Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 13.0s

Alternatives: 17

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 99.3

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

4. Applied rewrites 99.3%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 99.3

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

4. Applied rewrites 99.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-neg.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lift-exp.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. lift-*.f32 N/A

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

6. Applied rewrites 99.3%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. metadata-eval N/A

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

   9. lower-*.f32 99.3

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

8. Applied rewrites 99.3%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. times-frac N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

4. Applied rewrites 99.3%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

4. Applied rewrites 99.3%

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

Alternative 5: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 99.3

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

4. Applied rewrites 99.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-neg.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lift-exp.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. lift-*.f32 N/A

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

6. Applied rewrites 99.3%

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

8. Applied rewrites 99.2%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 99.3

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

4. Applied rewrites 99.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-neg.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lift-exp.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. lift-*.f32 N/A

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

6. Applied rewrites 99.3%

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

8. Applied rewrites 99.2%

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

Alternative 7: 10.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.3333333333333333 + \frac{0.05555555555555555 \cdot r}{s}}{s}, r, 1\right), \frac{0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. lower-fma.f32 N/A

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

4. Applied rewrites 97.0%

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

5. Taylor expanded in r around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{1 + r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right)}, \frac{\frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right) + 1}, \frac{\frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right) \cdot r} + 1, \frac{\frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}, r, 1\right)}, \frac{\frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

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

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{18} \cdot \frac{r}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{s}}, r, 1\right), \frac{\frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{s} + \frac{1}{18} \cdot \frac{r}{{s}^{2}}}, r, 1\right), \frac{\frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{3}} \cdot \frac{1}{s} + \frac{1}{18} \cdot \frac{r}{{s}^{2}}, r, 1\right), \frac{\frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{3} \cdot 1}{s}} + \frac{1}{18} \cdot \frac{r}{{s}^{2}}, r, 1\right), \frac{\frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{3}}}{s} + \frac{1}{18} \cdot \frac{r}{{s}^{2}}, r, 1\right), \frac{\frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{-1}{3}}{s} + \color{blue}{\frac{\frac{1}{18} \cdot r}{{s}^{2}}}, r, 1\right), \frac{\frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{-1}{3}}{s} + \frac{\frac{1}{18} \cdot r}{\color{blue}{s \cdot s}}, r, 1\right), \frac{\frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   11. associate-/r* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{-1}{3}}{s} + \color{blue}{\frac{\frac{\frac{1}{18} \cdot r}{s}}{s}}, r, 1\right), \frac{\frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   12. div-add-rev N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{3} + \frac{\frac{1}{18} \cdot r}{s}}{s}}, r, 1\right), \frac{\frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   13. lower-/.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{3} + \frac{\frac{1}{18} \cdot r}{s}}{s}}, r, 1\right), \frac{\frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   14. lower-+.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{3} + \frac{\frac{1}{18} \cdot r}{s}}}{s}, r, 1\right), \frac{\frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   15. lower-/.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{-1}{3} + \color{blue}{\frac{\frac{1}{18} \cdot r}{s}}}{s}, r, 1\right), \frac{\frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   16. lower-*.f32 10.9

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.3333333333333333 + \frac{\color{blue}{0.05555555555555555 \cdot r}}{s}}{s}, r, 1\right), \frac{0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

7. Applied rewrites 10.9%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-0.3333333333333333 + \frac{0.05555555555555555 \cdot r}{s}}{s}, r, 1\right)}, \frac{0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]
8. Add Preprocessing

Alternative 8: 10.4% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 99.3

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

4. Applied rewrites 99.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-neg.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lift-exp.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. lift-*.f32 N/A

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

6. Applied rewrites 99.3%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. metadata-eval N/A

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

   9. lower-*.f32 99.3

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

8. Applied rewrites 99.3%

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

9. Taylor expanded in r around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f32 N/A

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

    4. metadata-eval N/A

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

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

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

    6. associate-*r/ N/A

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

    7. unpow2 N/A

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

    8. associate-/r* N/A

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

    9. associate-*r/ N/A

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

    10. metadata-eval N/A

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

    11. associate-*r/ N/A

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

    12. metadata-eval N/A

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

    13. div-add-rev N/A

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

    14. lower-/.f32 N/A

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

    15. metadata-eval N/A

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

    16. lower-fma.f32 N/A

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

    17. lower-/.f32 10.9

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

11. Applied rewrites 10.9%

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

Alternative 9: 9.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.125 + \frac{-0.041666666666666664 \cdot r}{s}}{s \cdot s}, r, \frac{-0.25}{s}\right), r, 0.25\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{\frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, 0.0007716049382716049, -0.006944444444444444 \cdot r\right)}{\mathsf{PI}\left(\right)}}{s} + \frac{0.041666666666666664}{\mathsf{PI}\left(\right)}}{s} - \frac{0.125}{\mathsf{PI}\left(\right) \cdot r}}{-s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/
   (fma
    (fma (/ (+ 0.125 (/ (* -0.041666666666666664 r) s)) (* s s)) r (/ -0.25 s))
    r
    0.25)
   (* (* (* 2.0 (PI)) s) r))
  (/
   (-
    (/
     (+
      (/
       (/
        (fma (* r (/ r s)) 0.0007716049382716049 (* -0.006944444444444444 r))
        (PI))
       s)
      (/ 0.041666666666666664 (PI)))
     s)
    (/ 0.125 (* (PI) r)))
   (- s))))

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f32 N/A

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

5. Applied rewrites 10.8%

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

6. Taylor expanded in r around 0

   \[\leadsto \frac{\color{blue}{\frac{1}{4} + r \cdot \left(r \cdot \left(\frac{-1}{24} \cdot \frac{r}{{s}^{3}} + \frac{1}{8} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{4} \cdot \frac{1}{s}\right)}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{\frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, \frac{1}{1296}, \frac{-1}{144} \cdot r\right)}{\mathsf{PI}\left(\right)}}{-s} - \frac{\frac{1}{24}}{\mathsf{PI}\left(\right)}}{-s} - \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot r}}{-s} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{r \cdot \left(r \cdot \left(\frac{-1}{24} \cdot \frac{r}{{s}^{3}} + \frac{1}{8} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{4} \cdot \frac{1}{s}\right) + \frac{1}{4}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{\frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, \frac{1}{1296}, \frac{-1}{144} \cdot r\right)}{\mathsf{PI}\left(\right)}}{-s} - \frac{\frac{1}{24}}{\mathsf{PI}\left(\right)}}{-s} - \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot r}}{-s} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(r \cdot \left(\frac{-1}{24} \cdot \frac{r}{{s}^{3}} + \frac{1}{8} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{4} \cdot \frac{1}{s}\right) \cdot r} + \frac{1}{4}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{\frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, \frac{1}{1296}, \frac{-1}{144} \cdot r\right)}{\mathsf{PI}\left(\right)}}{-s} - \frac{\frac{1}{24}}{\mathsf{PI}\left(\right)}}{-s} - \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot r}}{-s} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(r \cdot \left(\frac{-1}{24} \cdot \frac{r}{{s}^{3}} + \frac{1}{8} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{\frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, \frac{1}{1296}, \frac{-1}{144} \cdot r\right)}{\mathsf{PI}\left(\right)}}{-s} - \frac{\frac{1}{24}}{\mathsf{PI}\left(\right)}}{-s} - \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot r}}{-s} \]

8. Applied rewrites 10.3%

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

9. Final simplification 10.3%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.125 + \frac{-0.041666666666666664 \cdot r}{s}}{s \cdot s}, r, \frac{-0.25}{s}\right), r, 0.25\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{\frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, 0.0007716049382716049, -0.006944444444444444 \cdot r\right)}{\mathsf{PI}\left(\right)}}{s} + \frac{0.041666666666666664}{\mathsf{PI}\left(\right)}}{s} - \frac{0.125}{\mathsf{PI}\left(\right) \cdot r}}{-s} \]
10. Add Preprocessing

Alternative 10: 9.6% accurate, 2.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 8.9%

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

7. Applied rewrites 8.9%

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

8. Taylor expanded in s around inf

   \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} + \left(\frac{-1}{24} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{144} \cdot \frac{{r}^{2}}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{16} \cdot \frac{{r}^{2}}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)\right)\right)}}{s \cdot r} \]
9. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{8} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} + \frac{-1}{24} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)}\right) + \left(\frac{1}{144} \cdot \frac{{r}^{2}}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{16} \cdot \frac{{r}^{2}}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)\right)}}{s \cdot r} \]

   2. distribute-rgt-out N/A

      \[\leadsto \frac{\color{blue}{\frac{r}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{-1}{8} + \frac{-1}{24}\right)} + \left(\frac{1}{144} \cdot \frac{{r}^{2}}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{16} \cdot \frac{{r}^{2}}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)\right)}{s \cdot r} \]

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. associate-+r+ N/A

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

10. Applied rewrites 10.1%

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

11. Taylor expanded in s around -inf

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

13. Applied rewrites 10.3%

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

14. Final simplification 10.3%

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

Alternative 11: 9.8% accurate, 3.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 8.9%

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

7. Applied rewrites 8.9%

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

8. Taylor expanded in r around 0

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

10. Applied rewrites 10.2%

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

Alternative 12: 9.8% accurate, 4.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 99.3

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

4. Applied rewrites 99.3%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-neg.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lift-exp.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. lift-*.f32 N/A

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

9. Applied rewrites 10.2%

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

Alternative 13: 8.9% accurate, 6.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 8.9%

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

7. Applied rewrites 8.9%

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

8. Taylor expanded in s around inf

   \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} + \left(\frac{-1}{24} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{144} \cdot \frac{{r}^{2}}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{16} \cdot \frac{{r}^{2}}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)\right)\right)}}{s \cdot r} \]
9. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{8} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} + \frac{-1}{24} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)}\right) + \left(\frac{1}{144} \cdot \frac{{r}^{2}}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{16} \cdot \frac{{r}^{2}}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)\right)}}{s \cdot r} \]

   2. distribute-rgt-out N/A

      \[\leadsto \frac{\color{blue}{\frac{r}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{-1}{8} + \frac{-1}{24}\right)} + \left(\frac{1}{144} \cdot \frac{{r}^{2}}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{16} \cdot \frac{{r}^{2}}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)\right)}{s \cdot r} \]

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. associate-+r+ N/A

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

10. Applied rewrites 10.1%

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

11. Taylor expanded in s around inf

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

    1. associate-+r+ N/A

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

    2. distribute-rgt-out N/A

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

    3. associate-/r* N/A

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

    4. metadata-eval N/A

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

    5. associate-*l/ N/A

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

    6. associate-*r/ N/A

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

    7. metadata-eval N/A

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

    8. div-add-rev N/A

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

    9. lower-/.f32 N/A

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

    10. lower-fma.f32 N/A

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

    11. lower-/.f32 N/A

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

    12. lower-PI.f32 8.9

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

13. Applied rewrites 8.9%

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

Alternative 14: 8.9% accurate, 8.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 8.7

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

5. Applied rewrites 8.7%

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

7. Applied rewrites 8.7%

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

Alternative 15: 8.9% 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 99.3%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 8.7

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

5. Applied rewrites 8.7%

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

7. Applied rewrites 8.7%

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

Alternative 16: 8.9% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 8.7

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

5. Applied rewrites 8.7%

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

7. Applied rewrites 8.7%

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

Alternative 17: 8.9% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 8.7

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

5. Applied rewrites 8.7%

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

7. Applied rewrites 8.7%

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

9. Applied rewrites 8.7%

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

Reproduce

herbie shell --seed 2024366 
(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))))