Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 12.4s

Alternatives: 19

Speedup: N/A×

Specification

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

Math

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

FPCore

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-*.f32 99.6

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

6. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

6. Applied rewrites 99.6%

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

Alternative 3: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-*.f32 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

8. Applied rewrites 99.6%

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

Alternative 4: 10.5% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 13.8%

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

6. Taylor expanded in s around 0

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

8. Applied rewrites 13.8%

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

Alternative 5: 10.5% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 13.8%

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

   1. lift-+.f32 N/A

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

7. Applied rewrites 13.8%

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

Alternative 6: 10.5% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. times-frac N/A

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

   8. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

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

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. unpow2 N/A

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

   11. associate-/r* N/A

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

   12. div-add-rev N/A

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

   13. lower-/.f32 N/A

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

   14. lower-+.f32 N/A

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

   15. lower-/.f32 N/A

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

   16. lower-*.f32 13.7

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

7. Applied rewrites 13.7%

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

9. Applied rewrites 13.7%

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

Alternative 7: 10.5% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. times-frac N/A

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

   8. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

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

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. unpow2 N/A

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

   11. associate-/r* N/A

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

   12. div-add-rev N/A

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

   13. lower-/.f32 N/A

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

   14. lower-+.f32 N/A

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

   15. lower-/.f32 N/A

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

   16. lower-*.f32 13.7

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

7. Applied rewrites 13.7%

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

9. Applied rewrites 13.7%

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

Alternative 8: 10.0% accurate, 2.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 13.8%

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

6. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

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

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. unpow2 N/A

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

   11. associate-/r* N/A

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

   12. div-add-rev N/A

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

   13. lower-/.f32 N/A

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

   14. lower-+.f32 N/A

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

   15. lower-/.f32 N/A

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

   16. lower-*.f32 13.0

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

8. Applied rewrites 13.0%

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

10. Applied rewrites 13.0%

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

Alternative 9: 10.0% accurate, 3.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f32 N/A

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

5. Applied rewrites 13.0%

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

7. Applied rewrites 13.0%

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

8. Final simplification 13.0%

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

Alternative 10: 10.0% accurate, 3.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f32 N/A

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

5. Applied rewrites 13.0%

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

6. Taylor expanded in s around inf

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

8. Applied rewrites 13.0%

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

Alternative 11: 10.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, 0.06944444444444445, \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{1}{s}, \frac{0.25}{r}\right)}{\mathsf{PI}\left(\right)}\right)}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (fma
   (/ r (* (* s s) (PI)))
   0.06944444444444445
   (/ (fma -0.16666666666666666 (/ 1.0 s) (/ 0.25 r)) (PI)))
  s))

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 12.9%

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

Alternative 12: 9.0% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 11.5%

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

Alternative 13: 9.0% accurate, 5.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-*.f32 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. associate-/l* N/A

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

   12. lower-/.f32 N/A

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

   13. associate-*r/ N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f32 N/A

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

   16. lower-PI.f32 11.5

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

9. Applied rewrites 11.5%

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

Alternative 14: 9.0% 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 99.5%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-/r* N/A

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

   6. associate-*r/ N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. associate-/r* N/A

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

   10. div-add-rev N/A

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

   11. lower-/.f32 N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. lower-PI.f32 11.5

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

5. Applied rewrites 11.5%

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

Alternative 15: 9.0% accurate, 5.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
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. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. times-frac N/A

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

   10. lower-fma.f32 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f32 N/A

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

7. Applied rewrites 11.5%

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

8. Final simplification 11.5%

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

Alternative 16: 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 99.5%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 10.9

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

5. Applied rewrites 10.9%

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

7. Applied rewrites 10.9%

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

Alternative 17: 9.0% accurate, 10.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 10.9

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

5. Applied rewrites 10.9%

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

7. Applied rewrites 10.9%

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

Alternative 18: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 10.9

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

5. Applied rewrites 10.9%

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

7. Applied rewrites 10.9%

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

9. Applied rewrites 10.9%

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

Alternative 19: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 10.9

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

5. Applied rewrites 10.9%

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

7. Applied rewrites 10.9%

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

9. Applied rewrites 10.9%

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

Reproduce

herbie shell --seed 2024352 
(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))))