Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 10.5s

Alternatives: 16

Speedup: N/A×

Specification

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

Math

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

FPCore

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

Alternative 2: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-r}{s}}\\ t_1 := \frac{0.25 \cdot t\_0}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}\\ \mathbf{if}\;t\_1 + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 0.0020000000949949026:\\ \;\;\;\;\frac{t\_0 + e^{\frac{\frac{r}{-3}}{s}}}{r} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \frac{\frac{\frac{\frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, 0.0007716049382716049, -0.006944444444444444 \cdot r\right)}{\mathsf{PI}\left(\right)}}{s} + \frac{0.041666666666666664}{\mathsf{PI}\left(\right)}}{s} - \frac{0.125}{\mathsf{PI}\left(\right) \cdot r}}{-s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (exp (/ (- r) s)))
        (t_1 (/ (* 0.25 t_0) (* (* (* 2.0 (PI)) s) r))))
   (if (<=
        (+ t_1 (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 (PI)) s) r)))
        0.0020000000949949026)
     (* (/ (+ t_0 (exp (/ (/ r -3.0) s))) r) 0.125)
     (+
      t_1
      (/
       (-
        (/
         (+
          (/
           (/
            (fma
             (* r (/ r s))
             0.0007716049382716049
             (* -0.006944444444444444 r))
            (PI))
           s)
          (/ 0.041666666666666664 (PI)))
         s)
        (/ 0.125 (* (PI) r)))
       (- s))))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.00200000009

   1. Initial program 99.7%

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

      1. lift-+.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f32 N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-fma.f32 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-fma.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. distribute-lft-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

   6. Applied rewrites 99.7%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f32 N/A

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

      10. lower-/.f32 99.7

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

      11. lift-+.f32 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f32 99.7

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

   8. Applied rewrites 99.7%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. metadata-eval N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f32 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. lower-*.f32 N/A

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

   10. Applied rewrites 99.3%

       \[\leadsto \color{blue}{\frac{e^{\frac{-r}{s}} + e^{\frac{\frac{r}{-3}}{s}}}{r} \cdot 0.125} \]

   if 0.00200000009 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 97.7%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 70.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 0.0020000000949949026:\\ \;\;\;\;\frac{e^{\frac{-r}{s}} + e^{\frac{\frac{r}{-3}}{s}}}{r} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{\frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, 0.0007716049382716049, -0.006944444444444444 \cdot r\right)}{\mathsf{PI}\left(\right)}}{s} + \frac{0.041666666666666664}{\mathsf{PI}\left(\right)}}{s} - \frac{0.125}{\mathsf{PI}\left(\right) \cdot r}}{-s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-r}{s}}\\ t_1 := \frac{0.25 \cdot t\_0}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}\\ \mathbf{if}\;t\_1 + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 0.0020000000949949026:\\ \;\;\;\;\frac{t\_0 + e^{\frac{\frac{r}{-3}}{s}}}{r} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \frac{\frac{0.125}{\mathsf{PI}\left(\right) \cdot r} + \frac{\frac{\mathsf{fma}\left(\frac{r}{s}, 0.006944444444444444, -0.041666666666666664\right)}{\mathsf{PI}\left(\right)}}{s}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (exp (/ (- r) s)))
        (t_1 (/ (* 0.25 t_0) (* (* (* 2.0 (PI)) s) r))))
   (if (<=
        (+ t_1 (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 (PI)) s) r)))
        0.0020000000949949026)
     (* (/ (+ t_0 (exp (/ (/ r -3.0) s))) r) 0.125)
     (+
      t_1
      (/
       (+
        (/ 0.125 (* (PI) r))
        (/
         (/ (fma (/ r s) 0.006944444444444444 -0.041666666666666664) (PI))
         s))
       s)))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.00200000009

   1. Initial program 99.7%

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

      1. lift-+.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f32 N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-fma.f32 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-fma.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. distribute-lft-out N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

   6. Applied rewrites 99.7%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f32 N/A

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

      10. lower-/.f32 99.7

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

      11. lift-+.f32 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f32 99.7

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

   8. Applied rewrites 99.7%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. metadata-eval N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f32 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. lower-*.f32 N/A

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

   10. Applied rewrites 99.3%

       \[\leadsto \color{blue}{\frac{e^{\frac{-r}{s}} + e^{\frac{\frac{r}{-3}}{s}}}{r} \cdot 0.125} \]

   if 0.00200000009 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 97.7%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

   5. Applied rewrites 67.2%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{0.125}{\mathsf{PI}\left(\right) \cdot r} + \frac{\frac{\mathsf{fma}\left(\frac{r}{s}, 0.006944444444444444, -0.041666666666666664\right)}{\mathsf{PI}\left(\right)}}{s}}{s}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

6. Applied rewrites 99.6%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

6. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. metadata-eval N/A

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

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

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

   6. lower-/.f32 N/A

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

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

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

   8. metadata-eval N/A

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

   9. lower-*.f32 99.5

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

8. Applied rewrites 99.5%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.5

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

9. Applied rewrites 99.5%

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

Alternative 7: 10.9% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 11.0%

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

Alternative 8: 10.3% accurate, 3.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f32 N/A

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

7. Applied rewrites 10.2%

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

8. Final simplification 10.2%

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

Alternative 9: 10.3% accurate, 3.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 10.2%

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

9. Applied rewrites 10.2%

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

10. Final simplification 10.2%

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

Alternative 10: 10.3% accurate, 4.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 10.2%

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

8. Final simplification 10.2%

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

Alternative 11: 10.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{2 - \frac{\mathsf{fma}\left(1.3333333333333333, r, \frac{\left(r \cdot r\right) \cdot 0.5555555555555556}{-s}\right)}{s}}{\left(s \cdot r\right) \cdot \mathsf{PI}\left(\right)} \cdot 0.125 \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/
   (-
    2.0
    (/ (fma 1.3333333333333333 r (/ (* (* r r) 0.5555555555555556) (- s))) s))
   (* (* s r) (PI)))
  0.125))

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in s around -inf

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

9. Applied rewrites 10.2%

   \[\leadsto \frac{\color{blue}{2 - \frac{\mathsf{fma}\left(1.3333333333333333, r, \frac{\left(r \cdot r\right) \cdot 0.5555555555555556}{-s}\right)}{s}}}{\left(s \cdot r\right) \cdot \mathsf{PI}\left(\right)} \cdot 0.125 \]
10. Add Preprocessing

Alternative 12: 10.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{1.3333333333333333 - \frac{r \cdot 0.5555555555555556}{s}}{s}, -1, \frac{2}{r}\right)}{\mathsf{PI}\left(\right) \cdot s} \cdot 0.125 \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/
   (fma
    (/ (- 1.3333333333333333 (/ (* r 0.5555555555555556) s)) s)
    -1.0
    (/ 2.0 r))
   (* (PI) s))
  0.125))

Derivation

1. Initial program 99.5%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

6. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

   10. lower-/.f32 99.5

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

   11. lift-+.f32 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f32 99.5

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in s around -inf

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

    1. *-commutative N/A

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

    2. lower-fma.f32 N/A

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

    3. lower-/.f32 N/A

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

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

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

    5. metadata-eval N/A

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

    6. *-lft-identity N/A

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

    7. lower--.f32 N/A

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

    8. lower-/.f32 N/A

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

    9. distribute-rgt-out N/A

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

    10. metadata-eval N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{4}{3} - \frac{r \cdot \color{blue}{\frac{5}{9}}}{s}}{s}, -1, 2 \cdot \frac{1}{r}\right)}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{1}{8} \]

    11. lower-*.f32 N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{4}{3} - \frac{\color{blue}{r \cdot \frac{5}{9}}}{s}}{s}, -1, 2 \cdot \frac{1}{r}\right)}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{1}{8} \]

    12. associate-*r/ N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{4}{3} - \frac{r \cdot \frac{5}{9}}{s}}{s}, -1, \color{blue}{\frac{2 \cdot 1}{r}}\right)}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{1}{8} \]

    13. metadata-eval N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{4}{3} - \frac{r \cdot \frac{5}{9}}{s}}{s}, -1, \frac{\color{blue}{2}}{r}\right)}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{1}{8} \]

    14. lower-/.f32 10.1

        \[\leadsto \frac{\mathsf{fma}\left(\frac{1.3333333333333333 - \frac{r \cdot 0.5555555555555556}{s}}{s}, -1, \color{blue}{\frac{2}{r}}\right)}{\mathsf{PI}\left(\right) \cdot s} \cdot 0.125 \]

11. Applied rewrites 10.1%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1.3333333333333333 - \frac{r \cdot 0.5555555555555556}{s}}{s}, -1, \frac{2}{r}\right)}}{\mathsf{PI}\left(\right) \cdot s} \cdot 0.125 \]
12. Add Preprocessing

Alternative 13: 9.2% accurate, 6.3× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (sqrt (PI)))) (/ 0.25 (* (* r t_0) (* t_0 s)))))

Derivation

1. Initial program 99.5%

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

3. 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 9.2

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

5. Applied rewrites 9.2%

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

7. Applied rewrites 9.2%

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

9. Applied rewrites 9.2%

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

11. Applied rewrites 9.2%

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

Alternative 14: 9.2% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 9.2

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

5. Applied rewrites 9.2%

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

7. Applied rewrites 9.2%

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

9. Applied rewrites 9.2%

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

Alternative 15: 9.2% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 9.2

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

5. Applied rewrites 9.2%

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

7. Applied rewrites 9.2%

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

9. Applied rewrites 9.2%

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

Alternative 16: 9.2% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 9.2

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

5. Applied rewrites 9.2%

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

7. Applied rewrites 9.2%

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

Reproduce

herbie shell --seed 2025009 
(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))))