Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 13.9s

Alternatives: 14

Speedup: N/A×

Specification

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

Math

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

FPCore

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 99.6

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. metadata-eval N/A

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

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

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

   6. lift-*.f32 N/A

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

   7. frac-2neg N/A

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

   8. lift-/.f32 99.6

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 94.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-r}{s}}\\ t_1 := \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r\\ \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{t\_0 \cdot 0.25}{t\_1} \leq 0.6000000238418579:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{r}{s}, \frac{0.25}{t\_1}, \frac{0.125}{r} \cdot \frac{\sqrt[3]{t\_0}}{\mathsf{PI}\left(\right) \cdot s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (exp (/ (- r) s))) (t_1 (* (* (* (PI) 2.0) s) r)))
   (if (<=
        (+
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))
         (/ (* t_0 0.25) t_1))
        0.6000000238418579)
     (fma
      (- 1.0 (/ r s))
      (/ 0.25 t_1)
      (* (/ 0.125 r) (/ (cbrt t_0) (* (PI) s))))
     (/
      (-
       (/
        (-
         (/ -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))) < 0.600000024

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 3.9

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

   5. Applied rewrites 3.9%

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

   7. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - \frac{r}{s}, \frac{0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r}, \frac{\sqrt[3]{e^{\frac{-r}{s}}}}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{0.125}{r}\right)} \]

   if 0.600000024 < (+.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 97.8%

      \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
   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 77.6%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 0.6000000238418579:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{r}{s}, \frac{0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r}, \frac{0.125}{r} \cdot \frac{\sqrt[3]{e^{\frac{-r}{s}}}}{\mathsf{PI}\left(\right) \cdot s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-r}{s}}\\ \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{t\_0 \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 0.6000000238418579:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{r}{s} \cdot r, 0.6666666666666666, r\right) + r}{\frac{\frac{r}{\frac{t\_0}{s}}}{\frac{0.125}{\mathsf{PI}\left(\right)}}}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (exp (/ (- r) s))))
   (if (<=
        (+
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))
         (/ (* t_0 0.25) (* (* (* (PI) 2.0) s) r)))
        0.6000000238418579)
     (/
      (/
       (+ (fma (* (/ r s) r) 0.6666666666666666 r) r)
       (/ (/ r (/ t_0 s)) (/ 0.125 (PI))))
      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))) < 0.600000024

   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

   4. Applied rewrites 3.3%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f32 N/A

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

      7. lower-/.f32 96.3

         \[\leadsto \frac{\frac{r + \mathsf{fma}\left(r \cdot \color{blue}{\frac{r}{s}}, 0.6666666666666666, r\right)}{\frac{\frac{r}{\frac{e^{\frac{-r}{s}}}{s}}}{\frac{0.125}{\mathsf{PI}\left(\right)}}}}{r} \]

   7. Applied rewrites 96.3%

      \[\leadsto \frac{\frac{r + \color{blue}{\mathsf{fma}\left(r \cdot \frac{r}{s}, 0.6666666666666666, r\right)}}{\frac{\frac{r}{\frac{e^{\frac{-r}{s}}}{s}}}{\frac{0.125}{\mathsf{PI}\left(\right)}}}}{r} \]

   if 0.600000024 < (+.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 97.8%

      \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
   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 77.6%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 0.6000000238418579:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{r}{s} \cdot r, 0.6666666666666666, r\right) + r}{\frac{\frac{r}{\frac{e^{\frac{-r}{s}}}{s}}}{\frac{0.125}{\mathsf{PI}\left(\right)}}}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{-0.021604938271604937}{s} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 5: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 99.6

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. metadata-eval N/A

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

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

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

   6. lift-*.f32 N/A

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

   7. frac-2neg N/A

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

   8. lift-/.f32 99.6

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

8. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-neg.f32 N/A

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

   11. lift-exp.f32 N/A

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

   12. lower-/.f32 97.3

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 97.3

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

10. Applied rewrites 97.3%

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

11. Final simplification 97.3%

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

Alternative 6: 44.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(\frac{r}{\mathsf{PI}\left(\right)} \cdot 0.125, \frac{-1.3333333333333333}{s}, \frac{\frac{\mathsf{fma}\left(0.06944444444444445, \frac{r \cdot r}{\mathsf{PI}\left(\right)}, \left(\frac{s}{\mathsf{PI}\left(\right)} \cdot s\right) \cdot 0.25\right)}{s}}{s}\right)}{s}}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (/
   (fma
    (* (/ r (PI)) 0.125)
    (/ -1.3333333333333333 s)
    (/
     (/ (fma 0.06944444444444445 (/ (* r r) (PI)) (* (* (/ s (PI)) s) 0.25)) s)
     s))
   s)
  r))

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 12.6%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 9.4%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(0.125, \mathsf{fma}\left(\frac{r}{\mathsf{PI}\left(\right)}, \frac{-1.3333333333333333}{s}, \frac{\frac{\mathsf{fma}\left(r \cdot \frac{r}{\mathsf{PI}\left(\right)}, -0.6666666666666666, \mathsf{fma}\left(\left(r \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot -0.6666666666666666, -0.3333333333333333, r \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)\right)}{s}}{s}\right), \frac{0.25}{\mathsf{PI}\left(\right)}\right)}{s}}}{r} \]
8. Step-by-step derivation

9. Applied rewrites 9.4%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(0.125 \cdot \frac{r}{\mathsf{PI}\left(\right)}, \frac{-1.3333333333333333}{s}, \mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(\frac{r}{\mathsf{PI}\left(\right)} \cdot r, -0.4444444444444444, \frac{r}{\mathsf{PI}\left(\right)} \cdot r\right)}{s}}{s}, 0.125, \frac{0.25}{\mathsf{PI}\left(\right)}\right)\right)}{s}}{r} \]

10. Taylor expanded in s around 0

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

12. Applied rewrites 40.9%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(0.125 \cdot \frac{r}{\mathsf{PI}\left(\right)}, \frac{-1.3333333333333333}{s}, \frac{\frac{\mathsf{fma}\left(0.06944444444444445, \frac{r \cdot r}{\mathsf{PI}\left(\right)}, \left(s \cdot \frac{s}{\mathsf{PI}\left(\right)}\right) \cdot 0.25\right)}{s}}{s}\right)}{s}}{r} \]

13. Final simplification 42.5%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{\mathsf{PI}\left(\right)} \cdot 0.125, \frac{-1.3333333333333333}{s}, \frac{\frac{\mathsf{fma}\left(0.06944444444444445, \frac{r \cdot r}{\mathsf{PI}\left(\right)}, \left(\frac{s}{\mathsf{PI}\left(\right)} \cdot s\right) \cdot 0.25\right)}{s}}{s}\right)}{s}}{r} \]
14. Add Preprocessing

Alternative 7: 44.3% accurate, 2.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 12.6%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 9.4%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(0.125, \mathsf{fma}\left(\frac{r}{\mathsf{PI}\left(\right)}, \frac{-1.3333333333333333}{s}, \frac{\frac{\mathsf{fma}\left(r \cdot \frac{r}{\mathsf{PI}\left(\right)}, -0.6666666666666666, \mathsf{fma}\left(\left(r \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) \cdot -0.6666666666666666, -0.3333333333333333, r \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)\right)}{s}}{s}\right), \frac{0.25}{\mathsf{PI}\left(\right)}\right)}{s}}}{r} \]
8. Step-by-step derivation

9. Applied rewrites 9.4%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(0.125 \cdot \frac{r}{\mathsf{PI}\left(\right)}, \frac{-1.3333333333333333}{s}, \mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(\frac{r}{\mathsf{PI}\left(\right)} \cdot r, -0.4444444444444444, \frac{r}{\mathsf{PI}\left(\right)} \cdot r\right)}{s}}{s}, 0.125, \frac{0.25}{\mathsf{PI}\left(\right)}\right)\right)}{s}}{r} \]

10. Taylor expanded in s around 0

    \[\leadsto \frac{\frac{\frac{\frac{1}{8} \cdot \left(\frac{-4}{9} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{{r}^{2}}{\mathsf{PI}\left(\right)}\right) + s \cdot \left(\frac{-1}{6} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{s}{\mathsf{PI}\left(\right)}\right)}{{s}^{2}}}{s}}{r} \]
11. Step-by-step derivation

12. Applied rewrites 42.9%

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

Alternative 8: 10.3% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac N/A

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

7. Applied rewrites 10.8%

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

8. Final simplification 10.8%

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

Alternative 9: 9.2% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{s}{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right) \cdot s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ 1.0 (/ s (- (/ -0.16666666666666666 (* (PI) s)) (/ -0.25 (* (PI) r))))))

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]
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 9.5%

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

7. Applied rewrites 9.5%

   \[\leadsto \frac{1}{\color{blue}{\frac{s}{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right) \cdot s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}}} \]
8. Add Preprocessing

Alternative 10: 9.2% 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 99.6%

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

3. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]
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 9.5%

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

Alternative 11: 9.1% accurate, 6.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.4

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

5. Applied rewrites 9.4%

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

7. Applied rewrites 9.4%

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

9. Applied rewrites 9.4%

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

11. Applied rewrites 9.5%

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

Alternative 12: 9.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(4 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 1.0 (* (* 4.0 (* (PI) s)) r)))

Derivation

1. Initial program 99.6%

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

3. 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 9.4

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

5. Applied rewrites 9.4%

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

7. Applied rewrites 9.5%

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

8. Final simplification 9.5%

   \[\leadsto \frac{1}{\left(4 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)\right) \cdot r} \]
9. Add Preprocessing

Alternative 13: 9.1% 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 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.4

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

5. Applied rewrites 9.4%

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

7. Applied rewrites 9.4%

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

9. Applied rewrites 9.5%

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

Alternative 14: 9.1% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.4

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

5. Applied rewrites 9.4%

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

7. Applied rewrites 9.4%

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

Reproduce

herbie shell --seed 2024331 
(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))))