Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 14.4s

Alternatives: 30

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.125 (exp (/ (- r) s))) (* (* (PI) r) s))
  (/ (* (/ 0.125 (* (PI) s)) (exp (/ (/ r -3.0) 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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{0.125}{\mathsf{PI}\left(\right) \cdot s} \cdot e^{\frac{\frac{r}{-3}}{s}}}{r}} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-neg.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lift-exp.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. lift-*.f32 N/A

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

6. Applied rewrites 99.5%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* 0.125 (/ (exp (/ (- r) s)) (* (* (PI) s) r)))
  (/ (* 0.75 (exp (/ r (* -3.0 s)))) (* (* (* 6.0 (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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-*l* N/A

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

   8. times-frac N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. metadata-eval N/A

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

   9. lower-*.f32 99.5

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

6. Applied rewrites 99.5%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* 0.125 (/ (exp (/ (- r) s)) (* (* (PI) s) r)))
  (/ (* 0.75 (exp (/ r (* -3.0 s)))) (* (* s r) (* 6.0 (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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-*l* N/A

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

   8. times-frac N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. metadata-eval N/A

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

   9. lower-*.f32 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 99.5

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

8. Applied rewrites 99.5%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. associate-*l* N/A

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

   10. times-frac N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

4. Applied rewrites 99.5%

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

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  0.125
  (/ (exp (/ (/ r -3.0) s)) (* (PI) (* s r)))
  (* 0.125 (/ (exp (/ (- r) s)) (* (* (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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. times-frac N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

4. Applied rewrites 99.5%

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

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (fma
   (/ (exp (/ (/ r -3.0) s)) (PI))
   0.125
   (* (/ (exp (/ (- r) s)) (PI)) 0.125))
  (* 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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/r* N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*l* N/A

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

   12. associate-/r* N/A

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

4. Applied rewrites 99.4%

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

Alternative 7: 10.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{\frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, 0.0007716049382716049, -0.006944444444444444 \cdot r\right)}{\mathsf{PI}\left(\right)}}{s} + \frac{0.041666666666666664}{\mathsf{PI}\left(\right)}}{s} - \frac{0.125}{\mathsf{PI}\left(\right) \cdot r}}{-s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* 0.125 (/ (exp (/ (- r) s)) (* (* (PI) s) r)))
  (/
   (-
    (/
     (+
      (/
       (/
        (fma (* r (/ r s)) 0.0007716049382716049 (* -0.006944444444444444 r))
        (PI))
       s)
      (/ 0.041666666666666664 (PI)))
     s)
    (/ 0.125 (* (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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-*l* N/A

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

   8. times-frac N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. metadata-eval N/A

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

   9. lower-*.f32 99.5

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f32 N/A

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

9. Applied rewrites 9.5%

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

10. Final simplification 9.5%

    \[\leadsto 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{\frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, 0.0007716049382716049, -0.006944444444444444 \cdot r\right)}{\mathsf{PI}\left(\right)}}{s} + \frac{0.041666666666666664}{\mathsf{PI}\left(\right)}}{s} - \frac{0.125}{\mathsf{PI}\left(\right) \cdot r}}{-s} \]
11. Add Preprocessing

Alternative 8: 10.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.041666666666666664 + \frac{-0.004629629629629629 \cdot r}{s}}{s \cdot s}, r, \frac{-0.25}{s}\right), r, 0.75\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* 0.125 (/ (exp (/ (- r) s)) (* (* (PI) s) r)))
  (/
   (fma
    (fma
     (/ (+ 0.041666666666666664 (/ (* -0.004629629629629629 r) s)) (* s s))
     r
     (/ -0.25 s))
    r
    0.75)
   (* (* (* 6.0 (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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-*l* N/A

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

   8. times-frac N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 8.2%

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

8. Taylor expanded in r around 0

   \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4} + r \cdot \left(r \cdot \left(\frac{-1}{216} \cdot \frac{r}{{s}^{3}} + \frac{1}{24} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{4} \cdot \frac{1}{s}\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{r \cdot \left(r \cdot \left(\frac{-1}{216} \cdot \frac{r}{{s}^{3}} + \frac{1}{24} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{4} \cdot \frac{1}{s}\right) + \frac{3}{4}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\left(r \cdot \left(\frac{-1}{216} \cdot \frac{r}{{s}^{3}} + \frac{1}{24} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{4} \cdot \frac{1}{s}\right) \cdot r} + \frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(r \cdot \left(\frac{-1}{216} \cdot \frac{r}{{s}^{3}} + \frac{1}{24} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

10. Applied rewrites 9.5%

    \[\leadsto 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.041666666666666664 + \frac{-0.004629629629629629 \cdot r}{s}}{s \cdot s}, r, \frac{-0.25}{s}\right), r, 0.75\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
11. Add Preprocessing

Alternative 9: 10.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{0.125}{\mathsf{PI}\left(\right) \cdot s}, e^{\frac{-r}{s}}, \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r}{s}, 0.041666666666666664, -0.25\right)}{s}, r, 0.75\right)}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s}\right)}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (fma
   (/ 0.125 (* (PI) s))
   (exp (/ (- r) s))
   (/
    (fma (/ (fma (/ r s) 0.041666666666666664 -0.25) s) r 0.75)
    (* (* 6.0 (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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-*l* N/A

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

   8. times-frac N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 8.2%

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

8. Taylor expanded in r around 0

   \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4} + r \cdot \left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{r \cdot \left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) + \frac{3}{4}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) \cdot r} + \frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   4. fp-cancel-sub-sign-inv N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot \frac{r}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \frac{1}{s}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   5. +-commutative N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \frac{1}{s} + \frac{1}{24} \cdot \frac{r}{{s}^{2}}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{4}} \cdot \frac{1}{s} + \frac{1}{24} \cdot \frac{r}{{s}^{2}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   7. associate-*r/ N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} \cdot 1}{s}} + \frac{1}{24} \cdot \frac{r}{{s}^{2}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   8. metadata-eval N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{4}}}{s} + \frac{1}{24} \cdot \frac{r}{{s}^{2}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   9. associate-*r/ N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4}}{s} + \color{blue}{\frac{\frac{1}{24} \cdot r}{{s}^{2}}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   10. unpow2 N/A

       \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4}}{s} + \frac{\frac{1}{24} \cdot r}{\color{blue}{s \cdot s}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   11. associate-/r* N/A

       \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4}}{s} + \color{blue}{\frac{\frac{\frac{1}{24} \cdot r}{s}}{s}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   12. div-add-rev N/A

       \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}{s}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   13. lower-/.f32 N/A

       \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}{s}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   14. lower-+.f32 N/A

       \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   15. lower-/.f32 N/A

       \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \color{blue}{\frac{\frac{1}{24} \cdot r}{s}}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   16. lower-*.f32 9.4

       \[\leadsto 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{-0.25 + \frac{\color{blue}{0.041666666666666664 \cdot r}}{s}}{s}, r, 0.75\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

10. Applied rewrites 9.4%

    \[\leadsto 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(\frac{-0.25 + \frac{0.041666666666666664 \cdot r}{s}}{s}, r, 0.75\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

12. Applied rewrites 9.4%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{0.125}{\mathsf{PI}\left(\right) \cdot s}, e^{\frac{-r}{s}}, \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r}{s}, 0.041666666666666664, -0.25\right)}{s}, r, 0.75\right)}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s}\right)}{r}} \]
13. Add Preprocessing

Alternative 10: 10.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, 0.125, \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r}{s}, 0.041666666666666664, -0.25\right)}{s}, r, 0.75\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ (exp (/ (- r) s)) (* (* (PI) s) r))
  0.125
  (/
   (fma (/ (fma (/ r s) 0.041666666666666664 -0.25) s) r 0.75)
   (* (* (* 6.0 (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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-*l* N/A

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

   8. times-frac N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 8.2%

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

8. Taylor expanded in r around 0

   \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4} + r \cdot \left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{r \cdot \left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) + \frac{3}{4}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) \cdot r} + \frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   4. fp-cancel-sub-sign-inv N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot \frac{r}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \frac{1}{s}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   5. +-commutative N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \frac{1}{s} + \frac{1}{24} \cdot \frac{r}{{s}^{2}}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{4}} \cdot \frac{1}{s} + \frac{1}{24} \cdot \frac{r}{{s}^{2}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   7. associate-*r/ N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} \cdot 1}{s}} + \frac{1}{24} \cdot \frac{r}{{s}^{2}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   8. metadata-eval N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{4}}}{s} + \frac{1}{24} \cdot \frac{r}{{s}^{2}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   9. associate-*r/ N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4}}{s} + \color{blue}{\frac{\frac{1}{24} \cdot r}{{s}^{2}}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   10. unpow2 N/A

       \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4}}{s} + \frac{\frac{1}{24} \cdot r}{\color{blue}{s \cdot s}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   11. associate-/r* N/A

       \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4}}{s} + \color{blue}{\frac{\frac{\frac{1}{24} \cdot r}{s}}{s}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   12. div-add-rev N/A

       \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}{s}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   13. lower-/.f32 N/A

       \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}{s}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   14. lower-+.f32 N/A

       \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   15. lower-/.f32 N/A

       \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \color{blue}{\frac{\frac{1}{24} \cdot r}{s}}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   16. lower-*.f32 9.4

       \[\leadsto 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{-0.25 + \frac{\color{blue}{0.041666666666666664 \cdot r}}{s}}{s}, r, 0.75\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

10. Applied rewrites 9.4%

    \[\leadsto 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(\frac{-0.25 + \frac{0.041666666666666664 \cdot r}{s}}{s}, r, 0.75\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
11. Step-by-step derivation

    1. lift-+.f32 N/A

       \[\leadsto \color{blue}{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} \]

    2. lift-*.f32 N/A

       \[\leadsto \color{blue}{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

    3. *-commutative N/A

       \[\leadsto \color{blue}{\frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \cdot \frac{1}{8}} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

    4. lower-fma.f32 9.4

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, 0.125, \frac{\mathsf{fma}\left(\frac{-0.25 + \frac{0.041666666666666664 \cdot r}{s}}{s}, r, 0.75\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}\right)} \]

12. Applied rewrites 9.4%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, 0.125, \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r}{s}, 0.041666666666666664, -0.25\right)}{s}, r, 0.75\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}\right)} \]
13. Add Preprocessing

Alternative 11: 10.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{\frac{-r}{s}} \cdot 0.125}{\mathsf{PI}\left(\right)} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r}{s}, 0.041666666666666664, -0.25\right)}{s}, r, 0.75\right)}{6 \cdot \mathsf{PI}\left(\right)}}{s \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/ (* (exp (/ (- r) s)) 0.125) (PI))
   (/
    (fma (/ (fma (/ r s) 0.041666666666666664 -0.25) s) r 0.75)
    (* 6.0 (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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-*l* N/A

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

   8. times-frac N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 8.2%

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

8. Taylor expanded in r around 0

   \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4} + r \cdot \left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{r \cdot \left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) + \frac{3}{4}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) \cdot r} + \frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   4. fp-cancel-sub-sign-inv N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot \frac{r}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \frac{1}{s}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   5. +-commutative N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \frac{1}{s} + \frac{1}{24} \cdot \frac{r}{{s}^{2}}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{4}} \cdot \frac{1}{s} + \frac{1}{24} \cdot \frac{r}{{s}^{2}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   7. associate-*r/ N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} \cdot 1}{s}} + \frac{1}{24} \cdot \frac{r}{{s}^{2}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   8. metadata-eval N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{4}}}{s} + \frac{1}{24} \cdot \frac{r}{{s}^{2}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   9. associate-*r/ N/A

      \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4}}{s} + \color{blue}{\frac{\frac{1}{24} \cdot r}{{s}^{2}}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   10. unpow2 N/A

       \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4}}{s} + \frac{\frac{1}{24} \cdot r}{\color{blue}{s \cdot s}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   11. associate-/r* N/A

       \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4}}{s} + \color{blue}{\frac{\frac{\frac{1}{24} \cdot r}{s}}{s}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   12. div-add-rev N/A

       \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}{s}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   13. lower-/.f32 N/A

       \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}{s}}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   14. lower-+.f32 N/A

       \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   15. lower-/.f32 N/A

       \[\leadsto \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \color{blue}{\frac{\frac{1}{24} \cdot r}{s}}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   16. lower-*.f32 9.4

       \[\leadsto 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{-0.25 + \frac{\color{blue}{0.041666666666666664 \cdot r}}{s}}{s}, r, 0.75\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

10. Applied rewrites 9.4%

    \[\leadsto 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(\frac{-0.25 + \frac{0.041666666666666664 \cdot r}{s}}{s}, r, 0.75\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
11. Step-by-step derivation

    1. lift-+.f32 N/A

       \[\leadsto \color{blue}{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} \]

    2. lift-*.f32 N/A

       \[\leadsto \color{blue}{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

    3. lift-/.f32 N/A

       \[\leadsto \frac{1}{8} \cdot \color{blue}{\frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

    4. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

    5. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot e^{\frac{-r}{s}}}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

    6. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot e^{\frac{-r}{s}}}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot s\right)} \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

    7. associate-*l* N/A

       \[\leadsto \frac{\frac{1}{8} \cdot e^{\frac{-r}{s}}}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

    8. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot e^{\frac{-r}{s}}}{\mathsf{PI}\left(\right) \cdot \color{blue}{\left(s \cdot r\right)}} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

    9. associate-/r* N/A

       \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8} \cdot e^{\frac{-r}{s}}}{\mathsf{PI}\left(\right)}}{s \cdot r}} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

    10. lift-/.f32 N/A

        \[\leadsto \frac{\frac{\frac{1}{8} \cdot e^{\frac{-r}{s}}}{\mathsf{PI}\left(\right)}}{s \cdot r} + \color{blue}{\frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{24} \cdot r}{s}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} \]

12. Applied rewrites 9.3%

    \[\leadsto \color{blue}{\frac{\frac{e^{\frac{-r}{s}} \cdot 0.125}{\mathsf{PI}\left(\right)} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r}{s}, 0.041666666666666664, -0.25\right)}{s}, r, 0.75\right)}{6 \cdot \mathsf{PI}\left(\right)}}{s \cdot r}} \]
13. Add Preprocessing

Alternative 12: 9.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot r\\ t_1 := \frac{r}{\mathsf{PI}\left(\right)}\\ \frac{\frac{\left(\frac{\mathsf{fma}\left(t\_1, -0.06944444444444445, \frac{\left(t\_1 \cdot r\right) \cdot 0.021604938271604937}{s}\right)}{s} + \frac{0.16666666666666666}{\mathsf{PI}\left(\right)}\right) \cdot t\_0 - s \cdot 0.25}{t\_0 \cdot s}}{-s} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (PI) r)) (t_1 (/ r (PI))))
   (/
    (/
     (-
      (*
       (+
        (/
         (fma
          t_1
          -0.06944444444444445
          (/ (* (* t_1 r) 0.021604938271604937) s))
         s)
        (/ 0.16666666666666666 (PI)))
       t_0)
      (* s 0.25))
     (* t_0 s))
    (- 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

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

5. Applied rewrites 8.7%

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

7. Applied rewrites 8.8%

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

8. Final simplification 8.8%

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

Alternative 13: 9.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\mathsf{PI}\left(\right)}\\ \frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot r, \frac{\mathsf{fma}\left(t\_0, -0.06944444444444445, \frac{\left(t\_0 \cdot r\right) \cdot 0.021604938271604937}{s}\right)}{-s} - \frac{0.16666666666666666}{\mathsf{PI}\left(\right)}, s \cdot 0.25\right)}{\left(\left(s \cdot r\right) \cdot \mathsf{PI}\left(\right)\right) \cdot s} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ r (PI))))
   (/
    (fma
     (* (PI) r)
     (-
      (/
       (fma t_0 -0.06944444444444445 (/ (* (* t_0 r) 0.021604938271604937) s))
       (- s))
      (/ 0.16666666666666666 (PI)))
     (* s 0.25))
    (* (* (* s r) (PI)) 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

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

5. Applied rewrites 8.7%

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

7. Applied rewrites 8.7%

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

8. Final simplification 8.7%

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

Alternative 14: 9.7% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/
    (-
     (/
      (/
       (/
        (fma (* 0.021604938271604937 r) r (* (* -0.06944444444444445 r) s))
        (PI))
       s)
      (- s))
     (/ 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

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

5. Applied rewrites 8.7%

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

6. Taylor expanded in s around 0

   \[\leadsto \frac{\frac{\frac{\frac{\frac{-5}{72} \cdot \frac{r \cdot s}{\mathsf{PI}\left(\right)} + \frac{7}{324} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s}}{-s} - \frac{\frac{1}{6}}{\mathsf{PI}\left(\right)}}{-s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{-s} \]
7. Step-by-step derivation

8. Applied rewrites 8.7%

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

9. Final simplification 8.7%

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

Alternative 15: 9.9% accurate, 3.3× speedup

Math

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

FPCore

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

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

5. Applied rewrites 8.7%

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

6. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{r \cdot \left(\frac{-5}{72} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{r}}{-\color{blue}{s}} \]
7. Step-by-step derivation

8. Applied rewrites 8.6%

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

Alternative 16: 9.9% accurate, 3.6× speedup

Math

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

FPCore

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

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f32 N/A

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

5. Applied rewrites 8.6%

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

6. Final simplification 8.6%

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

Alternative 17: 9.9% accurate, 3.7× speedup

Math

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

FPCore

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

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{0.125}{\mathsf{PI}\left(\right) \cdot s} \cdot e^{\frac{\frac{r}{-3}}{s}}}{r}} \]

5. Taylor expanded in s around inf

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

7. Applied rewrites 8.6%

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

Alternative 18: 9.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, 0.06944444444444445, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{1}{s}, \frac{0.25}{r}\right)}{\mathsf{PI}\left(\right)}\right)}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (fma
   (/ r (* (* s s) (PI)))
   0.06944444444444445
   (/ (fma -0.16666666666666666 (/ 1.0 s) (/ 0.25 r)) (PI)))
  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

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 8.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, 0.06944444444444445, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{1}{s}, \frac{0.25}{r}\right)}{\mathsf{PI}\left(\right)}\right)}{s}} \]
6. Add Preprocessing

Alternative 19: 9.0% accurate, 4.8× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (/ (/ (fma (/ (/ r s) s) -0.16666666666666666 (/ 0.25 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. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 7.9%

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

Alternative 20: 9.0% accurate, 5.4× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (/ (fma (/ r (* (* s s) (PI))) -0.16666666666666666 (/ 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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{0.125}{\mathsf{PI}\left(\right) \cdot s} \cdot e^{\frac{\frac{r}{-3}}{s}}}{r}} \]

5. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 7.9

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

7. Applied rewrites 7.9%

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

Alternative 21: 9.0% accurate, 5.6× speedup

Math

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

FPCore

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

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{0.125}{\mathsf{PI}\left(\right) \cdot s} \cdot e^{\frac{\frac{r}{-3}}{s}}}{r}} \]

5. Taylor expanded in s around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. lower-/.f32 N/A

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

7. Applied rewrites 7.9%

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

Alternative 22: 9.0% accurate, 5.8× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (/ (/ (/ (fma (/ r s) -0.16666666666666666 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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{0.125}{\mathsf{PI}\left(\right) \cdot s} \cdot e^{\frac{\frac{r}{-3}}{s}}}{r}} \]

5. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 7.9

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

7. Applied rewrites 7.9%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 7.9%

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

Alternative 23: 9.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{1}{s}, \frac{0.25}{r}\right)}{\mathsf{PI}\left(\right)}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ (/ (fma -0.16666666666666666 (/ 1.0 s) (/ 0.25 r)) (PI)) 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

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

   2. fp-cancel-sub-sign-inv N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-/r* N/A

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

   6. associate-*r/ N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. associate-/r* N/A

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

   10. div-add-rev N/A

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

   11. lower-/.f32 N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. lower-PI.f32 7.9

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

5. Applied rewrites 7.9%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{1}{s}, \frac{0.25}{r}\right)}{\mathsf{PI}\left(\right)}}{s}} \]
6. Add Preprocessing

Alternative 24: 9.0% accurate, 5.9× speedup

Math

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

FPCore

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

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-*l* N/A

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

   8. times-frac N/A

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

   9. lower-*.f32 N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. metadata-eval N/A

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

   9. lower-*.f32 99.5

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f32 N/A

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

9. Applied rewrites 7.9%

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

10. Final simplification 7.9%

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

Alternative 25: 9.0% accurate, 6.6× speedup

Math

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

FPCore

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

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{0.125}{\mathsf{PI}\left(\right) \cdot s} \cdot e^{\frac{\frac{r}{-3}}{s}}}{r}} \]

5. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 7.9

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

7. Applied rewrites 7.9%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 7.9%

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

Alternative 26: 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. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 7.8

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

5. Applied rewrites 7.8%

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

7. Applied rewrites 7.8%

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

Alternative 27: 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. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 7.8

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

5. Applied rewrites 7.8%

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

Alternative 28: 9.0% accurate, 10.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 7.8

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

5. Applied rewrites 7.8%

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

7. Applied rewrites 7.8%

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

Alternative 29: 9.0% accurate, 13.5× speedup

Math

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

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. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 7.8

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

5. Applied rewrites 7.8%

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

7. Applied rewrites 7.8%

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

9. Applied rewrites 7.8%

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

Alternative 30: 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. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 7.8

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

5. Applied rewrites 7.8%

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

7. Applied rewrites 7.8%

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

9. Applied rewrites 7.8%

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

Reproduce

herbie shell --seed 2024350 
(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))))