Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 16.1s

Alternatives: 16

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 11.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}\\ t_1 := \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r\\ \mathbf{if}\;t\_0 + \frac{e^{\frac{-r}{s}} \cdot 0.25}{t\_1} \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;t\_0 + \frac{\mathsf{fma}\left(r, -0.25 \cdot \frac{1}{s}, 0.25\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r)))
        (t_1 (* (* (* (PI) 2.0) s) r)))
   (if (<= (+ t_0 (/ (* (exp (/ (- r) s)) 0.25) t_1)) 9.999999747378752e-5)
     (+ t_0 (/ (fma r (* -0.25 (/ 1.0 s)) 0.25) t_1))
     (/
      (-
       (/
        (-
         (/ -0.16666666666666666 (PI))
         (/
          (*
           (-
            (/ -0.06944444444444445 (PI))
            (* (/ -0.021604938271604937 s) (/ r (PI))))
           r)
          s))
        s)
       (/ -0.25 (* (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))) < 9.99999975e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. associate-+l+ N/A

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

   5. Applied rewrites 5.0%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{0.5}{s}, r, -1\right), 1\right)}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   6. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 5.0

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

   8. Applied rewrites 5.0%

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

   10. Applied rewrites 5.1%

       \[\leadsto \frac{\mathsf{fma}\left(r, \color{blue}{\frac{1}{s} \cdot -0.25}, 0.25\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   if 9.99999975e-5 < (+.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 98.1%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(r, -0.25 \cdot \frac{1}{s}, 0.25\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 12.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-{s}^{-2}\right) \cdot \frac{\mathsf{fma}\left(-0.06944444444444445, t\_0, \frac{s}{\mathsf{PI}\left(\right)} \cdot 0.16666666666666666\right)}{s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot t\_0\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ r (PI))))
   (if (<=
        (+
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* (PI) 2.0) s) r)))
        4.999999873689376e-5)
     (/
      (fma
       (*
        (- (pow s -2.0))
        (/
         (fma -0.06944444444444445 t_0 (* (/ s (PI)) 0.16666666666666666))
         s))
       r
       (/ 0.25 (* (PI) s)))
      r)
     (/
      (-
       (/
        (-
         (/ -0.16666666666666666 (PI))
         (/
          (*
           (-
            (/ -0.06944444444444445 (PI))
            (* (/ -0.021604938271604937 s) t_0))
           r)
          s))
        s)
       (/ -0.25 (* (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))) < 4.99999987e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   5. Applied rewrites 5.0%

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

   7. Applied rewrites 5.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(-\mathsf{fma}\left(\frac{0.06944444444444445}{\mathsf{PI}\left(\right)}, \frac{r}{s}, \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\right)\right) \cdot 1}{\left(-s\right) \cdot s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} \]
   8. Step-by-step derivation

   9. Applied rewrites 5.5%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.06944444444444445}{\mathsf{PI}\left(\right)}, \frac{-r}{s}, \frac{0.16666666666666666}{\mathsf{PI}\left(\right)}\right) \cdot \left(-{s}^{-2}\right), r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} \]

   10. Taylor expanded in s around 0

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

   12. Applied rewrites 5.8%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.06944444444444445, \frac{r}{\mathsf{PI}\left(\right)}, 0.16666666666666666 \cdot \frac{s}{\mathsf{PI}\left(\right)}\right)}{s} \cdot \left(-{s}^{-2}\right), r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} \]

   if 4.99999987e-5 < (+.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 98.0%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 11.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-{s}^{-2}\right) \cdot \frac{\mathsf{fma}\left(-0.06944444444444445, \frac{r}{\mathsf{PI}\left(\right)}, \frac{s}{\mathsf{PI}\left(\right)} \cdot 0.16666666666666666\right)}{s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 14.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(r, {\left(\left(14.4 \cdot s\right) \cdot \mathsf{PI}\left(\right)\right)}^{-1}, t\_0\right)}{s}}{s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ -0.16666666666666666 (PI))))
   (if (<=
        (+
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* (PI) 2.0) s) r)))
        9.999999747378752e-5)
     (/
      (fma
       (/ (/ (fma r (pow (* (* 14.4 s) (PI)) -1.0) t_0) s) s)
       r
       (/ 0.25 (* (PI) s)))
      r)
     (/
      (-
       (/
        (-
         t_0
         (/
          (*
           (-
            (/ -0.06944444444444445 (PI))
            (* (/ -0.021604938271604937 s) (/ r (PI))))
           r)
          s))
        s)
       (/ -0.25 (* (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))) < 9.99999975e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   5. Applied rewrites 5.0%

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

   7. Applied rewrites 7.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(r, {\left(\mathsf{PI}\left(\right) \cdot \left(s \cdot 14.4\right)\right)}^{-1}, \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\right)}{s}}{s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} \]

   if 9.99999975e-5 < (+.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 98.1%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 12.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(r, {\left(\left(14.4 \cdot s\right) \cdot \mathsf{PI}\left(\right)\right)}^{-1}, \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\right)}{s}}{s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 14.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-r, {\left(\left(14.4 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right)}^{-1}, \frac{0.16666666666666666}{\mathsf{PI}\left(\right)}\right)}{\left(-s\right) \cdot s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<=
      (+
       (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))
       (/ (* (exp (/ (- r) s)) 0.25) (* (* (* (PI) 2.0) s) r)))
      4.999999873689376e-5)
   (/
    (fma
     (/
      (fma (- r) (pow (* (* 14.4 (PI)) s) -1.0) (/ 0.16666666666666666 (PI)))
      (* (- s) s))
     r
     (/ 0.25 (* (PI) s)))
    r)
   (/
    (-
     (/
      (-
       (/ -0.16666666666666666 (PI))
       (/
        (*
         (-
          (/ -0.06944444444444445 (PI))
          (* (/ -0.021604938271604937 s) (/ r (PI))))
         r)
        s))
      s)
     (/ -0.25 (* (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))) < 4.99999987e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   5. Applied rewrites 5.0%

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

   7. Applied rewrites 5.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(-\mathsf{fma}\left(\frac{0.06944444444444445}{\mathsf{PI}\left(\right)}, \frac{r}{s}, \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\right)\right) \cdot 1}{\left(-s\right) \cdot s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} \]
   8. Step-by-step derivation

   9. Applied rewrites 5.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(-\mathsf{fma}\left(\frac{0.06944444444444445}{\mathsf{PI}\left(\right)}, \frac{r}{s}, \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\right)\right) \cdot 1}{\left(-s\right) \cdot s}, r, \frac{0.25}{\left(\mathsf{PI}\left(\right) \cdot 1\right) \cdot s}\right)}{r} \]
   10. Step-by-step derivation

   11. Applied rewrites 8.9%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-r, {\left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 14.4\right)\right)}^{-1}, \frac{0.16666666666666666}{\mathsf{PI}\left(\right)}\right)}{\left(-s\right) \cdot s}, r, \frac{0.25}{\left(\mathsf{PI}\left(\right) \cdot 1\right) \cdot s}\right)}{r} \]

   if 4.99999987e-5 < (+.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 98.0%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-r, {\left(\left(14.4 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right)}^{-1}, \frac{0.16666666666666666}{\mathsf{PI}\left(\right)}\right)}{\left(-s\right) \cdot s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 14.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{r}{s}, \frac{0.06944444444444445}{\mathsf{PI}\left(\right)}, t\_0\right) \cdot {s}^{-2}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ -0.16666666666666666 (PI))))
   (if (<=
        (+
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* (PI) 2.0) s) r)))
        9.999999747378752e-5)
     (/
      (fma
       (* (fma (/ r s) (/ 0.06944444444444445 (PI)) t_0) (pow s -2.0))
       r
       (/ 0.25 (* (PI) s)))
      r)
     (/
      (-
       (/
        (-
         t_0
         (/
          (*
           (-
            (/ -0.06944444444444445 (PI))
            (* (/ -0.021604938271604937 s) (/ r (PI))))
           r)
          s))
        s)
       (/ -0.25 (* (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))) < 9.99999975e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   5. Applied rewrites 5.0%

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

   7. Applied rewrites 5.0%

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

   9. Applied rewrites 5.5%

      \[\leadsto \frac{\mathsf{fma}\left(-1 \cdot \left(\mathsf{fma}\left(\frac{r}{s}, \frac{0.06944444444444445}{\mathsf{PI}\left(\right)}, \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\right) \cdot \left(-{s}^{-2}\right)\right), r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} \]

   if 9.99999975e-5 < (+.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 98.1%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{r}{s}, \frac{0.06944444444444445}{\mathsf{PI}\left(\right)}, \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\right) \cdot {s}^{-2}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 99.6

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.6

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 8: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 99.6

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.6

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in s around 0

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

    1. associate-*r/ N/A

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

    2. lower-/.f32 N/A

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

    3. *-commutative N/A

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

    4. mul-1-neg N/A

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

    5. exp-neg N/A

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

    6. associate-*l/ N/A

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

    7. metadata-eval N/A

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

    8. lower-/.f32 N/A

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

    9. lower-exp.f32 N/A

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

    10. lower-/.f32 N/A

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

    11. *-commutative N/A

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

    12. lower-*.f32 N/A

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

    13. *-commutative N/A

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

    14. lower-*.f32 N/A

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

    15. lower-PI.f32 99.6

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

11. Applied rewrites 99.6%

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

12. Final simplification 99.6%

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

Alternative 9: 12.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 8.000000134899068 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.06944444444444445}{\mathsf{PI}\left(\right)}, \frac{r}{s}, \frac{-0.16666666666666666}{\log \mathsf{E}\left(\right) \cdot \mathsf{PI}\left(\right)}\right)}{s \cdot s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(r, -0.25 \cdot \frac{1}{s}, 0.25\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= s 8.000000134899068e-16)
   (/
    (fma
     (/
      (fma
       (/ 0.06944444444444445 (PI))
       (/ r s)
       (/ -0.16666666666666666 (* (log (E)) (PI))))
      (* s s))
     r
     (/ 0.25 (* (PI) s)))
    r)
   (+
    (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))
    (/ (fma r (* -0.25 (/ 1.0 s)) 0.25) (* (* (* (PI) 2.0) s) r)))))

Derivation

1. Split input into 2 regimes

2. if s < 8.00000013e-16

   1. Initial program 100.0%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

   5. Applied rewrites 4.0%

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

   7. Applied rewrites 4.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(-\mathsf{fma}\left(\frac{0.06944444444444445}{\mathsf{PI}\left(\right)}, \frac{r}{s}, \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\right)\right) \cdot 1}{\left(-s\right) \cdot s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} \]
   8. Step-by-step derivation

   9. Applied rewrites 14.4%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(-\mathsf{fma}\left(\frac{0.06944444444444445}{\mathsf{PI}\left(\right)}, \frac{r}{s}, \frac{-0.16666666666666666}{\mathsf{PI}\left(\right) \cdot \log \mathsf{E}\left(\right)}\right)\right) \cdot 1}{\left(-s\right) \cdot s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} \]

   if 8.00000013e-16 < s

   1. Initial program 99.2%

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

   3. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l/ N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. associate-+l+ N/A

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

   5. Applied rewrites 13.2%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\mathsf{fma}\left(\frac{r}{s}, \mathsf{fma}\left(\frac{0.5}{s}, r, -1\right), 1\right)}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   6. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 15.1

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

   8. Applied rewrites 15.1%

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

   10. Applied rewrites 15.0%

       \[\leadsto \frac{\mathsf{fma}\left(r, \color{blue}{\frac{1}{s} \cdot -0.25}, 0.25\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 12.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 8.000000134899068 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.06944444444444445}{\mathsf{PI}\left(\right)}, \frac{r}{s}, \frac{-0.16666666666666666}{\log \mathsf{E}\left(\right) \cdot \mathsf{PI}\left(\right)}\right)}{s \cdot s}, r, \frac{0.25}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(r, -0.25 \cdot \frac{1}{s}, 0.25\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 9.3% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 99.6

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.6

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in s around inf

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

    1. *-commutative N/A

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

    2. associate-/r* N/A

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

    3. metadata-eval N/A

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

    4. associate-*r/ N/A

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

    5. lower-/.f32 N/A

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

    6. associate-*r/ N/A

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

    7. metadata-eval N/A

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

    8. lower-/.f32 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f32 N/A

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

    11. lower-PI.f32 8.5

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

11. Applied rewrites 8.5%

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

12. Final simplification 8.5%

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

Alternative 11: 9.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (-
   (/
    (-
     (/ -0.16666666666666666 (PI))
     (/
      (*
       (-
        (/ -0.06944444444444445 (PI))
        (* (/ -0.021604938271604937 s) (/ r (PI))))
       r)
      s))
    s)
   (/ -0.25 (* (PI) r)))
  s))

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 8.4%

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

6. Final simplification 8.4%

   \[\leadsto \frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s} \]
7. Add Preprocessing

Alternative 12: 9.1% accurate, 5.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 8.4

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

5. Applied rewrites 8.4%

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

7. Applied rewrites 8.4%

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

8. Final simplification 8.4%

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

Alternative 13: 9.1% accurate, 6.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 8.4

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

5. Applied rewrites 8.4%

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

7. Applied rewrites 8.4%

   \[\leadsto \frac{\frac{1}{\frac{s}{\frac{0.25}{\mathsf{PI}\left(\right)}}}}{r} \]
8. Add Preprocessing

Alternative 14: 9.1% accurate, 10.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 8.4

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

5. Applied rewrites 8.4%

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

Alternative 15: 9.1% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 8.4

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

5. Applied rewrites 8.4%

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

7. Applied rewrites 8.3%

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

9. Applied rewrites 8.3%

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

Alternative 16: 9.1% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 8.4

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

5. Applied rewrites 8.4%

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

7. Applied rewrites 8.3%

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

Reproduce

herbie shell --seed 2024332 
(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))))