Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 12.8s

Alternatives: 21

Speedup: N/A×

Specification

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

Math

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

FPCore

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. times-frac N/A

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

   7. associate-*l/ N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. times-frac N/A

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

4. Applied rewrites 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot s\\ \mathsf{fma}\left(\frac{e^{\frac{\frac{r}{-3}}{s}}}{\left(t\_0 \cdot 6\right) \cdot r}, 0.75, \frac{0.125}{r} \cdot \frac{e^{\frac{-r}{s}}}{t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (PI) s)))
   (fma
    (/ (exp (/ (/ r -3.0) s)) (* (* t_0 6.0) r))
    0.75
    (* (/ 0.125 r) (/ (exp (/ (- r) s)) t_0)))))

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. times-frac N/A

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

   7. associate-*l/ N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. times-frac N/A

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.6%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. times-frac N/A

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

   7. associate-*l/ N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. times-frac N/A

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.6%

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

Alternative 4: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. times-frac N/A

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

   7. associate-*l/ N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. times-frac N/A

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.6%

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

8. Applied rewrites 99.6%

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

Alternative 5: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. +-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 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. distribute-rgt-out N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

6. Applied rewrites 99.6%

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

Alternative 6: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. +-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 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. +-commutative N/A

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

   4. div-add N/A

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

6. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 99.6

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

8. Applied rewrites 99.6%

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

Alternative 7: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/r* N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*l* N/A

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

   12. associate-/r* N/A

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. +-commutative N/A

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

   4. div-add N/A

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

6. Applied rewrites 99.5%

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

   1. lift-fma.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-out N/A

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

   6. lower-*.f32 N/A

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

   7. lower-+.f32 99.5

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

8. Applied rewrites 99.5%

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

Alternative 8: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   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 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. +-commutative N/A

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

   4. div-add N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.5

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

9. Applied rewrites 99.5%

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

Alternative 9: 10.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r\\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{0.05555555555555555}{s}}{s}, r, \frac{-0.3333333333333333}{s}\right), r, 1\right)}{6}, \frac{0.75}{t\_0}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (* (PI) s) r)))
   (fma
    (/
     (fma
      (fma (/ (/ 0.05555555555555555 s) s) r (/ -0.3333333333333333 s))
      r
      1.0)
     6.0)
    (/ 0.75 t_0)
    (* 0.125 (/ (exp (/ (- r) s)) t_0)))))

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-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. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. associate-*l* N/A

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

   11. times-frac N/A

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

4. Applied rewrites 97.6%

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

5. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

7. Applied rewrites 11.0%

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

Alternative 10: 10.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r\\ \mathsf{fma}\left(\frac{1 - \frac{\mathsf{fma}\left(0.3333333333333333, r, \left(r \cdot \frac{r}{s}\right) \cdot -0.05555555555555555\right)}{s}}{6}, \frac{0.75}{t\_0}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (* (PI) s) r)))
   (fma
    (/
     (-
      1.0
      (/ (fma 0.3333333333333333 r (* (* r (/ r s)) -0.05555555555555555)) s))
     6.0)
    (/ 0.75 t_0)
    (* 0.125 (/ (exp (/ (- r) s)) t_0)))))

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-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. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. associate-*l* N/A

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

   11. times-frac N/A

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

4. Applied rewrites 97.6%

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

5. Taylor expanded in s around inf

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

   1. associate-+r+ N/A

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

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

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

   3. associate-+l- N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. associate-*r/ N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. div-sub N/A

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

   10. associate-/l* N/A

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

   11. metadata-eval N/A

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

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

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

   13. +-commutative N/A

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

   14. lower--.f32 N/A

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

   15. lower-/.f32 N/A

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

7. Applied rewrites 11.0%

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

Alternative 11: 10.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r\\ \mathsf{fma}\left(0.16666666666666666 - \frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, -0.009259259259259259, 0.05555555555555555 \cdot r\right)}{s}, \frac{0.75}{t\_0}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (* (PI) s) r)))
   (fma
    (-
     0.16666666666666666
     (/ (fma (* r (/ r s)) -0.009259259259259259 (* 0.05555555555555555 r)) s))
    (/ 0.75 t_0)
    (* 0.125 (/ (exp (/ (- r) s)) t_0)))))

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-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. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. associate-*l* N/A

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

   11. times-frac N/A

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

4. Applied rewrites 97.6%

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

5. Taylor expanded in s around inf

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6} + \left(\frac{-1}{18} \cdot \frac{r}{s} + \frac{1}{108} \cdot \frac{{r}^{2}}{{s}^{2}}\right)}, \frac{\frac{3}{4}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]
6. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} + \frac{-1}{18} \cdot \frac{r}{s}\right) + \frac{1}{108} \cdot \frac{{r}^{2}}{{s}^{2}}}, \frac{\frac{3}{4}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} - \left(\mathsf{neg}\left(\frac{-1}{18}\right)\right) \cdot \frac{r}{s}\right)} + \frac{1}{108} \cdot \frac{{r}^{2}}{{s}^{2}}, \frac{\frac{3}{4}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   3. associate-+l- N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6} - \left(\left(\mathsf{neg}\left(\frac{-1}{18}\right)\right) \cdot \frac{r}{s} - \frac{1}{108} \cdot \frac{{r}^{2}}{{s}^{2}}\right)}, \frac{\frac{3}{4}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} - \left(\color{blue}{\frac{1}{18}} \cdot \frac{r}{s} - \frac{1}{108} \cdot \frac{{r}^{2}}{{s}^{2}}\right), \frac{\frac{3}{4}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} - \left(\color{blue}{\frac{\frac{1}{18} \cdot r}{s}} - \frac{1}{108} \cdot \frac{{r}^{2}}{{s}^{2}}\right), \frac{\frac{3}{4}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} - \left(\frac{\frac{1}{18} \cdot r}{s} - \color{blue}{\frac{\frac{1}{108} \cdot {r}^{2}}{{s}^{2}}}\right), \frac{\frac{3}{4}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} - \left(\frac{\frac{1}{18} \cdot r}{s} - \frac{\frac{1}{108} \cdot {r}^{2}}{\color{blue}{s \cdot s}}\right), \frac{\frac{3}{4}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   8. associate-/r* N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} - \left(\frac{\frac{1}{18} \cdot r}{s} - \color{blue}{\frac{\frac{\frac{1}{108} \cdot {r}^{2}}{s}}{s}}\right), \frac{\frac{3}{4}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   9. div-sub N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} - \color{blue}{\frac{\frac{1}{18} \cdot r - \frac{\frac{1}{108} \cdot {r}^{2}}{s}}{s}}, \frac{\frac{3}{4}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   10. associate-/l* N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{6} - \frac{\frac{1}{18} \cdot r - \color{blue}{\frac{1}{108} \cdot \frac{{r}^{2}}{s}}}{s}, \frac{\frac{3}{4}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{6} - \frac{\frac{1}{18} \cdot r - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{108}\right)\right)} \cdot \frac{{r}^{2}}{s}}{s}, \frac{\frac{3}{4}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

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

       \[\leadsto \mathsf{fma}\left(\frac{1}{6} - \frac{\color{blue}{\frac{1}{18} \cdot r + \frac{-1}{108} \cdot \frac{{r}^{2}}{s}}}{s}, \frac{\frac{3}{4}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{6} - \frac{\color{blue}{\frac{-1}{108} \cdot \frac{{r}^{2}}{s} + \frac{1}{18} \cdot r}}{s}, \frac{\frac{3}{4}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   14. lower--.f32 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6} - \frac{\frac{-1}{108} \cdot \frac{{r}^{2}}{s} + \frac{1}{18} \cdot r}{s}}, \frac{\frac{3}{4}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   15. lower-/.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{6} - \color{blue}{\frac{\frac{-1}{108} \cdot \frac{{r}^{2}}{s} + \frac{1}{18} \cdot r}{s}}, \frac{\frac{3}{4}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

7. Applied rewrites 10.9%

   \[\leadsto \mathsf{fma}\left(\color{blue}{0.16666666666666666 - \frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, -0.009259259259259259, 0.05555555555555555 \cdot r\right)}{s}}, \frac{0.75}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]
8. Add Preprocessing

Alternative 12: 10.8% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/r* N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*l* N/A

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

   12. associate-/r* N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

   1. associate-+r+ N/A

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

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

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

   3. associate-+l- N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. associate-*r/ N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. div-sub N/A

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

   10. associate-/l* N/A

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

   11. metadata-eval N/A

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

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

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

   13. +-commutative N/A

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

   14. lower--.f32 N/A

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

   15. lower-/.f32 N/A

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

7. Applied rewrites 10.9%

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

Alternative 13: 10.2% accurate, 4.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   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 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. +-commutative N/A

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

   4. div-add N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in s around -inf

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

9. Applied rewrites 10.3%

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

Alternative 14: 10.2% accurate, 5.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   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 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. +-commutative N/A

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

   4. div-add N/A

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

6. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. metadata-eval N/A

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

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

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

   6. lower-/.f32 N/A

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

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

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

   8. metadata-eval N/A

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

   9. lower-*.f32 99.5

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in r around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f32 N/A

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

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

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

    5. +-commutative N/A

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

    6. metadata-eval N/A

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

    7. associate-*r/ N/A

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

    8. metadata-eval N/A

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

    9. associate-*r/ N/A

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

    10. unpow2 N/A

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

    11. associate-/r* N/A

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

    12. div-add-rev N/A

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

    13. lower-/.f32 N/A

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

    14. lower-+.f32 N/A

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

    15. lower-/.f32 N/A

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

    16. lower-*.f32 10.3

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

11. Applied rewrites 10.3%

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

12. Final simplification 10.3%

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

Alternative 15: 10.2% accurate, 5.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   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 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. +-commutative N/A

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

   4. div-add N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

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

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

   5. associate-*r/ N/A

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

   6. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. div-add-rev N/A

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

   13. lower-/.f32 N/A

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

   14. lower-fma.f32 N/A

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

   15. lower-/.f32 10.3

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

9. Applied rewrites 10.3%

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

Alternative 16: 9.2% accurate, 5.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 9.1%

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

7. Applied rewrites 9.1%

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

Alternative 17: 9.2% accurate, 7.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   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 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. +-commutative N/A

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

   4. div-add N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-out N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-/.f32 9.1

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

9. Applied rewrites 9.1%

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

Alternative 18: 9.1% accurate, 8.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 8.8

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

5. Applied rewrites 8.8%

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

7. Applied rewrites 8.8%

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

Alternative 19: 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 99.5%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 8.8

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

5. Applied rewrites 8.8%

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

7. Applied rewrites 8.8%

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

Alternative 20: 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 99.5%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 8.8

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

5. Applied rewrites 8.8%

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

Alternative 21: 9.1% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 8.8

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

5. Applied rewrites 8.8%

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

7. Applied rewrites 8.8%

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

9. Applied rewrites 8.8%

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

Reproduce

herbie shell --seed 2024359 
(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))))