Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 19.5s

Alternatives: 26

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

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

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 99.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 11.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(t\_0 \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125, \frac{\frac{\frac{r}{s}}{s}}{\mathsf{PI}\left(\right)}, \frac{0.125}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} + \frac{e^{\frac{r}{-3 \cdot s}} \cdot 0.75}{t\_0 \cdot \left(s \cdot r\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* 6.0 (PI))))
   (if (<=
        (+
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* t_0 s) r))
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* (PI) 2.0) s) r)))
        1.9999999494757503e-5)
     (+
      (/ (fma -0.125 (/ (/ (/ r s) s) (PI)) (/ 0.125 (* (PI) s))) r)
      (/ (* (exp (/ r (* -3.0 s))) 0.75) (* t_0 (* 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))) < 1.99999995e-5

   1. Initial program 99.7%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-*.f32 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f32 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f32 N/A

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

      6. lift-*.f32 N/A

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

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

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

      8. lower-*.f32 N/A

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

      9. metadata-eval 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in r around 0

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lower-PI.f32 5.4

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

   9. Applied rewrites 5.4%

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

   if 1.99999995e-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 99.4%

      \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
   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 82.7%

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

4. Final simplification 11.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125, \frac{\frac{\frac{r}{s}}{s}}{\mathsf{PI}\left(\right)}, \frac{0.125}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} + \frac{e^{\frac{r}{-3 \cdot s}} \cdot 0.75}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(s \cdot r\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 11.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r\\ \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{t\_0} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{\left(\frac{1}{s} \cdot -0.3333333333333333\right) \cdot r} \cdot 0.75}{t\_0} + \frac{\mathsf{fma}\left(-0.125, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{0.125}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (* (* 6.0 (PI)) s) r)))
   (if (<=
        (+
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) t_0)
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* (PI) 2.0) s) r)))
        1.9999999494757503e-5)
     (+
      (/ (* (exp (* (* (/ 1.0 s) -0.3333333333333333) r)) 0.75) t_0)
      (/ (fma -0.125 (/ r (* (* s s) (PI))) (/ 0.125 (* (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))) < 1.99999995e-5

   1. Initial program 99.7%

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

      1. lift-/.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f32 N/A

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

      5. times-frac N/A

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

      6. lower-*.f32 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f32 99.7

         \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{-0.3333333333333333 \cdot \color{blue}{\frac{r}{s}}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f32 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-/.f32 99.7

         \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{r \cdot \left(\color{blue}{\frac{1}{s}} \cdot -0.3333333333333333\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   6. Applied rewrites 99.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{r \cdot \left(\frac{1}{s} \cdot -0.3333333333333333\right)}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   7. Taylor expanded in r around 0

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-PI.f32 5.4

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

   9. Applied rewrites 5.4%

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

   if 1.99999995e-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 99.4%

      \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
   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 82.7%

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

4. Final simplification 11.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{\left(\frac{1}{s} \cdot -0.3333333333333333\right) \cdot r} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(-0.125, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{0.125}{\mathsf{PI}\left(\right) \cdot s}\right)}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 11.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}\\ \mathbf{if}\;t\_0 + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{0.125}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} + t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))))
   (if (<=
        (+ t_0 (/ (* (exp (/ (- r) s)) 0.25) (* (* (* (PI) 2.0) s) r)))
        1.9999999494757503e-5)
     (+ (/ (fma -0.125 (/ r (* (* s s) (PI))) (/ 0.125 (* (PI) s))) r) t_0)
     (/
      (-
       (/
        (-
         (/ -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))) < 1.99999995e-5

   1. Initial program 99.7%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in r around 0

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-PI.f32 5.3

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

   7. Applied rewrites 5.5%

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

   if 1.99999995e-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 99.4%

      \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
   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 82.7%

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

4. Final simplification 11.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{0.125}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 11.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r\\ \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{t\_0} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{0.125}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333} \cdot 0.75}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (* (* 6.0 (PI)) s) r)))
   (if (<=
        (+
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) t_0)
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* (PI) 2.0) s) r)))
        1.9999999494757503e-5)
     (+
      (/ (fma -0.125 (/ r (* (* s s) (PI))) (/ 0.125 (* (PI) s))) r)
      (/ (* (exp (* (/ r s) -0.3333333333333333)) 0.75) t_0))
     (/
      (-
       (/
        (-
         (/ -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))) < 1.99999995e-5

   1. Initial program 99.7%

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

      1. lift-/.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f32 N/A

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

      5. times-frac N/A

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

      6. lower-*.f32 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f32 99.7

         \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{-0.3333333333333333 \cdot \color{blue}{\frac{r}{s}}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   5. Taylor expanded in r around 0

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

      1. lower-/.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-PI.f32 5.4

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

   7. Applied rewrites 5.4%

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

   if 1.99999995e-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 99.4%

      \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
   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 82.7%

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

4. Final simplification 11.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{0.125}{\mathsf{PI}\left(\right) \cdot s}\right)}{r} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 13.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-r}{s}\\ t_1 := \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}\\ \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{t\_0} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 1.000000045813705 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right), t\_0, -0.16666666666666666\right)} \cdot s\right)}^{-1}, \frac{1}{s}, \frac{t\_1}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{r}{\mathsf{PI}\left(\right)} \cdot -0.06944444444444445}{s}}{s} - t\_1}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ (- r) s)) (t_1 (/ -0.25 (* (PI) r))))
   (if (<=
        (+
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))
         (/ (* (exp t_0) 0.25) (* (* (* (PI) 2.0) s) r)))
        1.000000045813705e-18)
     (fma
      (pow
       (*
        (/
         (PI)
         (fma
          (fma (/ 0.021604938271604937 s) r -0.06944444444444445)
          t_0
          -0.16666666666666666))
        s)
       -1.0)
      (/ 1.0 s)
      (/ t_1 (- s)))
     (/
      (-
       (/
        (-
         (/ -0.16666666666666666 (PI))
         (/ (* (/ r (PI)) -0.06944444444444445) s))
        s)
       t_1)
      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))) < 1.00000005e-18

   1. Initial program 99.9%

      \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
   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 3.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 6.7%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \color{blue}{\frac{1}{s}}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 6.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666 - \frac{r}{s} \cdot \mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right)}{\mathsf{PI}\left(\right)} \cdot 1}{s}, \frac{1}{s}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 7.2%

       \[\leadsto \mathsf{fma}\left({\left(\frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right), \frac{-r}{s}, -0.16666666666666666\right)} \cdot s\right)}^{-1}, \frac{\color{blue}{1}}{s}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]

   if 1.00000005e-18 < (+.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 98.3%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{\frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)} - \frac{\frac{-5}{72} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s}}{s} - \frac{\frac{-1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]
   7. Step-by-step derivation

   8. Applied rewrites 61.7%

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

4. Final simplification 13.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 1.000000045813705 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right), \frac{-r}{s}, -0.16666666666666666\right)} \cdot s\right)}^{-1}, \frac{1}{s}, \frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{r}{\mathsf{PI}\left(\right)} \cdot -0.06944444444444445}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 13.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}\\ \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 2.0000000390829628 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(r \cdot r\right) \cdot \left(\frac{-0.021604938271604937}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)} - \frac{\frac{\frac{0.16666666666666666}{\mathsf{PI}\left(\right)}}{r} - \frac{0.06944444444444445}{\mathsf{PI}\left(\right) \cdot s}}{r}\right)}{s}, \frac{1}{s}, \frac{t\_0}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{r}{\mathsf{PI}\left(\right)} \cdot -0.06944444444444445}{s}}{s} - t\_0}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ -0.25 (* (PI) r))))
   (if (<=
        (+
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* (PI) 2.0) s) r)))
        2.0000000390829628e-25)
     (fma
      (/
       (*
        (* r r)
        (-
         (/ -0.021604938271604937 (* (* s s) (PI)))
         (/
          (-
           (/ (/ 0.16666666666666666 (PI)) r)
           (/ 0.06944444444444445 (* (PI) s)))
          r)))
       s)
      (/ 1.0 s)
      (/ t_0 (- s)))
     (/
      (-
       (/
        (-
         (/ -0.16666666666666666 (PI))
         (/ (* (/ r (PI)) -0.06944444444444445) s))
        s)
       t_0)
      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))) < 2.00000004e-25

   1. Initial program 99.9%

      \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
   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 3.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 5.9%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \color{blue}{\frac{1}{s}}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 5.0%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666 - \frac{r}{s} \cdot \mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right)}{\mathsf{PI}\left(\right)} \cdot 1}{s}, \frac{1}{s}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]

   10. Taylor expanded in r around -inf

       \[\leadsto \mathsf{fma}\left(\frac{\left({r}^{2} \cdot \left(-1 \cdot \frac{\frac{1}{6} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} - \frac{5}{72} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r} - \frac{7}{324} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot 1}{s}, \frac{1}{s}, -\frac{\frac{\frac{-1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 5.1%

       \[\leadsto \mathsf{fma}\left(\frac{\left(\left(\frac{-0.021604938271604937}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)} - \frac{\frac{\frac{0.16666666666666666}{\mathsf{PI}\left(\right)}}{r} - \frac{0.06944444444444445}{\mathsf{PI}\left(\right) \cdot s}}{r}\right) \cdot \left(r \cdot r\right)\right) \cdot 1}{s}, \frac{1}{s}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]

   if 2.00000004e-25 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 97.9%

      \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
   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 58.2%

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

   6. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{\frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)} - \frac{\frac{-5}{72} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s}}{s} - \frac{\frac{-1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.0%

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

4. Final simplification 11.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 2.0000000390829628 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(r \cdot r\right) \cdot \left(\frac{-0.021604938271604937}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)} - \frac{\frac{\frac{0.16666666666666666}{\mathsf{PI}\left(\right)}}{r} - \frac{0.06944444444444445}{\mathsf{PI}\left(\right) \cdot s}}{r}\right)}{s}, \frac{1}{s}, \frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{r}{\mathsf{PI}\left(\right)} \cdot -0.06944444444444445}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 13.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}\\ t_1 := \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 3.999999886872274 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_1 - \left(\left(\frac{\frac{0.021604938271604937}{s}}{\mathsf{PI}\left(\right)} - \frac{\frac{0.06944444444444445}{r}}{\mathsf{PI}\left(\right)}\right) \cdot r\right) \cdot \frac{r}{s}}{s}, \frac{1}{s}, \frac{t\_0}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1 - \frac{\frac{r}{\mathsf{PI}\left(\right)} \cdot -0.06944444444444445}{s}}{s} - t\_0}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ -0.25 (* (PI) r))) (t_1 (/ -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)))
        3.999999886872274e-9)
     (fma
      (/
       (-
        t_1
        (*
         (*
          (-
           (/ (/ 0.021604938271604937 s) (PI))
           (/ (/ 0.06944444444444445 r) (PI)))
          r)
         (/ r s)))
       s)
      (/ 1.0 s)
      (/ t_0 (- s)))
     (/ (- (/ (- t_1 (/ (* (/ r (PI)) -0.06944444444444445) s)) s) t_0) 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))) < 3.99999989e-9

   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

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

   5. Applied rewrites 3.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 5.1%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \color{blue}{\frac{1}{s}}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]

   8. Taylor expanded in r around inf

      \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)} - \left(r \cdot \left(\frac{7}{324} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)} - \frac{5}{72} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot \frac{r}{s}}{s}, \frac{1}{s}, -\frac{\frac{\frac{-1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 5.6%

       \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \left(\left(\frac{\frac{0.021604938271604937}{s}}{\mathsf{PI}\left(\right)} - \frac{\frac{0.06944444444444445}{r}}{\mathsf{PI}\left(\right)}\right) \cdot r\right) \cdot \frac{r}{s}}{s}, \frac{1}{s}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]

   if 3.99999989e-9 < (+.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 98.9%

      \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
   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 66.7%

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

   6. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{\frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)} - \frac{\frac{-5}{72} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s}}{s} - \frac{\frac{-1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.6%

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

4. Final simplification 11.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 3.999999886872274 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \left(\left(\frac{\frac{0.021604938271604937}{s}}{\mathsf{PI}\left(\right)} - \frac{\frac{0.06944444444444445}{r}}{\mathsf{PI}\left(\right)}\right) \cdot r\right) \cdot \frac{r}{s}}{s}, \frac{1}{s}, \frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{r}{\mathsf{PI}\left(\right)} \cdot -0.06944444444444445}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 13.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}\\ \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 3.999999886872274 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-0.16666666666666666 \cdot \mathsf{PI}\left(\right) - \left(\mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right) \cdot \frac{r}{s}\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{s}, \frac{1}{s}, \frac{t\_0}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{r}{\mathsf{PI}\left(\right)} \cdot -0.06944444444444445}{s}}{s} - t\_0}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ -0.25 (* (PI) r))))
   (if (<=
        (+
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* (PI) 2.0) s) r)))
        3.999999886872274e-9)
     (fma
      (/
       (/
        (-
         (* -0.16666666666666666 (PI))
         (*
          (* (fma (/ 0.021604938271604937 s) r -0.06944444444444445) (/ r s))
          (PI)))
        (* (PI) (PI)))
       s)
      (/ 1.0 s)
      (/ t_0 (- s)))
     (/
      (-
       (/
        (-
         (/ -0.16666666666666666 (PI))
         (/ (* (/ r (PI)) -0.06944444444444445) s))
        s)
       t_0)
      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))) < 3.99999989e-9

   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

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

   5. Applied rewrites 3.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 5.9%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \color{blue}{\frac{1}{s}}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 5.9%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666 \cdot \mathsf{PI}\left(\right) - \mathsf{PI}\left(\right) \cdot \left(\frac{r}{s} \cdot \mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right)\right)}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{s}, \frac{1}{s}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 5.9%

       \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666 \cdot {\mathsf{PI}\left(\right)}^{1} - \mathsf{PI}\left(\right) \cdot \left(\frac{r}{s} \cdot \mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right)\right)}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{s}, \frac{1}{s}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]

   if 3.99999989e-9 < (+.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 98.9%

      \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
   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 66.7%

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

   6. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{\frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)} - \frac{\frac{-5}{72} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s}}{s} - \frac{\frac{-1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.6%

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

4. Final simplification 11.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 3.999999886872274 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-0.16666666666666666 \cdot \mathsf{PI}\left(\right) - \left(\mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right) \cdot \frac{r}{s}\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{s}, \frac{1}{s}, \frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{r}{\mathsf{PI}\left(\right)} \cdot -0.06944444444444445}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 99.6

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

9. Applied rewrites 99.6%

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

10. Final simplification 99.6%

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

Alternative 11: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{r}{s} \cdot -0.3333333333333333} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* (exp (* (/ r s) -0.3333333333333333)) 0.75) (* (* (* 6.0 (PI)) s) r))
  (/ (* (exp (/ (- r) s)) 0.25) (* (* (* (PI) 2.0) s) r))))

Derivation

1. Initial program 99.7%

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

   1. lift-/.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. neg-mul-1 N/A

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

   4. lift-*.f32 N/A

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

   5. times-frac N/A

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

   6. lower-*.f32 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 99.6

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{-0.3333333333333333 \cdot \color{blue}{\frac{r}{s}}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

4. Applied rewrites 99.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

5. Final simplification 99.6%

   \[\leadsto \frac{e^{\frac{r}{s} \cdot -0.3333333333333333} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \]
6. Add Preprocessing

Alternative 12: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. neg-mul-1 N/A

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

   4. lift-*.f32 N/A

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

   5. times-frac N/A

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

   6. lower-*.f32 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 99.6

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{-0.3333333333333333 \cdot \color{blue}{\frac{r}{s}}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

4. Applied rewrites 99.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

5. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. mul-1-neg N/A

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

   6. rec-exp N/A

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

   7. associate-/l/ N/A

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

   8. associate-*r* N/A

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

   9. associate-*l/ N/A

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

   10. metadata-eval N/A

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

   11. associate-/r* N/A

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

   12. associate-/l/ N/A

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

   13. lower-/.f32 N/A

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

   14. lower-/.f32 N/A

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

   \[\leadsto \frac{\frac{0.125}{r}}{e^{\frac{r}{s}} \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
9. Add Preprocessing

Alternative 13: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r\right) \cdot e^{\frac{r}{s}}} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ 0.25 (* (* (* (* (PI) 2.0) s) r) (exp (/ r s))))
  (/ (* (exp (* (/ r s) -0.3333333333333333)) 0.75) (* (* (* 6.0 (PI)) s) r))))

Derivation

1. Initial program 99.7%

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

   1. lift-/.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. neg-mul-1 N/A

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

   4. lift-*.f32 N/A

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

   5. times-frac N/A

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

   6. lower-*.f32 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 99.6

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{-0.3333333333333333 \cdot \color{blue}{\frac{r}{s}}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

4. Applied rewrites 99.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. distribute-frac-neg N/A

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

   7. lift-/.f32 N/A

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

   8. rec-exp N/A

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

   9. lift-exp.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. un-div-inv N/A

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

   13. associate-/l/ N/A

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

   14. lower-/.f32 N/A

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

   15. lower-*.f32 99.6

       \[\leadsto \frac{0.25}{\color{blue}{\left(\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r\right) \cdot e^{\frac{r}{s}}}} + \frac{0.75 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{0.25}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r\right) \cdot e^{\frac{r}{s}}}} + \frac{0.75 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

7. Final simplification 99.6%

   \[\leadsto \frac{0.25}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r\right) \cdot e^{\frac{r}{s}}} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
8. Add Preprocessing

Alternative 14: 14.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}\\ \mathbf{if}\;s \leq 1.999999943436137 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-0.16666666666666666 \cdot \mathsf{PI}\left(\right) - \left(\mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right) \cdot \frac{r}{s}\right) \cdot {\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{s}, \frac{1}{s}, \frac{t\_0}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{r}{\mathsf{PI}\left(\right)} \cdot -0.06944444444444445}{s}}{s} - t\_0}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ -0.25 (* (PI) r))))
   (if (<= s 1.999999943436137e-9)
     (fma
      (/
       (/
        (-
         (* -0.16666666666666666 (PI))
         (*
          (* (fma (/ 0.021604938271604937 s) r -0.06944444444444445) (/ r s))
          (pow (sqrt (PI)) 2.0)))
        (* (PI) (PI)))
       s)
      (/ 1.0 s)
      (/ t_0 (- s)))
     (/
      (-
       (/
        (-
         (/ -0.16666666666666666 (PI))
         (/ (* (/ r (PI)) -0.06944444444444445) s))
        s)
       t_0)
      s))))

Derivation

1. Split input into 2 regimes

2. if s < 1.99999994e-9

   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 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 3.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 5.3%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \color{blue}{\frac{1}{s}}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 5.2%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666 \cdot \mathsf{PI}\left(\right) - \mathsf{PI}\left(\right) \cdot \left(\frac{r}{s} \cdot \mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right)\right)}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{s}, \frac{1}{s}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 5.9%

       \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666 \cdot \mathsf{PI}\left(\right) - {\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\frac{r}{s} \cdot \mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right)\right)}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{s}, \frac{1}{s}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]

   if 1.99999994e-9 < s

   1. Initial program 98.8%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 29.7%

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

   6. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{\frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)} - \frac{\frac{-5}{72} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s}}{s} - \frac{\frac{-1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.8%

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

4. Final simplification 11.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.999999943436137 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-0.16666666666666666 \cdot \mathsf{PI}\left(\right) - \left(\mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right) \cdot \frac{r}{s}\right) \cdot {\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2}}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{s}, \frac{1}{s}, \frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{r}{\mathsf{PI}\left(\right)} \cdot -0.06944444444444445}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 13.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}\\ \mathbf{if}\;s \leq 1.999999943436137 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-0.16666666666666666 \cdot \mathsf{PI}\left(\right) - \left(\mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right) \cdot \frac{r}{s}\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{s}, \frac{1}{s}, \frac{t\_0}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{r}{\mathsf{PI}\left(\right)} \cdot -0.06944444444444445}{s}}{s} - t\_0}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ -0.25 (* (PI) r))))
   (if (<= s 1.999999943436137e-9)
     (fma
      (/
       (/
        (-
         (* -0.16666666666666666 (PI))
         (*
          (* (fma (/ 0.021604938271604937 s) r -0.06944444444444445) (/ r s))
          (PI)))
        (* (PI) (PI)))
       s)
      (/ 1.0 s)
      (/ t_0 (- s)))
     (/
      (-
       (/
        (-
         (/ -0.16666666666666666 (PI))
         (/ (* (/ r (PI)) -0.06944444444444445) s))
        s)
       t_0)
      s))))

Derivation

1. Split input into 2 regimes

2. if s < 1.99999994e-9

   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 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 3.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 6.4%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \color{blue}{\frac{1}{s}}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 5.3%

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666 \cdot \mathsf{PI}\left(\right) - \mathsf{PI}\left(\right) \cdot \left(\frac{r}{s} \cdot \mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right)\right)}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{s}, \frac{1}{s}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 5.3%

       \[\leadsto \mathsf{fma}\left(\frac{\frac{-0.16666666666666666 \cdot \mathsf{PI}\left(\right) - \mathsf{PI}\left(\right) \cdot \left(\frac{r}{s} \cdot \mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right)\right)}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{s}, \frac{1}{s}, -\frac{\frac{-0.25}{{\mathsf{PI}\left(\right)}^{1} \cdot r}}{s}\right) \]

   if 1.99999994e-9 < s

   1. Initial program 98.8%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 29.7%

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

   6. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{\frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)} - \frac{\frac{-5}{72} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s}}{s} - \frac{\frac{-1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.8%

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

4. Final simplification 10.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.999999943436137 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{-0.16666666666666666 \cdot \mathsf{PI}\left(\right) - \left(\mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right) \cdot \frac{r}{s}\right) \cdot \mathsf{PI}\left(\right)}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}{s}, \frac{1}{s}, \frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{r}{\mathsf{PI}\left(\right)} \cdot -0.06944444444444445}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 10.1% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. 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 9.8%

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

6. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{\frac{\frac{-1}{6}}{\mathsf{PI}\left(\right)} - \frac{\frac{-5}{72} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s}}{s} - \frac{\frac{-1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]
7. Step-by-step derivation

8. Applied rewrites 10.1%

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

Alternative 17: 8.4% accurate, 3.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. 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 9.2

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

5. Applied rewrites 9.2%

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

7. Applied rewrites 9.2%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 9.2%

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

11. Final simplification 9.2%

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

Alternative 18: 9.1% accurate, 6.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 9.2%

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

Alternative 19: 9.0% accurate, 6.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. 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 9.2

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

5. Applied rewrites 9.2%

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

7. Applied rewrites 9.2%

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

9. Applied rewrites 9.2%

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

10. Final simplification 9.2%

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

Alternative 20: 9.0% accurate, 6.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. 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 9.2

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

5. Applied rewrites 9.2%

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

7. Applied rewrites 9.2%

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

9. Applied rewrites 9.2%

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

10. Final simplification 9.2%

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

Alternative 21: 9.0% accurate, 7.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. 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 9.2

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

5. Applied rewrites 9.2%

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

7. Applied rewrites 9.2%

   \[\leadsto \frac{\frac{0.25}{\mathsf{PI}\left(\right)} \cdot \frac{1}{r}}{\color{blue}{s}} \]

8. Final simplification 9.2%

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

Alternative 22: 9.0% accurate, 7.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. 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 9.2

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

5. Applied rewrites 9.2%

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

7. Applied rewrites 9.2%

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

9. Applied rewrites 9.2%

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

Alternative 23: 9.0% accurate, 9.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. 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 9.2

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

5. Applied rewrites 9.2%

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

7. Applied rewrites 9.2%

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

9. Applied rewrites 9.2%

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

Alternative 24: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
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 9.2

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

5. Applied rewrites 9.2%

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

7. Applied rewrites 9.2%

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

9. Applied rewrites 9.2%

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

Alternative 25: 9.0% 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 99.7%

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

3. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
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 9.2

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

5. Applied rewrites 9.2%

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

7. Applied rewrites 9.2%

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

Alternative 26: 9.0% 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 99.7%

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

3. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
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 9.2

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

5. Applied rewrites 9.2%

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

7. Applied rewrites 9.2%

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

9. Applied rewrites 9.2%

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

Reproduce

herbie shell --seed 2024284 
(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))))