Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 5.2s

Alternatives: 13

Speedup: N/A×

Specification

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

Math

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

FPCore

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-*.f32 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 99.7

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

7. Applied rewrites 99.7%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-*.f32 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 99.7

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

7. Applied rewrites 99.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-*.f32 99.7

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

9. Applied rewrites 99.7%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-*.f32 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lift-/.f32 99.7

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

10. Applied rewrites 99.7%

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

Alternative 4: 10.4% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-*.f32 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in s around -inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. pow2 N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lift-/.f32 9.3

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

10. Applied rewrites 9.3%

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

Alternative 5: 10.4% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-*.f32 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 9.3

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

10. Applied rewrites 9.3%

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

Alternative 6: 9.8% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 8.6%

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

6. Taylor expanded in r around 0

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lift-PI.f32 8.8

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

8. Applied rewrites 8.8%

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

   1. lift-/.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lift-/.f32 N/A

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

   12. lift-PI.f32 8.8

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

10. Applied rewrites 8.8%

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

11. Final simplification 8.8%

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

Alternative 7: 9.8% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 8.6%

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

6. Taylor expanded in r around 0

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lift-PI.f32 8.8

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

8. Applied rewrites 8.8%

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

9. Taylor expanded in s around 0

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

    1. lower-/.f32 N/A

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

    4. lower-fma.f32 N/A

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

    5. lower-/.f32 N/A

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

    6. lift-PI.f32 N/A

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

    7. lower-*.f32 N/A

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

    8. lower-/.f32 N/A

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

    9. lift-PI.f32 N/A

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

    10. pow2 N/A

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

    11. lift-*.f32 8.8

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

11. Applied rewrites 8.8%

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

12. Final simplification 8.8%

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

Alternative 8: 9.8% accurate, 3.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 8.6%

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

6. Taylor expanded in r around 0

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lift-PI.f32 8.8

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

8. Applied rewrites 8.8%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 8.8

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

10. Applied rewrites 8.8%

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

11. Final simplification 8.8%

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

Alternative 9: 9.8% accurate, 4.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 8.6%

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

6. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

8. Applied rewrites 8.8%

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

Alternative 10: 8.9% accurate, 4.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 8.6%

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

6. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lift-PI.f32 8.2

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

8. Applied rewrites 8.2%

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

9. Final simplification 8.2%

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

Alternative 11: 8.9% accurate, 10.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 8.2

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

5. Applied rewrites 8.2%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. lift-PI.f32 8.2

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

7. Applied rewrites 8.2%

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

Alternative 12: 8.9% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 8.2

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

5. Applied rewrites 8.2%

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

Alternative 13: 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 99.7%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 8.2

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

5. Applied rewrites 8.2%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 8.2

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

7. Applied rewrites 8.2%

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

Reproduce

herbie shell --seed 2025040 
(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))))