Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.4%

Time: 8.3s

Alternatives: 18

Speedup: N/A×

Specification

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

Math

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

FPCore

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

Initial Program: 99.5% 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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*r* N/A

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

   10. times-frac N/A

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

   11. lower-fma.f32 N/A

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

4. Applied rewrites 99.3%

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

   1. lift-/.f32 N/A

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

6. Applied rewrites 99.4%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*r* N/A

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

   10. times-frac N/A

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

   11. lower-fma.f32 N/A

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

4. Applied rewrites 99.3%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/r* N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*l* N/A

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

   12. associate-/r* N/A

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

4. Applied rewrites 99.3%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/r* N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*l* N/A

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

   12. associate-/r* N/A

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

4. Applied rewrites 99.3%

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

   1. lift-fma.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-*l/ N/A

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

   7. lift-/.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. frac-2neg N/A

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

   10. lift-neg.f32 N/A

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

   11. metadata-eval N/A

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

   12. associate-/r* N/A

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

   13. lift-*.f32 N/A

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

   14. lift-/.f32 N/A

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

6. Applied rewrites 99.3%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/r* N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*l* N/A

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

   12. associate-/r* N/A

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

4. Applied rewrites 99.3%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. metadata-eval N/A

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

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

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

   6. lift-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. distribute-rgt-neg-in N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f32 99.3

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

6. Applied rewrites 99.3%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/r* N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*l* N/A

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

   12. associate-/r* N/A

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

4. Applied rewrites 99.3%

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

6. Applied rewrites 99.2%

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

Alternative 7: 10.8% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*r* N/A

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

   10. times-frac N/A

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

   11. lower-fma.f32 N/A

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in s around -inf

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f32 N/A

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

   12. lower-/.f32 10.8

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

7. Applied rewrites 10.8%

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

Alternative 8: 10.8% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/r* N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*l* N/A

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

   12. associate-/r* N/A

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 10.8

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

7. Applied rewrites 10.8%

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

Alternative 9: 10.2% accurate, 3.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 10.4%

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

Alternative 10: 10.2% 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 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*r* N/A

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

   10. times-frac N/A

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

   11. lower-fma.f32 N/A

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 10.4%

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

Alternative 11: 9.2% accurate, 5.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-/r* N/A

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

   6. associate-*r/ N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. associate-/r* N/A

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

   10. div-add-rev N/A

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

   11. lower-/.f32 N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. lower-PI.f32 9.6

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

5. Applied rewrites 9.6%

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

Alternative 12: 9.2% accurate, 6.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*r* N/A

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

   10. times-frac N/A

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

   11. lower-fma.f32 N/A

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower--.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 9.5

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

7. Applied rewrites 9.5%

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

Alternative 13: 9.1% accurate, 8.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 9.3

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

5. Applied rewrites 9.3%

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

7. Applied rewrites 9.3%

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

Alternative 14: 9.1% 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 99.3%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 9.3

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

5. Applied rewrites 9.3%

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

Alternative 15: 9.1% accurate, 10.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 9.3

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

5. Applied rewrites 9.3%

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

7. Applied rewrites 9.3%

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

Alternative 16: 9.1% 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 99.3%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 9.3

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

5. Applied rewrites 9.3%

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

7. Applied rewrites 9.3%

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

Alternative 17: 9.1% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 9.3

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

5. Applied rewrites 9.3%

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

7. Applied rewrites 9.3%

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

9. Applied rewrites 9.3%

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

Alternative 18: 9.1% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 9.3

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

5. Applied rewrites 9.3%

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

7. Applied rewrites 9.3%

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

9. Applied rewrites 9.3%

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

Reproduce

herbie shell --seed 2025017 
(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))))