Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 12.2s

Alternatives: 17

Speedup: N/A×

Specification

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

Math

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

FPCore

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.5

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

7. Applied rewrites 99.5%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. lift-/.f32 N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

9. Applied rewrites 99.5%

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

11. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.5

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

7. Applied rewrites 99.5%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. lift-/.f32 N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

9. Applied rewrites 99.5%

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

    1. lift-/.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. rem-square-sqrt N/A

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

    5. lift-sqrt.f32 N/A

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

    6. lift-sqrt.f32 N/A

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

    7. associate-*l* N/A

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

    8. lift-*.f32 N/A

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

    9. lift-*.f32 N/A

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

    10. associate-/l/ N/A

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

11. Applied rewrites 99.5%

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

Alternative 3: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.5

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

7. Applied rewrites 99.5%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. lift-/.f32 N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

9. Applied rewrites 99.5%

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

    1. lift-/.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. rem-square-sqrt N/A

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

    5. lift-sqrt.f32 N/A

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

    6. lift-sqrt.f32 N/A

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

    7. associate-*l* N/A

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

    8. lift-*.f32 N/A

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

    9. lift-*.f32 N/A

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

    10. associate-/l/ N/A

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

11. Applied rewrites 96.0%

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

Alternative 4: 10.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-r}{s}\\ \mathsf{fma}\left(\frac{e^{t\_0}}{r}, \frac{0.125}{\mathsf{PI}\left(\right) \cdot s}, \frac{\frac{\frac{\mathsf{fma}\left(0.0007716049382716049, \frac{r}{s}, -0.006944444444444444\right)}{\mathsf{PI}\left(\right)} \cdot t\_0 - \frac{0.041666666666666664}{\mathsf{PI}\left(\right)}}{s} + \frac{0.125}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ (- r) s)))
   (fma
    (/ (exp t_0) r)
    (/ 0.125 (* (PI) s))
    (/
     (+
      (/
       (-
        (*
         (/ (fma 0.0007716049382716049 (/ r s) -0.006944444444444444) (PI))
         t_0)
        (/ 0.041666666666666664 (PI)))
       s)
      (/ 0.125 (* (PI) r)))
     s))))

Derivation

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

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f32 N/A

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

5. Applied rewrites 8.2%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{\frac{\frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, 0.0007716049382716049, -0.006944444444444444 \cdot r\right)}{\mathsf{PI}\left(\right)}}{-s} - \frac{0.041666666666666664}{\mathsf{PI}\left(\right)}}{-s} - \frac{0.125}{\mathsf{PI}\left(\right) \cdot r}}{-s}} \]
6. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

7. Applied rewrites 8.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, 0.25, \frac{\frac{\frac{\frac{r \cdot \mathsf{fma}\left(0.0007716049382716049, \frac{r}{s}, -0.006944444444444444\right)}{\mathsf{PI}\left(\right)}}{-s} - \frac{0.041666666666666664}{\mathsf{PI}\left(\right)}}{-s} - \frac{0.125}{r \cdot \mathsf{PI}\left(\right)}}{-s}\right)} \]

9. Applied rewrites 8.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{r}, \frac{0.125}{\mathsf{PI}\left(\right) \cdot s}, \frac{\frac{\frac{\mathsf{fma}\left(0.0007716049382716049, \frac{r}{s}, -0.006944444444444444\right)}{\mathsf{PI}\left(\right)} \cdot \frac{-r}{s} - \frac{0.041666666666666664}{\mathsf{PI}\left(\right)}}{-s} - \frac{0.125}{\mathsf{PI}\left(\right) \cdot r}}{-s}\right)} \]

10. Final simplification 8.2%

    \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{r}, \frac{0.125}{\mathsf{PI}\left(\right) \cdot s}, \frac{\frac{\frac{\mathsf{fma}\left(0.0007716049382716049, \frac{r}{s}, -0.006944444444444444\right)}{\mathsf{PI}\left(\right)} \cdot \frac{-r}{s} - \frac{0.041666666666666664}{\mathsf{PI}\left(\right)}}{s} + \frac{0.125}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
11. Add Preprocessing

Alternative 5: 10.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{r}, \frac{0.125}{s \cdot \mathsf{PI}\left(\right)}, \frac{\frac{\frac{\frac{r \cdot \mathsf{fma}\left(0.0007716049382716049, \frac{r}{s}, -0.006944444444444444\right)}{\mathsf{PI}\left(\right)}}{s} + \frac{0.041666666666666664}{\mathsf{PI}\left(\right)}}{s} - \frac{0.125}{r \cdot \mathsf{PI}\left(\right)}}{-s}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ (exp (/ (- r) s)) r)
  (/ 0.125 (* s (PI)))
  (/
   (-
    (/
     (+
      (/
       (/ (* r (fma 0.0007716049382716049 (/ r s) -0.006944444444444444)) (PI))
       s)
      (/ 0.041666666666666664 (PI)))
     s)
    (/ 0.125 (* r (PI))))
   (- s))))

Derivation

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

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f32 N/A

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

5. Applied rewrites 8.2%

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

7. Applied rewrites 8.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{r}, \frac{0.125}{s \cdot \mathsf{PI}\left(\right)}, \frac{\frac{\frac{\frac{r \cdot \mathsf{fma}\left(0.0007716049382716049, \frac{r}{s}, -0.006944444444444444\right)}{\mathsf{PI}\left(\right)}}{-s} - \frac{0.041666666666666664}{\mathsf{PI}\left(\right)}}{-s} - \frac{0.125}{r \cdot \mathsf{PI}\left(\right)}}{-s}\right)} \]

8. Final simplification 8.2%

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{r}, \frac{0.125}{s \cdot \mathsf{PI}\left(\right)}, \frac{\frac{\frac{\frac{r \cdot \mathsf{fma}\left(0.0007716049382716049, \frac{r}{s}, -0.006944444444444444\right)}{\mathsf{PI}\left(\right)}}{s} + \frac{0.041666666666666664}{\mathsf{PI}\left(\right)}}{s} - \frac{0.125}{r \cdot \mathsf{PI}\left(\right)}}{-s}\right) \]
9. Add Preprocessing

Alternative 6: 10.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, 0.25, \frac{\frac{\frac{\frac{r \cdot \mathsf{fma}\left(0.0007716049382716049, \frac{r}{s}, -0.006944444444444444\right)}{\mathsf{PI}\left(\right)}}{s} + \frac{0.041666666666666664}{\mathsf{PI}\left(\right)}}{s} - \frac{0.125}{r \cdot \mathsf{PI}\left(\right)}}{-s}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ (exp (/ (- r) s)) (* (* (* 2.0 (PI)) s) r))
  0.25
  (/
   (-
    (/
     (+
      (/
       (/ (* r (fma 0.0007716049382716049 (/ r s) -0.006944444444444444)) (PI))
       s)
      (/ 0.041666666666666664 (PI)))
     s)
    (/ 0.125 (* r (PI))))
   (- s))))

Derivation

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

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f32 N/A

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

5. Applied rewrites 8.2%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{\frac{\frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, 0.0007716049382716049, -0.006944444444444444 \cdot r\right)}{\mathsf{PI}\left(\right)}}{-s} - \frac{0.041666666666666664}{\mathsf{PI}\left(\right)}}{-s} - \frac{0.125}{\mathsf{PI}\left(\right) \cdot r}}{-s}} \]
6. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

7. Applied rewrites 8.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, 0.25, \frac{\frac{\frac{\frac{r \cdot \mathsf{fma}\left(0.0007716049382716049, \frac{r}{s}, -0.006944444444444444\right)}{\mathsf{PI}\left(\right)}}{-s} - \frac{0.041666666666666664}{\mathsf{PI}\left(\right)}}{-s} - \frac{0.125}{r \cdot \mathsf{PI}\left(\right)}}{-s}\right)} \]

8. Final simplification 8.2%

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, 0.25, \frac{\frac{\frac{\frac{r \cdot \mathsf{fma}\left(0.0007716049382716049, \frac{r}{s}, -0.006944444444444444\right)}{\mathsf{PI}\left(\right)}}{s} + \frac{0.041666666666666664}{\mathsf{PI}\left(\right)}}{s} - \frac{0.125}{r \cdot \mathsf{PI}\left(\right)}}{-s}\right) \]
9. Add Preprocessing

Alternative 7: 10.5% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.5

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

7. Applied rewrites 99.5%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. lift-/.f32 N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

9. Applied rewrites 99.5%

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

10. Taylor expanded in s around -inf

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

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

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

    2. metadata-eval N/A

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

    3. *-lft-identity N/A

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

    4. lower--.f32 N/A

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

    5. lower-/.f32 N/A

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

    6. lower-fma.f32 N/A

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

    7. unpow2 N/A

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

    8. associate-/l* N/A

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

    9. lower-*.f32 N/A

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

    10. lower-/.f32 N/A

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

    11. lower-*.f32 8.1

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

12. Applied rewrites 8.1%

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

Alternative 8: 9.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.125}{\mathsf{PI}\left(\right) \cdot s}\\ \frac{\mathsf{fma}\left(t\_0, 1, t\_0 \cdot e^{\frac{-r}{s}}\right)}{r} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ 0.125 (* (PI) s))))
   (/ (fma t_0 1.0 (* t_0 (exp (/ (- r) s)))) r)))

Derivation

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 7.9%

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

Alternative 9: 9.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r\\ \mathsf{fma}\left(0.125, \frac{1}{t\_0}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (* (PI) s) r)))
   (fma 0.125 (/ 1.0 t_0) (* 0.125 (/ (exp (/ (- r) s)) t_0)))))

Derivation

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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. associate-*l* N/A

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

   10. times-frac N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 7.9%

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

Alternative 10: 9.6% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.5

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

7. Applied rewrites 99.5%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. lift-/.f32 N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

9. Applied rewrites 99.5%

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

10. Taylor expanded in s around inf

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

    1. +-commutative N/A

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

    2. lower-fma.f32 N/A

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

    3. lower-/.f32 7.7

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

12. Applied rewrites 7.7%

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

Alternative 11: 9.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \mathsf{PI}\left(\right)\\ t_1 := \frac{r}{\mathsf{PI}\left(\right)}\\ \frac{\frac{\frac{\mathsf{fma}\left(t\_0, \frac{\mathsf{fma}\left(t\_1, -0.06944444444444445, \frac{\left(t\_1 \cdot r\right) \cdot 0.021604938271604937}{s}\right)}{-s} - \frac{0.16666666666666666}{\mathsf{PI}\left(\right)}, s \cdot 0.25\right)}{s}}{t\_0}}{s} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* r (PI))) (t_1 (/ r (PI))))
   (/
    (/
     (/
      (fma
       t_0
       (-
        (/
         (fma
          t_1
          -0.06944444444444445
          (/ (* (* t_1 r) 0.021604938271604937) s))
         (- s))
        (/ 0.16666666666666666 (PI)))
       (* s 0.25))
      s)
     t_0)
    s)))

Derivation

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 7.6%

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

7. Applied rewrites 7.7%

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

8. Final simplification 7.7%

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

Alternative 12: 10.0% accurate, 3.7× speedup

Math

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

FPCore

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

Derivation

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

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 7.7%

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

6. Taylor expanded in s around 0

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

8. Applied rewrites 7.7%

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

Alternative 13: 10.0% accurate, 4.0× speedup

Math

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

FPCore

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

Derivation

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

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 7.7%

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

7. Applied rewrites 7.7%

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

Alternative 14: 10.0% accurate, 4.2× speedup

Math

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

FPCore

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

Derivation

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.5

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

7. Applied rewrites 99.5%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. lift-/.f32 N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

9. Applied rewrites 99.5%

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

10. Taylor expanded in s around -inf

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

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

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

    2. metadata-eval N/A

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

    3. *-lft-identity N/A

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

    4. lower--.f32 N/A

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

    5. lower-/.f32 N/A

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

12. Applied rewrites 7.7%

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

Alternative 15: 8.9% accurate, 6.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 7.5

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

5. Applied rewrites 7.5%

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

7. Applied rewrites 7.5%

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

9. Applied rewrites 7.5%

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

11. Applied rewrites 7.5%

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

Alternative 16: 8.9% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

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

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 7.5

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

5. Applied rewrites 7.5%

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

7. Applied rewrites 7.5%

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

9. Applied rewrites 7.5%

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

Alternative 17: 8.9% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

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

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 7.5

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

5. Applied rewrites 7.5%

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

7. Applied rewrites 7.5%

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

9. Applied rewrites 7.5%

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

Reproduce

herbie shell --seed 2025006 
(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))))