Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 9.8s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

Alternative 3: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 99.5%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. lift-/.f32 N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-+.f32 99.5

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

6. Applied rewrites 99.5%

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

Alternative 4: 10.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\frac{\frac{\mathsf{fma}\left(\frac{0.00102880658436214}{\mathsf{PI}\left(\right)}, \frac{r \cdot r}{s}, -0.009259259259259259 \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{-s} - \frac{0.05555555555555555}{\mathsf{PI}\left(\right)}}{s} + \frac{0.16666666666666666}{\mathsf{PI}\left(\right) \cdot r}}{s}, 0.75, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/
   (+
    (/
     (-
      (/
       (fma
        (/ 0.00102880658436214 (PI))
        (/ (* r r) s)
        (* -0.009259259259259259 (/ r (PI))))
       (- s))
      (/ 0.05555555555555555 (PI)))
     s)
    (/ 0.16666666666666666 (* (PI) r)))
   s)
  0.75
  (* 0.125 (/ (exp (/ (- r) s)) (* (* (PI) s) r)))))

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 10.7

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

7. Applied rewrites 10.7%

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

8. Taylor expanded in s around -inf

   \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{108} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{972} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}}{s} - \frac{1}{18} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{108} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{972} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}}{s} - \frac{1}{18} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}\right)}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{108} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{972} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}}{s} - \frac{1}{18} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(s\right)}}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   3. lower-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{108} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{972} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}}{s} - \frac{1}{18} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{\mathsf{neg}\left(s\right)}}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

10. Applied rewrites 11.6%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(\frac{0.00102880658436214}{\mathsf{PI}\left(\right)}, \frac{r \cdot r}{s}, -0.009259259259259259 \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{-s} - \frac{0.05555555555555555}{\mathsf{PI}\left(\right)}}{-s} - \frac{0.16666666666666666}{\mathsf{PI}\left(\right) \cdot r}}{-s}}, 0.75, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

11. Final simplification 11.6%

    \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\mathsf{fma}\left(\frac{0.00102880658436214}{\mathsf{PI}\left(\right)}, \frac{r \cdot r}{s}, -0.009259259259259259 \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{-s} - \frac{0.05555555555555555}{\mathsf{PI}\left(\right)}}{s} + \frac{0.16666666666666666}{\mathsf{PI}\left(\right) \cdot r}}{s}, 0.75, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]
12. Add Preprocessing

Alternative 5: 10.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.041666666666666664 + \frac{-0.004629629629629629 \cdot r}{s}}{s \cdot s}, r, \frac{-0.25}{s}\right), r, 0.75\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.125 (exp (/ (- r) s))) (* (* (PI) s) r))
  (/
   (fma
    (fma
     (/ (+ 0.041666666666666664 (/ (* -0.004629629629629629 r) s)) (* s s))
     r
     (/ -0.25 s))
    r
    0.75)
   (* (* (* 6.0 (PI)) s) r))))

Derivation

1. Initial program 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f32 N/A

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

   6. lower-/.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. times-frac N/A

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

   12. lift-*.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. lower-*.f32 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{8} \cdot e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4} + r \cdot \left(r \cdot \left(\frac{-1}{216} \cdot \frac{r}{{s}^{3}} + \frac{1}{24} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{4} \cdot \frac{1}{s}\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{8} \cdot e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{r \cdot \left(r \cdot \left(\frac{-1}{216} \cdot \frac{r}{{s}^{3}} + \frac{1}{24} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{4} \cdot \frac{1}{s}\right) + \frac{3}{4}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{8} \cdot e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\left(r \cdot \left(\frac{-1}{216} \cdot \frac{r}{{s}^{3}} + \frac{1}{24} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{4} \cdot \frac{1}{s}\right) \cdot r} + \frac{3}{4}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(r \cdot \left(\frac{-1}{216} \cdot \frac{r}{{s}^{3}} + \frac{1}{24} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

9. Applied rewrites 11.6%

   \[\leadsto \frac{0.125 \cdot e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.041666666666666664 + \frac{-0.004629629629629629 \cdot r}{s}}{s \cdot s}, r, \frac{-0.25}{s}\right), r, 0.75\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
10. Add Preprocessing

Alternative 6: 10.9% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

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

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

   5. associate-*r/ N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

   8. lower-fma.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. metadata-eval N/A

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

   12. associate-*r/ N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f32 11.5

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

7. Applied rewrites 11.5%

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

Alternative 7: 10.9% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 10.7

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

7. Applied rewrites 10.7%

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

8. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 11.5

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

10. Applied rewrites 11.5%

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

Alternative 8: 10.9% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around -inf

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. associate-/l* N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 11.5

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

7. Applied rewrites 11.5%

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

Alternative 9: 10.9% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f32 N/A

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

   6. lower-/.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. times-frac N/A

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

   12. lift-*.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. lower-*.f32 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

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

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. unpow2 N/A

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

   11. associate-/r* N/A

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

   12. div-add-rev N/A

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

   13. lower-/.f32 N/A

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

   14. lower-+.f32 N/A

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

   15. lower-/.f32 N/A

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

   16. lower-*.f32 11.5

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

9. Applied rewrites 11.5%

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

Alternative 10: 10.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{1.5 - \frac{\mathsf{fma}\left(r, 1, \frac{\left(\left(-r\right) \cdot r\right) \cdot 0.4166666666666667}{s}\right)}{s}}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (- 1.5 (/ (fma r 1.0 (/ (* (* (- r) r) 0.4166666666666667) s)) s))
  (* (* (* (PI) 6.0) s) r)))

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 9.9%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. frac-add N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

7. Applied rewrites 9.9%

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

8. Taylor expanded in s around -inf

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

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

      \[\leadsto \frac{\color{blue}{\frac{3}{2} - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{\frac{-1}{2} \cdot \left(\frac{-3}{2} \cdot r + \frac{-1}{2} \cdot r\right) + \frac{1}{2} \cdot \frac{\frac{-3}{4} \cdot {r}^{2} + \frac{-1}{12} \cdot {r}^{2}}{s}}{s}}}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{3}{2} - \color{blue}{1} \cdot \frac{\frac{-1}{2} \cdot \left(\frac{-3}{2} \cdot r + \frac{-1}{2} \cdot r\right) + \frac{1}{2} \cdot \frac{\frac{-3}{4} \cdot {r}^{2} + \frac{-1}{12} \cdot {r}^{2}}{s}}{s}}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]

   3. *-lft-identity N/A

      \[\leadsto \frac{\frac{3}{2} - \color{blue}{\frac{\frac{-1}{2} \cdot \left(\frac{-3}{2} \cdot r + \frac{-1}{2} \cdot r\right) + \frac{1}{2} \cdot \frac{\frac{-3}{4} \cdot {r}^{2} + \frac{-1}{12} \cdot {r}^{2}}{s}}{s}}}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]

   4. lower--.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{3}{2} - \frac{\frac{-1}{2} \cdot \left(\frac{-3}{2} \cdot r + \frac{-1}{2} \cdot r\right) + \frac{1}{2} \cdot \frac{\frac{-3}{4} \cdot {r}^{2} + \frac{-1}{12} \cdot {r}^{2}}{s}}{s}}}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{\frac{3}{2} - \color{blue}{\frac{\frac{-1}{2} \cdot \left(\frac{-3}{2} \cdot r + \frac{-1}{2} \cdot r\right) + \frac{1}{2} \cdot \frac{\frac{-3}{4} \cdot {r}^{2} + \frac{-1}{12} \cdot {r}^{2}}{s}}{s}}}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]

10. Applied rewrites 10.9%

    \[\leadsto \frac{\color{blue}{1.5 - \frac{\mathsf{fma}\left(r, 1, \frac{-\left(r \cdot r\right) \cdot 0.4166666666666667}{s}\right)}{s}}}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]

11. Final simplification 10.9%

    \[\leadsto \frac{1.5 - \frac{\mathsf{fma}\left(r, 1, \frac{\left(\left(-r\right) \cdot r\right) \cdot 0.4166666666666667}{s}\right)}{s}}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]
12. Add Preprocessing

Alternative 11: 10.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.4166666666666667 \cdot \frac{r}{s \cdot s} - \frac{1}{s}, r, 1.5\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (fma (- (* 0.4166666666666667 (/ r (* s s))) (/ 1.0 s)) r 1.5)
  (* (* (* (PI) 6.0) s) r)))

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 9.9%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. frac-add N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

7. Applied rewrites 9.9%

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

8. Taylor expanded in r around 0

   \[\leadsto \frac{\color{blue}{\frac{3}{2} + r \cdot \left(\frac{5}{12} \cdot \frac{r}{{s}^{2}} - \frac{1}{s}\right)}}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{r \cdot \left(\frac{5}{12} \cdot \frac{r}{{s}^{2}} - \frac{1}{s}\right) + \frac{3}{2}}}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{5}{12} \cdot \frac{r}{{s}^{2}} - \frac{1}{s}\right) \cdot r} + \frac{3}{2}}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{5}{12} \cdot \frac{r}{{s}^{2}} - \frac{1}{s}, r, \frac{3}{2}\right)}}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]

   4. lower--.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{5}{12} \cdot \frac{r}{{s}^{2}} - \frac{1}{s}}, r, \frac{3}{2}\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{5}{12} \cdot \frac{r}{{s}^{2}}} - \frac{1}{s}, r, \frac{3}{2}\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]

   6. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{5}{12} \cdot \color{blue}{\frac{r}{{s}^{2}}} - \frac{1}{s}, r, \frac{3}{2}\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]

   7. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{5}{12} \cdot \frac{r}{\color{blue}{s \cdot s}} - \frac{1}{s}, r, \frac{3}{2}\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]

   8. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{5}{12} \cdot \frac{r}{\color{blue}{s \cdot s}} - \frac{1}{s}, r, \frac{3}{2}\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]

   9. lower-/.f32 10.9

      \[\leadsto \frac{\mathsf{fma}\left(0.4166666666666667 \cdot \frac{r}{s \cdot s} - \color{blue}{\frac{1}{s}}, r, 1.5\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]

10. Applied rewrites 10.9%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.4166666666666667 \cdot \frac{r}{s \cdot s} - \frac{1}{s}, r, 1.5\right)}}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]
11. Add Preprocessing

Alternative 12: 9.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \frac{1.5 - \frac{r}{s}}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ (- 1.5 (/ r s)) (* (* (* (PI) 6.0) s) r)))

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 9.9%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. frac-add N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

7. Applied rewrites 9.9%

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

8. Taylor expanded in r around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 10.1

      \[\leadsto \frac{1.5 - \color{blue}{\frac{r}{s}}}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]

10. Applied rewrites 10.1%

    \[\leadsto \frac{\color{blue}{1.5 - \frac{r}{s}}}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r} \]
11. Add Preprocessing

Alternative 13: 9.1% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 9.8

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

5. Applied rewrites 9.8%

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

7. Applied rewrites 9.8%

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

9. Applied rewrites 9.8%

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

Reproduce

herbie shell --seed 2025011 
(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))))