 12.5s

Alternatives: 15

Speedup: N/A×

Specification

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

Math

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

FPCore

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \frac{0.125}{e^{\frac{r}{s}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r\right)} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(t\_0 \cdot s\right) \cdot \left(\left(t\_0 \cdot r\right) \cdot 6\right)} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (sqrt (PI))))
   (+
    (/ 0.125 (* (exp (/ r s)) (* (* (PI) s) r)))
    (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* t_0 s) (* (* t_0 r) 6.0))))))

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. times-frac N/A

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

6. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-sqr-sqrt N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-sqrt.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lower-sqrt.f32 99.6

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

8. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-*.f32 99.6

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

10. Applied rewrites 99.6%

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

Alternative 2: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \frac{0.125}{e^{\frac{r}{s}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r\right)} + \frac{0.75 \cdot e^{\frac{r}{-3 \cdot s}}}{6 \cdot \left(\left(\left(s \cdot t\_0\right) \cdot t\_0\right) \cdot r\right)} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (sqrt (PI))))
   (+
    (/ 0.125 (* (exp (/ r s)) (* (* (PI) s) r)))
    (/ (* 0.75 (exp (/ r (* -3.0 s)))) (* 6.0 (* (* (* s t_0) t_0) r))))))

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. times-frac N/A

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

6. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-sqr-sqrt N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-sqrt.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lower-sqrt.f32 99.6

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

8. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.6

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

10. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. times-frac N/A

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

6. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-sqr-sqrt N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-sqrt.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lower-sqrt.f32 99.6

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

8. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lift-sqrt.f32 N/A

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

   7. lift-sqrt.f32 N/A

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

   8. rem-square-sqrt N/A

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

   9. associate-*r* N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 99.6

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

10. Applied rewrites 99.6%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r\\ \frac{0.125}{e^{\frac{r}{s}} \cdot t\_0} + \frac{0.75 \cdot e^{\frac{r}{-3 \cdot s}}}{6 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (* (PI) s) r)))
   (+
    (/ 0.125 (* (exp (/ r s)) t_0))
    (/ (* 0.75 (exp (/ r (* -3.0 s)))) (* 6.0 t_0)))))

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. times-frac N/A

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

6. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.6

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

8. Applied rewrites 99.6%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r\\ \frac{0.125}{e^{\frac{r}{s}} \cdot t\_0} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}} \cdot 0.125}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (* (PI) s) r)))
   (+
    (/ 0.125 (* (exp (/ r s)) t_0))
    (/ (* (exp (* -0.3333333333333333 (/ r s))) 0.125) t_0))))

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. times-frac N/A

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

6. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

8. Applied rewrites 99.6%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r\\ 0.125 \cdot \left(\frac{e^{\frac{-r}{s}}}{t\_0} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (* (PI) s) r)))
   (*
    0.125
    (+
     (/ (exp (/ (- r) s)) t_0)
     (/ (exp (* -0.3333333333333333 (/ r s))) t_0)))))

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. times-frac N/A

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

6. Applied rewrites 99.6%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. div-inv N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. times-frac N/A

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

   8. metadata-eval N/A

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

   9. distribute-lft-out N/A

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

   10. lower-*.f32 N/A

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

8. Applied rewrites 99.5%

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

Alternative 7: 9.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot s\\ \frac{0.125}{e^{\frac{r}{s}} \cdot \left(t\_0 \cdot r\right)} + \frac{\frac{0.125}{\mathsf{PI}\left(\right) \cdot r} - \frac{0.041666666666666664}{t\_0}}{s} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (PI) s)))
   (+
    (/ 0.125 (* (exp (/ r s)) (* t_0 r)))
    (/ (- (/ 0.125 (* (PI) r)) (/ 0.041666666666666664 t_0)) s))))

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. times-frac N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 12.0

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

9. Applied rewrites 12.0%

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

Alternative 8: 9.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot s\\ \frac{0.125}{e^{\frac{r}{s}} \cdot \left(t\_0 \cdot r\right)} + \frac{\frac{0.125}{t\_0}}{r} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (PI) s)))
   (+ (/ 0.125 (* (exp (/ r s)) (* t_0 r))) (/ (/ 0.125 t_0) r))))

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. times-frac N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 11.4

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

9. Applied rewrites 11.4%

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

Alternative 9: 9.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r\\ \frac{0.125}{e^{\frac{r}{s}} \cdot t\_0} + \frac{0.75}{6 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (* (PI) s) r)))
   (+ (/ 0.125 (* (exp (/ r s)) t_0)) (/ 0.75 (* 6.0 t_0)))))

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. times-frac N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in s around inf

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

9. Applied rewrites 11.3%

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

Alternative 10: 9.1% accurate, 6.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 11.3%

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

Alternative 11: 9.0% accurate, 8.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 10.8

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

5. Applied rewrites 10.8%

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

7. Applied rewrites 10.8%

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

Alternative 12: 9.0% accurate, 9.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 10.8

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

5. Applied rewrites 10.8%

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

7. Applied rewrites 10.8%

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

Alternative 13: 9.0% accurate, 10.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 10.8

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

5. Applied rewrites 10.8%

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

Alternative 14: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 10.8

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

5. Applied rewrites 10.8%

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

7. Applied rewrites 10.8%

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

9. Applied rewrites 10.8%

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

Alternative 15: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 10.8

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

5. Applied rewrites 10.8%

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

7. Applied rewrites 10.8%

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

9. Applied rewrites 10.8%

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

Reproduce

herbie shell --seed 2024324 
(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))))