Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 10.5s

Alternatives: 20

Speedup: N/A×

Specification

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

Math

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

FPCore

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. metadata-eval N/A

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

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

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

   6. lift-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. distribute-rgt-neg-in N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f32 99.7

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

6. Applied rewrites 99.7%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. distribute-rgt-out N/A

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

   7. times-frac N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

6. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f32 N/A

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

6. Applied rewrites 99.6%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. distribute-rgt-out N/A

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

   7. times-frac N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

6. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 99.6

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

   4. lift-/.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. associate-/l/ N/A

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

   7. lift-*.f32 N/A

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

   8. lower-/.f32 99.6

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

   9. lift-+.f32 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f32 99.6

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

8. Applied rewrites 99.6%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. distribute-rgt-out N/A

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

   7. times-frac N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

6. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. lift-*.f32 N/A

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

   10. lower-/.f32 N/A

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

8. Applied rewrites 99.6%

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

Alternative 6: 11.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot s\\ \mathsf{fma}\left(\mathsf{fma}\left(\frac{r}{s \cdot s}, 0.05555555555555555, \frac{1}{r}\right) - \frac{0.3333333333333333}{s}, \frac{0.125}{t\_0}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{t\_0 \cdot r}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (PI) s)))
   (fma
    (-
     (fma (/ r (* s s)) 0.05555555555555555 (/ 1.0 r))
     (/ 0.3333333333333333 s))
    (/ 0.125 t_0)
    (* 0.125 (/ (exp (/ (- r) s)) (* t_0 r))))))

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 11.3%

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

Alternative 7: 11.1% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. distribute-rgt-out N/A

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

   7. times-frac N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in s around -inf

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

9. Applied rewrites 11.3%

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

Alternative 8: 11.1% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. distribute-rgt-out N/A

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

   7. times-frac N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in r around 0

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

9. Applied rewrites 11.3%

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

Alternative 9: 10.5% accurate, 3.1× speedup

Math

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

FPCore

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around -inf

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \left(\frac{1}{24} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right) + \left(\frac{1}{144} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)} + \frac{1}{16} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s}}}{r} \]
6. Step-by-step derivation

7. Applied rewrites 11.0%

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

8. Final simplification 11.0%

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

Alternative 10: 10.5% accurate, 3.5× speedup

Math

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

FPCore

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 9.9%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 9.8%

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

11. Taylor expanded in r around 0

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

13. Applied rewrites 11.0%

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

Alternative 11: 10.5% accurate, 3.7× speedup

Math

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

FPCore

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around -inf

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

7. Applied rewrites 10.9%

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

8. Final simplification 10.9%

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

Alternative 12: 10.5% accurate, 4.0× speedup

Math

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

FPCore

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 10.9%

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

Alternative 13: 9.3% accurate, 4.9× speedup

Math

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

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in r around 0

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

7. Applied rewrites 9.9%

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

Alternative 14: 9.4% accurate, 5.8× speedup

Math

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

FPCore

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 9.9%

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

9. Applied rewrites 9.9%

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

Alternative 15: 9.4% 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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 9.9%

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

Alternative 16: 9.3% accurate, 6.6× speedup

Math

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

FPCore

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 9.9%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f32 N/A

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

9. Applied rewrites 9.8%

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

Alternative 17: 9.2% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s} \cdot \frac{2}{\mathsf{PI}\left(\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (* (/ 0.125 s) (/ 2.0 (* (PI) r))))

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. distribute-rgt-out N/A

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

   7. times-frac N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in s around inf

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

9. Applied rewrites 9.8%

   \[\leadsto \frac{0.125}{s} \cdot \color{blue}{\frac{2}{\mathsf{PI}\left(\right) \cdot r}} \]
10. Add Preprocessing

Alternative 18: 9.2% accurate, 10.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 9.8%

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

7. Applied rewrites 9.8%

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

9. Applied rewrites 9.8%

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

Alternative 19: 9.2% 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

5. Applied rewrites 9.8%

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

7. Applied rewrites 9.8%

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

Alternative 20: 9.3% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 9.8%

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

7. Applied rewrites 9.8%

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

9. Applied rewrites 9.8%

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

Reproduce

herbie shell --seed 2025019 
(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))))