Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 5.4s

Alternatives: 15

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

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

FPCore

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in r around 0

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 99.8%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.75, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  0.75
  (/ (exp (* -0.3333333333333333 (/ r s))) (* (* (* (PI) 6.0) s) r))
  (* 0.25 (/ (exp (/ (- r) s)) (* (* (* (PI) 2.0) s) r)))))

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around 0

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

7. Applied rewrites 99.7%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in r around 0

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 99.8%

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

11. Taylor expanded in s around 0

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

13. Applied rewrites 99.7%

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

Alternative 4: 10.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.75, \frac{\frac{\mathsf{fma}\left(-0.00102880658436214, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{0.009259259259259259}{\mathsf{PI}\left(\right) \cdot s}\right) \cdot r - \frac{0.05555555555555555}{\mathsf{PI}\left(\right)}}{s} + \frac{0.16666666666666666}{\mathsf{PI}\left(\right) \cdot r}}{s}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around -inf

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

7. Applied rewrites 13.2%

   \[\leadsto \mathsf{fma}\left(0.75, \color{blue}{-\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(0.00102880658436214, \frac{\frac{r \cdot r}{s}}{\mathsf{PI}\left(\right)}, -0.009259259259259259 \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s}\right) - \frac{0.05555555555555555}{\mathsf{PI}\left(\right)}}{s}\right) - \frac{0.16666666666666666}{\mathsf{PI}\left(\right) \cdot r}}{s}}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r}\right) \]

8. Taylor expanded in r around 0

   \[\leadsto \mathsf{fma}\left(\frac{3}{4}, -\frac{\left(-\frac{r \cdot \left(\frac{-1}{972} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{108} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{18}}{\mathsf{PI}\left(\right)}}{s}\right) - \frac{\frac{1}{6}}{\mathsf{PI}\left(\right) \cdot r}}{s}, \frac{1}{4} \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r}\right) \]
9. Step-by-step derivation

10. Applied rewrites 13.2%

    \[\leadsto \mathsf{fma}\left(0.75, -\frac{\left(-\frac{\mathsf{fma}\left(-0.00102880658436214, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{0.009259259259259259}{\mathsf{PI}\left(\right) \cdot s}\right) \cdot r - \frac{0.05555555555555555}{\mathsf{PI}\left(\right)}}{s}\right) - \frac{0.16666666666666666}{\mathsf{PI}\left(\right) \cdot r}}{s}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r}\right) \]
11. Step-by-step derivation

    1. lift-PI.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. *-commutative N/A

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

    4. count-2-rev N/A

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

    5. lower-+.f32 N/A

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

    6. lift-PI.f32 N/A

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

    7. lift-PI.f32 13.2

       \[\leadsto \mathsf{fma}\left(0.75, -\frac{\left(-\frac{\mathsf{fma}\left(-0.00102880658436214, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{0.009259259259259259}{\mathsf{PI}\left(\right) \cdot s}\right) \cdot r - \frac{0.05555555555555555}{\mathsf{PI}\left(\right)}}{s}\right) - \frac{0.16666666666666666}{\mathsf{PI}\left(\right) \cdot r}}{s}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\mathsf{PI}\left(\right) + \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot s\right) \cdot r}\right) \]

12. Applied rewrites 13.2%

    \[\leadsto \mathsf{fma}\left(0.75, -\frac{\left(-\frac{\mathsf{fma}\left(-0.00102880658436214, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{0.009259259259259259}{\mathsf{PI}\left(\right) \cdot s}\right) \cdot r - \frac{0.05555555555555555}{\mathsf{PI}\left(\right)}}{s}\right) - \frac{0.16666666666666666}{\mathsf{PI}\left(\right) \cdot r}}{s}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)} \cdot s\right) \cdot r}\right) \]

13. Final simplification 13.2%

    \[\leadsto \mathsf{fma}\left(0.75, \frac{\frac{\mathsf{fma}\left(-0.00102880658436214, \frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{0.009259259259259259}{\mathsf{PI}\left(\right) \cdot s}\right) \cdot r - \frac{0.05555555555555555}{\mathsf{PI}\left(\right)}}{s} + \frac{0.16666666666666666}{\mathsf{PI}\left(\right) \cdot r}}{s}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}\right) \]
14. Add Preprocessing

Alternative 5: 10.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{0.75}{\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s}, \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.006172839506172839, 0.05555555555555555 \cdot r\right)}{s}, -1, 0.3333333333333333\right)}{s}, -1, \frac{1}{r}\right), 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ 0.75 (* (* (PI) 6.0) s))
  (fma
   (/
    (fma
     (/ (fma (/ (* r r) s) -0.006172839506172839 (* 0.05555555555555555 r)) s)
     -1.0
     0.3333333333333333)
    s)
   -1.0
   (/ 1.0 r))
  (* 0.25 (/ (exp (/ (- r) s)) (* (* (* (PI) 2.0) s) r)))))

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in s around -inf

   \[\leadsto \mathsf{fma}\left(\frac{\frac{3}{4}}{\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s}, \color{blue}{-1 \cdot \frac{\frac{1}{3} + -1 \cdot \frac{\frac{-1}{162} \cdot \frac{{r}^{2}}{s} + \frac{1}{18} \cdot r}{s}}{s} + \frac{1}{r}}, \frac{1}{4} \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r}\right) \]
6. Step-by-step derivation

7. Applied rewrites 13.2%

   \[\leadsto \mathsf{fma}\left(\frac{0.75}{\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s}, \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.006172839506172839, 0.05555555555555555 \cdot r\right)}{s}, -1, 0.3333333333333333\right)}{s}, -1, \frac{1}{r}\right)}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r}\right) \]
8. Add Preprocessing

Alternative 6: 10.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{0.75}{\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s}, \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, -0.006172839506172839, 0.05555555555555555 \cdot r\right)}{s}, -1, 0.3333333333333333\right)}{s}, -1, \frac{1}{r}\right), \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \cdot 0.125\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ 0.75 (* (* (PI) 6.0) s))
  (fma
   (/
    (fma
     (/ (fma (* r (/ r s)) -0.006172839506172839 (* 0.05555555555555555 r)) s)
     -1.0
     0.3333333333333333)
    s)
   -1.0
   (/ 1.0 r))
  (* (/ (exp (/ (- r) s)) (* (* (PI) s) r)) 0.125)))

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in r around 0

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 99.8%

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

11. Taylor expanded in s around -inf

    \[\leadsto \mathsf{fma}\left(\frac{\frac{3}{4}}{\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s}, \color{blue}{-1 \cdot \frac{\frac{1}{3} + -1 \cdot \frac{\frac{-1}{162} \cdot \frac{{r}^{2}}{s} + \frac{1}{18} \cdot r}{s}}{s} + \frac{1}{r}}, \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \cdot \frac{1}{8}\right) \]
12. Step-by-step derivation

13. Applied rewrites 13.2%

    \[\leadsto \mathsf{fma}\left(\frac{0.75}{\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s}, \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(r \cdot \frac{r}{s}, -0.006172839506172839, 0.05555555555555555 \cdot r\right)}{s}, -1, 0.3333333333333333\right)}{s}, -1, \frac{1}{r}\right)}, \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \cdot 0.125\right) \]
14. Add Preprocessing

Alternative 7: 10.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.75, \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.05555555555555555 - \frac{0.3333333333333333}{s}, r, 1\right)}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  0.75
  (/
   (fma
    (- (* (/ r (* s s)) 0.05555555555555555) (/ 0.3333333333333333 s))
    r
    1.0)
   (* (* (* (PI) 6.0) s) r))
  (* 0.25 (/ (exp (/ (- r) s)) (* (* (* (PI) 2.0) s) r)))))

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in r around 0

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

7. Applied rewrites 13.1%

   \[\leadsto \mathsf{fma}\left(0.75, \frac{\color{blue}{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.05555555555555555 - \frac{0.3333333333333333}{s}, r, 1\right)}}{\left(\left(\mathsf{PI}\left(\right) \cdot 6\right) \cdot s\right) \cdot r}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r}\right) \]
8. Add Preprocessing

Alternative 8: 10.7% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in r around 0

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 99.8%

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

11. Taylor expanded in s around inf

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

13. Applied rewrites 13.1%

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

Alternative 9: 10.7% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in r around 0

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 99.8%

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

11. Taylor expanded in s around -inf

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

13. Applied rewrites 13.1%

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

Alternative 10: 10.1% accurate, 3.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 12.4%

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

6. Taylor expanded in r around 0

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

8. Applied rewrites 12.4%

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

9. Final simplification 12.4%

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

Alternative 11: 9.1% accurate, 5.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 12.4%

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

6. Taylor expanded in s around inf

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

8. Applied rewrites 10.8%

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

Alternative 12: 9.1% accurate, 6.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

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

5. Applied rewrites 10.8%

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

Alternative 13: 9.0% accurate, 9.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 12.4%

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

6. Taylor expanded in s around inf

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

8. Applied rewrites 10.0%

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

9. Final simplification 10.0%

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

Alternative 14: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.7%

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

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 10.0%

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

Alternative 15: 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 99.7%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 10.0%

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

7. Applied rewrites 10.0%

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

Reproduce

herbie shell --seed 2025026 
(FPCore (s r)
  :name "Disney BSSRDF, PDF of scattering profile"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (< 1e-6 r) (< r 1000000.0)))
  (+ (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 (PI)) s) r)) (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 (PI)) s) r))))