Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 9.4s

Alternatives: 17

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.6% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-cube-cbrt N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. pow2 N/A

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

   9. lower-pow.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lower-cbrt.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lower-cbrt.f32 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-cbrt.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-PI.f32 99.7

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

9. Applied rewrites 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

   1. lift-+.f32 N/A

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

   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. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

4. Applied rewrites 99.7%

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

Alternative 3: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. lift-/.f32 N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-+.f32 99.7

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

6. Applied rewrites 99.7%

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

Alternative 4: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

   1. lift-+.f32 N/A

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

   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. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.7%

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

Alternative 5: 10.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\frac{\frac{\mathsf{fma}\left(0.00102880658436214, r \cdot \frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, -0.009259259259259259 \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} + \frac{0.05555555555555555}{\mathsf{PI}\left(\right)}}{s} - \frac{0.16666666666666666}{\mathsf{PI}\left(\right) \cdot r}}{-s}, 0.75, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/
   (-
    (/
     (+
      (/
       (fma
        0.00102880658436214
        (* r (/ (/ r s) (PI)))
        (* -0.009259259259259259 (/ r (PI))))
       s)
      (/ 0.05555555555555555 (PI)))
     s)
    (/ 0.16666666666666666 (* (PI) r)))
   (- s))
  0.75
  (* 0.125 (/ (exp (/ (- r) s)) (* (* (PI) s) r)))))

Derivation

1. Initial program 99.7%

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

   1. lift-+.f32 N/A

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

   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. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around -inf

   \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{108} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{972} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}}{s} - \frac{1}{18} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

7. Applied rewrites 13.2%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(0.00102880658436214, r \cdot \frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, -0.009259259259259259 \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{-s} - \frac{0.05555555555555555}{\mathsf{PI}\left(\right)}}{-s} - \frac{0.16666666666666666}{\mathsf{PI}\left(\right) \cdot r}}{-s}}, 0.75, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

8. Final simplification 13.2%

   \[\leadsto \mathsf{fma}\left(\frac{\frac{\frac{\mathsf{fma}\left(0.00102880658436214, r \cdot \frac{\frac{r}{s}}{\mathsf{PI}\left(\right)}, -0.009259259259259259 \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} + \frac{0.05555555555555555}{\mathsf{PI}\left(\right)}}{s} - \frac{0.16666666666666666}{\mathsf{PI}\left(\right) \cdot r}}{-s}, 0.75, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]
9. Add Preprocessing

Alternative 6: 10.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.05555555555555555}{s}, \frac{r}{s}, \frac{-0.3333333333333333}{s}\right), r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, 0.75, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/
   (fma
    (fma (/ 0.05555555555555555 s) (/ r s) (/ -0.3333333333333333 s))
    r
    1.0)
   (* (* (* 6.0 (PI)) s) r))
  0.75
  (* 0.125 (/ (exp (/ (- r) s)) (* (* (PI) s) r)))))

Derivation

1. Initial program 99.7%

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

   1. lift-+.f32 N/A

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

   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. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in r around 0

   \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{1 + r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right) + \color{blue}{1}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right) \cdot r + 1}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}, \color{blue}{r}, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

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

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\frac{1}{18} \cdot r}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\frac{1}{18} \cdot r}{s \cdot s} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   7. times-frac N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\frac{1}{18}}{s} \cdot \frac{r}{s} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{18}}{s}, \frac{r}{s}, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{s}\right), r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   9. lower-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{18}}{s}, \frac{r}{s}, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{s}\right), r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   10. lower-/.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{18}}{s}, \frac{r}{s}, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{s}\right), r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{18}}{s}, \frac{r}{s}, \frac{-1}{3} \cdot \frac{1}{s}\right), r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   12. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{18}}{s}, \frac{r}{s}, \frac{\frac{-1}{3} \cdot 1}{s}\right), r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{18}}{s}, \frac{r}{s}, \frac{\frac{-1}{3}}{s}\right), r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   14. lower-/.f32 13.1

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.05555555555555555}{s}, \frac{r}{s}, \frac{-0.3333333333333333}{s}\right), r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, 0.75, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

7. Applied rewrites 13.1%

   \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.05555555555555555}{s}, \frac{r}{s}, \frac{-0.3333333333333333}{s}\right), r, 1\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, 0.75, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]
8. Add Preprocessing

Alternative 7: 10.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.05555555555555555}{s}, r, -0.3333333333333333\right)}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, 0.75, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/
   (fma (/ (fma (/ 0.05555555555555555 s) r -0.3333333333333333) s) r 1.0)
   (* (* (* 6.0 (PI)) s) r))
  0.75
  (* 0.125 (/ (exp (/ (- r) s)) (* (* (PI) s) r)))))

Derivation

1. Initial program 99.7%

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

   1. lift-+.f32 N/A

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

   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. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in r around 0

   \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{1 + r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right) + \color{blue}{1}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right) \cdot r + 1}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}, \color{blue}{r}, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

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

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\frac{1}{18} \cdot r}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\frac{1}{18} \cdot r}{s \cdot s} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   7. times-frac N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\frac{1}{18}}{s} \cdot \frac{r}{s} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{s}, r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{18}}{s}, \frac{r}{s}, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{s}\right), r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   9. lower-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{18}}{s}, \frac{r}{s}, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{s}\right), r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   10. lower-/.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{18}}{s}, \frac{r}{s}, \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot \frac{1}{s}\right), r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{18}}{s}, \frac{r}{s}, \frac{-1}{3} \cdot \frac{1}{s}\right), r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   12. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{18}}{s}, \frac{r}{s}, \frac{\frac{-1}{3} \cdot 1}{s}\right), r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{1}{18}}{s}, \frac{r}{s}, \frac{\frac{-1}{3}}{s}\right), r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

   14. lower-/.f32 13.1

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.05555555555555555}{s}, \frac{r}{s}, \frac{-0.3333333333333333}{s}\right), r, 1\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, 0.75, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

7. Applied rewrites 13.1%

   \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.05555555555555555}{s}, \frac{r}{s}, \frac{-0.3333333333333333}{s}\right), r, 1\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, 0.75, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

8. Taylor expanded in s around inf

   \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{1 + \left(\frac{-1}{3} \cdot \frac{r}{s} + \frac{1}{18} \cdot \frac{{r}^{2}}{{s}^{2}}\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{3}{4}, \frac{1}{8} \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

10. Applied rewrites 13.1%

    \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{0.05555555555555555}{s}, r, -0.3333333333333333\right)}{s}, r, 1\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, 0.75, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]
11. Add Preprocessing

Alternative 8: 10.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{-0.25 + \frac{0.125 \cdot r}{s}}{s}, r, 0.25\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 - \frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, -0.041666666666666664, 0.25 \cdot r\right)}{s}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

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

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. unpow2 N/A

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

   11. associate-/r* N/A

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

   12. div-add-rev N/A

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

   13. lower-/.f32 N/A

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

   14. lower-+.f32 N/A

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

   15. lower-/.f32 N/A

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

   16. lower-*.f32 12.4

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-0.25 + \frac{0.125 \cdot r}{s}}{s}, r, 0.25\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

5. Applied rewrites 12.4%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-0.25 + \frac{0.125 \cdot r}{s}}{s}, r, 0.25\right)}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

6. Taylor expanded in s around inf

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{8} \cdot r}{s}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4} + \left(\frac{-1}{4} \cdot \frac{r}{s} + \frac{1}{24} \cdot \frac{{r}^{2}}{{s}^{2}}\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
7. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{8} \cdot r}{s}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(\frac{3}{4} + \frac{-1}{4} \cdot \frac{r}{s}\right) + \color{blue}{\frac{1}{24} \cdot \frac{{r}^{2}}{{s}^{2}}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

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

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{8} \cdot r}{s}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(\frac{3}{4} - \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \frac{r}{s}\right) + \color{blue}{\frac{1}{24}} \cdot \frac{{r}^{2}}{{s}^{2}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. associate-+l- N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{8} \cdot r}{s}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} - \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \frac{r}{s} - \frac{1}{24} \cdot \frac{{r}^{2}}{{s}^{2}}\right)}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{8} \cdot r}{s}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} - \left(\frac{1}{4} \cdot \frac{r}{s} - \frac{1}{24} \cdot \frac{{r}^{2}}{{s}^{2}}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{8} \cdot r}{s}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} - \left(\frac{\frac{1}{4} \cdot r}{s} - \color{blue}{\frac{1}{24}} \cdot \frac{{r}^{2}}{{s}^{2}}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   6. associate-*r/ N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{8} \cdot r}{s}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} - \left(\frac{\frac{1}{4} \cdot r}{s} - \frac{\frac{1}{24} \cdot {r}^{2}}{\color{blue}{{s}^{2}}}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   7. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{8} \cdot r}{s}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} - \left(\frac{\frac{1}{4} \cdot r}{s} - \frac{\frac{1}{24} \cdot {r}^{2}}{s \cdot \color{blue}{s}}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   8. associate-/r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{8} \cdot r}{s}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} - \left(\frac{\frac{1}{4} \cdot r}{s} - \frac{\frac{\frac{1}{24} \cdot {r}^{2}}{s}}{\color{blue}{s}}\right)}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   9. div-sub N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{8} \cdot r}{s}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} - \frac{\frac{1}{4} \cdot r - \frac{\frac{1}{24} \cdot {r}^{2}}{s}}{\color{blue}{s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   10. associate-/l* N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{8} \cdot r}{s}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} - \frac{\frac{1}{4} \cdot r - \frac{1}{24} \cdot \frac{{r}^{2}}{s}}{s}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{8} \cdot r}{s}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} - \frac{\frac{1}{4} \cdot r - \left(\mathsf{neg}\left(\frac{-1}{24}\right)\right) \cdot \frac{{r}^{2}}{s}}{s}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

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

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{8} \cdot r}{s}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} - \frac{\frac{1}{4} \cdot r + \frac{-1}{24} \cdot \frac{{r}^{2}}{s}}{s}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   13. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{8} \cdot r}{s}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} - \frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   14. lower--.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-1}{4} + \frac{\frac{1}{8} \cdot r}{s}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} - \color{blue}{\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

8. Applied rewrites 12.4%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{-0.25 + \frac{0.125 \cdot r}{s}}{s}, r, 0.25\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{0.75 - \frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, -0.041666666666666664, 0.25 \cdot r\right)}{s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
9. Add Preprocessing

Alternative 9: 10.1% accurate, 3.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-cube-cbrt N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. pow2 N/A

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

   9. lower-pow.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lower-cbrt.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lower-cbrt.f32 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

9. Applied rewrites 12.4%

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

10. Taylor expanded in s around inf

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

    1. associate-/r* N/A

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

    2. unpow2 N/A

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

    3. associate-/l/ N/A

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

    4. *-commutative N/A

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

    5. associate-*l/ N/A

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

    6. associate-*r/ N/A

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

    7. metadata-eval N/A

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

    8. associate-/r* N/A

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

    9. div-add-rev N/A

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

    10. lower-/.f32 N/A

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

    11. *-commutative N/A

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

    12. lower-fma.f32 N/A

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

    13. lower-/.f32 N/A

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

    14. lower-/.f32 N/A

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

    15. lower-/.f32 N/A

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

    16. lower-PI.f32 12.4

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

12. Applied rewrites 12.4%

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

Alternative 10: 10.1% accurate, 4.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 12.4%

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

8. Final simplification 12.4%

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

Alternative 11: 9.1% accurate, 5.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 10.8%

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

Alternative 12: 9.1% accurate, 6.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 10.8

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

7. Applied rewrites 10.8%

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

8. Final simplification 10.8%

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

Alternative 13: 9.0% accurate, 6.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 10.0

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

7. Applied rewrites 10.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lift-PI.f32 N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-sqrt.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lower-sqrt.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 10.0

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

9. Applied rewrites 10.0%

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

10. Final simplification 10.0%

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

Alternative 14: 9.0% accurate, 8.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. lower-PI.f32 10.0

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

7. Applied rewrites 10.0%

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

Alternative 15: 9.0% accurate, 10.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 10.0

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

5. Applied rewrites 10.0%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-*.f32 10.0

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

7. Applied rewrites 10.0%

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

Alternative 16: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 10.0

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

7. Applied rewrites 10.0%

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

8. Final simplification 10.0%

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

Alternative 17: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 10.0

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

7. Applied rewrites 10.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 10.0

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

9. Applied rewrites 10.0%

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

10. Final simplification 10.0%

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

Reproduce

herbie shell --seed 2025026 
(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))))