Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 5.6s

Alternatives: 18

Speedup: N/A×

Specification

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

Math

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

FPCore

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot s\\ \frac{e^{\frac{-r}{s}}}{t\_0 \cdot r} \cdot 0.125 + \frac{0.75 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{t\_0 \cdot \left(6 \cdot r\right)} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (PI) s)))
   (+
    (* (/ (exp (/ (- r) s)) (* t_0 r)) 0.125)
    (/ (* 0.75 (exp (* -0.3333333333333333 (/ r s)))) (* t_0 (* 6.0 r))))))

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.8

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

5. Applied rewrites 99.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. lift-PI.f32 N/A

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

   19. lower-*.f32 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 99.8

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

10. Applied rewrites 99.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.8

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

5. Applied rewrites 99.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. lift-PI.f32 N/A

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

   19. lower-*.f32 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 99.8

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

10. Applied rewrites 99.8%

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

    1. lift-*.f32 N/A

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

    2. lift-PI.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. lift-*.f32 N/A

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

    5. associate-*l* N/A

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

    6. lower-*.f32 N/A

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

    7. lift-PI.f32 N/A

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

    8. lower-*.f32 N/A

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

    9. lift-*.f32 99.8

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

12. Applied rewrites 99.8%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot s\\ \mathsf{fma}\left(0.75, \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{\left(t\_0 \cdot 6\right) \cdot r}, \frac{e^{\frac{-r}{s}}}{t\_0 \cdot r} \cdot 0.125\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (PI) s)))
   (fma
    0.75
    (/ (exp (* (/ r s) -0.3333333333333333)) (* (* t_0 6.0) r))
    (* (/ (exp (/ (- r) s)) (* t_0 r)) 0.125))))

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.8

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

5. Applied rewrites 99.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. lift-PI.f32 N/A

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

   19. lower-*.f32 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 99.8

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

10. Applied rewrites 99.8%

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

    1. lift-+.f32 N/A

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

    2. +-commutative N/A

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

    3. lift-/.f32 N/A

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

    4. lift-*.f32 N/A

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

    5. associate-/l* N/A

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

12. Applied rewrites 99.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.75, \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{\left(\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot 6\right) \cdot r}, \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \cdot 0.125\right)} \]
13. Add Preprocessing

Alternative 4: 10.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.05555555555555555, r, -0.3333333333333333 \cdot s\right)}{s \cdot s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 (PI)) s) r))
  (/
   (*
    0.75
    (fma
     (/ (fma 0.05555555555555555 r (* -0.3333333333333333 s)) (* s s))
     r
     1.0))
   (* (* (* 6.0 (PI)) s) r))))

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \color{blue}{\left(1 + r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right)\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \left(r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right) + \color{blue}{1}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \left(\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right) \cdot r + 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}, \color{blue}{r}, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   4. lower--.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{18} - \frac{1}{3} \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{18} - \frac{1}{3} \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{18} - \frac{1}{3} \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{1}{3} \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{1}{3} \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{\frac{1}{3} \cdot 1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{\frac{1}{3}}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   12. lower-/.f32 8.2

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.05555555555555555 - \frac{0.3333333333333333}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

5. Applied rewrites 8.2%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \color{blue}{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.05555555555555555 - \frac{0.3333333333333333}{s}, r, 1\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

6. Taylor expanded in s around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{\frac{-1}{3} \cdot s + \frac{1}{18} \cdot r}{{s}^{2}}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
7. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{\frac{-1}{3} \cdot s + \frac{1}{18} \cdot r}{{s}^{2}}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{\frac{1}{18} \cdot r + \frac{-1}{3} \cdot s}{{s}^{2}}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{18}, r, \frac{-1}{3} \cdot s\right)}{{s}^{2}}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   4. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{18}, r, \frac{-1}{3} \cdot s\right)}{{s}^{2}}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   5. pow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{18}, r, \frac{-1}{3} \cdot s\right)}{s \cdot s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   6. lift-*.f32 8.2

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.05555555555555555, r, -0.3333333333333333 \cdot s\right)}{s \cdot s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

8. Applied rewrites 8.2%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.05555555555555555, r, -0.3333333333333333 \cdot s\right)}{s \cdot s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
9. Add Preprocessing

Alternative 5: 10.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot s\\ \frac{e^{\frac{-r}{s}}}{t\_0 \cdot r} \cdot 0.125 + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.041666666666666664 - \frac{0.25}{s}, r, 0.75\right)}{t\_0 \cdot \left(6 \cdot r\right)} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (PI) s)))
   (+
    (* (/ (exp (/ (- r) s)) (* t_0 r)) 0.125)
    (/
     (fma (- (* (/ r (* s s)) 0.041666666666666664) (/ 0.25 s)) r 0.75)
     (* t_0 (* 6.0 r))))))

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.8

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

5. Applied rewrites 99.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. lift-PI.f32 N/A

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

   19. lower-*.f32 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 99.8

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

10. Applied rewrites 99.8%

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

11. Taylor expanded in r around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f32 N/A

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

    4. lower--.f32 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f32 N/A

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

    7. lower-/.f32 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f32 N/A

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

    10. associate-*r/ N/A

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

    11. metadata-eval N/A

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

    12. lower-/.f32 8.2

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

13. Applied rewrites 8.2%

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

Alternative 6: 9.5% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \color{blue}{\left(1 + r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right)\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \left(r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right) + \color{blue}{1}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \left(\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right) \cdot r + 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}, \color{blue}{r}, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   4. lower--.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{18} - \frac{1}{3} \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{18} - \frac{1}{3} \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{18} - \frac{1}{3} \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{1}{3} \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{1}{3} \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{\frac{1}{3} \cdot 1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{\frac{1}{3}}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   12. lower-/.f32 8.2

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.05555555555555555 - \frac{0.3333333333333333}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

5. Applied rewrites 8.2%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \color{blue}{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.05555555555555555 - \frac{0.3333333333333333}{s}, r, 1\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{\frac{1}{3}}{s}, r, 1\right)}{\color{blue}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right)} \cdot r} \]

   2. lift-PI.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{\frac{1}{3}}{s}, r, 1\right)}{\left(\left(6 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot s\right) \cdot r} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{\frac{1}{3}}{s}, r, 1\right)}{\left(\color{blue}{\left(6 \cdot \mathsf{PI}\left(\right)\right)} \cdot s\right) \cdot r} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{\frac{1}{3}}{s}, r, 1\right)}{\color{blue}{\left(6 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)\right)} \cdot r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{\frac{1}{3}}{s}, r, 1\right)}{\left(6 \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot r} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{\frac{1}{3}}{s}, r, 1\right)}{\color{blue}{\left(\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot 6\right)} \cdot r} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{\frac{1}{3}}{s}, r, 1\right)}{\left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot s\right)} \cdot 6\right) \cdot r} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{\frac{1}{3}}{s}, r, 1\right)}{\left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot s\right)} \cdot 6\right) \cdot r} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{18} - \frac{\frac{1}{3}}{s}, r, 1\right)}{\left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot s\right) \cdot 6\right) \cdot r} \]

   10. lift-*.f32 8.2

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.05555555555555555 - \frac{0.3333333333333333}{s}, r, 1\right)}{\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot 6\right)} \cdot r} \]

7. Applied rewrites 8.2%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.05555555555555555 - \frac{0.3333333333333333}{s}, r, 1\right)}{\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot 6\right)} \cdot r} \]

8. Taylor expanded in s around inf

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

10. Applied rewrites 8.1%

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

Alternative 7: 9.5% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in s around inf

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

8. Applied rewrites 8.1%

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

10. Applied rewrites 8.1%

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

    1. lift-/.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. associate-/l/ N/A

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

    4. *-commutative N/A

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

    5. lift-PI.f32 N/A

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

    6. lift-*.f32 N/A

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

    7. *-commutative N/A

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

    8. associate-/r* N/A

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

    9. lower-/.f32 N/A

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

    10. lower-/.f32 N/A

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

    11. *-commutative N/A

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

    12. lift-*.f32 N/A

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

    13. lift-PI.f32 8.1

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

12. Applied rewrites 8.1%

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

13. Final simplification 8.1%

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

Alternative 8: 9.5% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.8

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

5. Applied rewrites 99.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. lift-PI.f32 N/A

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

   19. lower-*.f32 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 99.8

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

10. Applied rewrites 99.8%

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

11. Taylor expanded in s around inf

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

13. Applied rewrites 8.1%

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

Alternative 9: 9.5% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in s around inf

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

8. Applied rewrites 8.1%

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

9. Taylor expanded in s around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. mul-1-neg N/A

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

    4. distribute-frac-neg N/A

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

    5. lower-/.f32 N/A

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

    6. lift-/.f32 N/A

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

    7. lift-neg.f32 N/A

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

    8. lift-exp.f32 N/A

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

    9. *-commutative N/A

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

    10. lift-*.f32 N/A

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

    11. lift-PI.f32 N/A

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

    12. *-commutative N/A

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

    13. lift-*.f32 8.1

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

11. Applied rewrites 8.1%

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

Alternative 10: 10.0% accurate, 3.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in 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 7.4%

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

6. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot \mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   4. associate-/r* N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   5. lower-/.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   6. lift-/.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   7. lift-PI.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   8. associate-*r/ N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6} \cdot 1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   9. metadata-eval N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   10. lift-/.f32 N/A

       \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   11. lift-PI.f32 7.9

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

8. Applied rewrites 7.9%

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

   1. lift-/.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   2. lift--.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   3. div-sub N/A

      \[\leadsto -\left(\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{\mathsf{PI}\left(\right)}\right)}{s}}{s} - \frac{\frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]

10. Applied rewrites 7.9%

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

11. Final simplification 7.9%

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

Alternative 11: 10.0% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in 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 7.4%

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

6. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot \mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   4. associate-/r* N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   5. lower-/.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   6. lift-/.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   7. lift-PI.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   8. associate-*r/ N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6} \cdot 1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   9. metadata-eval N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   10. lift-/.f32 N/A

       \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   11. lift-PI.f32 7.9

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

8. Applied rewrites 7.9%

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

9. Final simplification 7.9%

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

Alternative 12: 10.0% accurate, 3.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in 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 7.4%

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

6. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot \mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   4. associate-/r* N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   5. lower-/.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   6. lift-/.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   7. lift-PI.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   8. associate-*r/ N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6} \cdot 1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   9. metadata-eval N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   10. lift-/.f32 N/A

       \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   11. lift-PI.f32 7.9

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

8. Applied rewrites 7.9%

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

9. Taylor expanded in r around 0

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

    1. *-commutative N/A

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

    2. associate-*r/ N/A

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

    3. metadata-eval N/A

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

    4. lower-fma.f32 N/A

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

    5. lower-/.f32 N/A

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

    6. lower-*.f32 N/A

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

    7. unpow2 N/A

       \[\leadsto -\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

    8. lower-*.f32 N/A

       \[\leadsto -\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

    9. lift-PI.f32 N/A

       \[\leadsto -\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

    10. lower-/.f32 N/A

        \[\leadsto -\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

    11. *-commutative N/A

        \[\leadsto -\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{\mathsf{PI}\left(\right) \cdot s}\right) - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

    12. lift-*.f32 N/A

        \[\leadsto -\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{\mathsf{PI}\left(\right) \cdot s}\right) - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

    13. lift-PI.f32 7.9

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

11. Applied rewrites 7.9%

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

12. Final simplification 7.9%

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

Alternative 13: 10.0% accurate, 4.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in 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 7.4%

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

6. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot \mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   4. associate-/r* N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   5. lower-/.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   6. lift-/.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   7. lift-PI.f32 N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   8. associate-*r/ N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6} \cdot 1}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   9. metadata-eval N/A

      \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   10. lift-/.f32 N/A

       \[\leadsto -\frac{\frac{\mathsf{fma}\left(\frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   11. lift-PI.f32 7.9

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

8. Applied rewrites 7.9%

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

9. Taylor expanded in s around inf

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

    1. lower-/.f32 N/A

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

11. Applied rewrites 7.8%

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

Alternative 14: 9.0% accurate, 10.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in 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. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 7.8

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

5. Applied rewrites 7.8%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 7.8

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

7. Applied rewrites 7.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 7.8

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

9. Applied rewrites 7.8%

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

Alternative 15: 9.0% accurate, 10.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in 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. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 7.8

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

5. Applied rewrites 7.8%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 7.8

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

7. Applied rewrites 7.8%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 7.8

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

9. Applied rewrites 7.8%

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

    1. lift-/.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-/r* N/A

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

    4. metadata-eval N/A

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

    5. associate-*r/ N/A

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

    6. lift-PI.f32 N/A

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

    7. lift-*.f32 N/A

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

    8. *-commutative N/A

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

    9. lower-/.f32 N/A

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

    10. associate-*r/ N/A

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

    11. metadata-eval N/A

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

    12. lower-/.f32 N/A

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

    13. *-commutative N/A

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

    14. lift-*.f32 N/A

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

    15. lift-PI.f32 7.8

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

11. Applied rewrites 7.8%

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

Alternative 16: 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 99.7%

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

3. Taylor expanded in 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. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 7.8

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

5. Applied rewrites 7.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 7.8

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

7. Applied rewrites 7.8%

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

Alternative 17: 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 99.7%

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

3. Taylor expanded in 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. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 7.8

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

5. Applied rewrites 7.8%

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

Alternative 18: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. 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. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 7.8

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

5. Applied rewrites 7.8%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 7.8

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

7. Applied rewrites 7.8%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 7.8

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

9. Applied rewrites 7.8%

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

Reproduce

herbie shell --seed 2025073 
(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))))