Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 11.1s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around 0

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

   1. associate-/r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in s around 0

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

10. Applied rewrites 99.7%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around 0

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

   1. associate-/r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in s around 0

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

10. Applied rewrites 99.7%

    \[\leadsto \mathsf{fma}\left(0.75, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \color{blue}{\frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \cdot 0.125}\right) \]
11. Step-by-step derivation

    1. associate-*l* N/A

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

    2. *-commutative N/A

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

    3. lower-*.f32 N/A

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

    4. *-commutative N/A

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

    5. lower-*.f32 N/A

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

    6. lower-PI.f32 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f32 99.6

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

12. Applied rewrites 99.6%

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

Alternative 4: 10.6% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around 0

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

   1. associate-/r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in s around 0

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

10. Applied rewrites 99.7%

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

11. Taylor expanded in r around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f32 N/A

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

    4. lower--.f32 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f32 N/A

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

    7. lower-/.f32 N/A

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

    8. pow2 N/A

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

    9. lower-*.f32 N/A

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

    10. associate-*r/ N/A

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

    11. metadata-eval N/A

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

    12. lower-/.f32 10.9

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

13. Applied rewrites 10.9%

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

Alternative 5: 10.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}\\ \frac{\mathsf{fma}\left(t\_0, 0.006944444444444444, \mathsf{fma}\left(0.0625, t\_0, \frac{0.25}{\mathsf{PI}\left(\right) \cdot r}\right)\right) - \frac{0.16666666666666666}{\mathsf{PI}\left(\right) \cdot s}}{s} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ r (* (* s s) (PI)))))
   (/
    (-
     (fma t_0 0.006944444444444444 (fma 0.0625 t_0 (/ 0.25 (* (PI) r))))
     (/ 0.16666666666666666 (* (PI) s)))
    s)))

Derivation

1. Initial program 99.6%

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

3. 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

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

5. Applied rewrites 10.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, 0.006944444444444444, \mathsf{fma}\left(0.0625, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{0.25}{\mathsf{PI}\left(\right) \cdot r}\right)\right) - \frac{0.16666666666666666}{\mathsf{PI}\left(\right) \cdot s}}{s}} \]
6. Add Preprocessing

Alternative 6: 10.0% accurate, 3.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around 0

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

   1. associate-/r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 10.5%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, 0.006944444444444444, \mathsf{fma}\left(0.0625, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{0.25}{\mathsf{PI}\left(\right) \cdot r}\right)\right) - \frac{0.16666666666666666}{\mathsf{PI}\left(\right) \cdot s}}{s}} \]

11. Taylor expanded in r around 0

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

    1. lower-/.f32 N/A

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

13. Applied rewrites 10.5%

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

14. Final simplification 10.5%

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

Alternative 7: 10.0% accurate, 3.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-/.f32 N/A

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

5. Applied rewrites 10.5%

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

6. Final simplification 10.5%

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

Alternative 8: 10.0% accurate, 3.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around 0

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

   1. associate-/r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 10.5%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, 0.006944444444444444, \mathsf{fma}\left(0.0625, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{0.25}{\mathsf{PI}\left(\right) \cdot r}\right)\right) - \frac{0.16666666666666666}{\mathsf{PI}\left(\right) \cdot s}}{s}} \]

11. Taylor expanded in r around 0

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

    1. lower-/.f32 N/A

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

13. Applied rewrites 10.5%

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

14. Final simplification 10.5%

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

Alternative 9: 9.0% accurate, 5.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in 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

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

5. Applied rewrites 9.7%

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

Alternative 10: 9.0% accurate, 6.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 9.7%

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

Alternative 11: 9.0% accurate, 8.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 9.4

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

5. Applied rewrites 9.4%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 9.4

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

7. Applied rewrites 9.4%

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

9. Applied rewrites 9.4%

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

10. Final simplification 9.4%

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

Alternative 12: 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 99.6%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 9.4

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

5. Applied rewrites 9.4%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 9.4

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

7. Applied rewrites 9.4%

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

Alternative 13: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 9.4

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

5. Applied rewrites 9.4%

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

Alternative 14: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 9.4

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

5. Applied rewrites 9.4%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 9.4

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

7. Applied rewrites 9.4%

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

Reproduce

herbie shell --seed 2025043 
(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))))