Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 12.1s

Alternatives: 16

Speedup: N/A×

Specification

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

Math

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

FPCore

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. lift-neg.f32 N/A

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

   5. remove-double-neg N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

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

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 N/A

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

   14. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 19.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot r\right) \cdot -0.6666666666666666\\ \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(r - \frac{\mathsf{fma}\left(\frac{t\_0 \cdot r}{s}, 0.3333333333333333, t\_0\right)}{s}\right) + r}{\frac{\left(\left(\frac{0.5}{s} \cdot r\right) \cdot r\right) \cdot r}{\frac{0.125}{\mathsf{PI}\left(\right)}} \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (* r r) -0.6666666666666666)))
   (if (<=
        (+
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r))
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* (PI) 2.0) s) r)))
        1.9999999494757503e-5)
     (/
      (+ (- r (/ (fma (/ (* t_0 r) s) 0.3333333333333333 t_0) s)) r)
      (* (/ (* (* (* (/ 0.5 s) r) r) r) (/ 0.125 (PI))) r))
     (/
      (-
       (/
        (-
         (/ -0.16666666666666666 (PI))
         (/ (* (/ -0.06944444444444445 (PI)) r) s))
        s)
       (/ -0.25 (* (PI) r)))
      s))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 1.99999995e-5

   1. Initial program 99.8%

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

   4. Applied rewrites 1.3%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   7. Applied rewrites 90.8%

      \[\leadsto \frac{r + \color{blue}{\left(r - \frac{\mathsf{fma}\left(\frac{\left(-0.6666666666666666 \cdot \left(r \cdot r\right)\right) \cdot r}{s}, 0.3333333333333333, -0.6666666666666666 \cdot \left(r \cdot r\right)\right)}{s}\right)}}{\frac{\frac{r}{\frac{e^{\frac{-r}{s}}}{s}}}{\frac{0.125}{\mathsf{PI}\left(\right)}} \cdot r} \]

   8. Taylor expanded in r around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-/.f32 4.0

         \[\leadsto \frac{r + \left(r - \frac{\mathsf{fma}\left(\frac{\left(-0.6666666666666666 \cdot \left(r \cdot r\right)\right) \cdot r}{s}, 0.3333333333333333, -0.6666666666666666 \cdot \left(r \cdot r\right)\right)}{s}\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \color{blue}{\frac{r}{s}}, 1\right), r, s\right) \cdot r}{\frac{0.125}{\mathsf{PI}\left(\right)}} \cdot r} \]

   10. Applied rewrites 4.0%

       \[\leadsto \frac{r + \left(r - \frac{\mathsf{fma}\left(\frac{\left(-0.6666666666666666 \cdot \left(r \cdot r\right)\right) \cdot r}{s}, 0.3333333333333333, -0.6666666666666666 \cdot \left(r \cdot r\right)\right)}{s}\right)}{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \frac{r}{s}, 1\right), r, s\right) \cdot r}}{\frac{0.125}{\mathsf{PI}\left(\right)}} \cdot r} \]

   11. Taylor expanded in s around 0

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

   13. Applied rewrites 18.4%

       \[\leadsto \frac{r + \left(r - \frac{\mathsf{fma}\left(\frac{\left(-0.6666666666666666 \cdot \left(r \cdot r\right)\right) \cdot r}{s}, 0.3333333333333333, -0.6666666666666666 \cdot \left(r \cdot r\right)\right)}{s}\right)}{\frac{\left(\left(\frac{0.5}{s} \cdot r\right) \cdot r\right) \cdot r}{\frac{0.125}{\mathsf{PI}\left(\right)}} \cdot r} \]

   if 1.99999995e-5 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 98.0%

      \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
   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 66.8%

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

   6. Taylor expanded in s around inf

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

   8. Applied rewrites 67.9%

      \[\leadsto \frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \frac{-0.06944444444444445}{\mathsf{PI}\left(\right)}}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r} \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(r - \frac{\mathsf{fma}\left(\frac{\left(\left(r \cdot r\right) \cdot -0.6666666666666666\right) \cdot r}{s}, 0.3333333333333333, \left(r \cdot r\right) \cdot -0.6666666666666666\right)}{s}\right) + r}{\frac{\left(\left(\frac{0.5}{s} \cdot r\right) \cdot r\right) \cdot r}{\frac{0.125}{\mathsf{PI}\left(\right)}} \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.9% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (/ (* 2.0 r) (* (/ (/ r (/ (exp (/ (- r) s)) s)) (/ 0.125 (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 11.4%

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

5. Taylor expanded in s around inf

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

   1. lower-*.f32 93.1

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

7. Applied rewrites 93.1%

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

Alternative 4: 15.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot r\right) \cdot -0.6666666666666666\\ \frac{\left(r - \frac{\mathsf{fma}\left(\frac{t\_0 \cdot r}{s}, 0.3333333333333333, t\_0\right)}{s}\right) + r}{\frac{\left(\left(\left(\frac{0.5}{s} - \frac{-1 - \frac{s}{r}}{r}\right) \cdot r\right) \cdot r\right) \cdot r}{\frac{0.125}{\mathsf{PI}\left(\right)}} \cdot r} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (* r r) -0.6666666666666666)))
   (/
    (+ (- r (/ (fma (/ (* t_0 r) s) 0.3333333333333333 t_0) s)) r)
    (*
     (/ (* (* (* (- (/ 0.5 s) (/ (- -1.0 (/ s r)) r)) r) r) r) (/ 0.125 (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 11.4%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 85.7%

   \[\leadsto \frac{r + \color{blue}{\left(r - \frac{\mathsf{fma}\left(\frac{\left(-0.6666666666666666 \cdot \left(r \cdot r\right)\right) \cdot r}{s}, 0.3333333333333333, -0.6666666666666666 \cdot \left(r \cdot r\right)\right)}{s}\right)}}{\frac{\frac{r}{\frac{e^{\frac{-r}{s}}}{s}}}{\frac{0.125}{\mathsf{PI}\left(\right)}} \cdot r} \]

8. Taylor expanded in r around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-/.f32 8.5

      \[\leadsto \frac{r + \left(r - \frac{\mathsf{fma}\left(\frac{\left(-0.6666666666666666 \cdot \left(r \cdot r\right)\right) \cdot r}{s}, 0.3333333333333333, -0.6666666666666666 \cdot \left(r \cdot r\right)\right)}{s}\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \color{blue}{\frac{r}{s}}, 1\right), r, s\right) \cdot r}{\frac{0.125}{\mathsf{PI}\left(\right)}} \cdot r} \]

10. Applied rewrites 8.5%

    \[\leadsto \frac{r + \left(r - \frac{\mathsf{fma}\left(\frac{\left(-0.6666666666666666 \cdot \left(r \cdot r\right)\right) \cdot r}{s}, 0.3333333333333333, -0.6666666666666666 \cdot \left(r \cdot r\right)\right)}{s}\right)}{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \frac{r}{s}, 1\right), r, s\right) \cdot r}}{\frac{0.125}{\mathsf{PI}\left(\right)}} \cdot r} \]

11. Taylor expanded in r around -inf

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

13. Applied rewrites 23.6%

    \[\leadsto \frac{r + \left(r - \frac{\mathsf{fma}\left(\frac{\left(-0.6666666666666666 \cdot \left(r \cdot r\right)\right) \cdot r}{s}, 0.3333333333333333, -0.6666666666666666 \cdot \left(r \cdot r\right)\right)}{s}\right)}{\frac{\left(\left(\left(\frac{0.5}{s} - \frac{-1 - \frac{s}{r}}{r}\right) \cdot r\right) \cdot r\right) \cdot r}{\frac{0.125}{\mathsf{PI}\left(\right)}} \cdot r} \]

14. Final simplification 23.6%

    \[\leadsto \frac{\left(r - \frac{\mathsf{fma}\left(\frac{\left(\left(r \cdot r\right) \cdot -0.6666666666666666\right) \cdot r}{s}, 0.3333333333333333, \left(r \cdot r\right) \cdot -0.6666666666666666\right)}{s}\right) + r}{\frac{\left(\left(\left(\frac{0.5}{s} - \frac{-1 - \frac{s}{r}}{r}\right) \cdot r\right) \cdot r\right) \cdot r}{\frac{0.125}{\mathsf{PI}\left(\right)}} \cdot r} \]
15. Add Preprocessing

Alternative 5: 6.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot r\right) \cdot -0.6666666666666666\\ \frac{\left(r - \frac{\mathsf{fma}\left(\frac{t\_0 \cdot r}{s}, 0.3333333333333333, t\_0\right)}{s}\right) + r}{\left(\left(\left(\left(s + r\right) \cdot \mathsf{PI}\left(\right)\right) \cdot r\right) \cdot 8\right) \cdot r} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (* r r) -0.6666666666666666)))
   (/
    (+ (- r (/ (fma (/ (* t_0 r) s) 0.3333333333333333 t_0) s)) r)
    (* (* (* (* (+ s r) (PI)) r) 8.0) 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 11.4%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 85.7%

   \[\leadsto \frac{r + \color{blue}{\left(r - \frac{\mathsf{fma}\left(\frac{\left(-0.6666666666666666 \cdot \left(r \cdot r\right)\right) \cdot r}{s}, 0.3333333333333333, -0.6666666666666666 \cdot \left(r \cdot r\right)\right)}{s}\right)}}{\frac{\frac{r}{\frac{e^{\frac{-r}{s}}}{s}}}{\frac{0.125}{\mathsf{PI}\left(\right)}} \cdot r} \]

8. Taylor expanded in r around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-/.f32 8.5

      \[\leadsto \frac{r + \left(r - \frac{\mathsf{fma}\left(\frac{\left(-0.6666666666666666 \cdot \left(r \cdot r\right)\right) \cdot r}{s}, 0.3333333333333333, -0.6666666666666666 \cdot \left(r \cdot r\right)\right)}{s}\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \color{blue}{\frac{r}{s}}, 1\right), r, s\right) \cdot r}{\frac{0.125}{\mathsf{PI}\left(\right)}} \cdot r} \]

10. Applied rewrites 8.5%

    \[\leadsto \frac{r + \left(r - \frac{\mathsf{fma}\left(\frac{\left(-0.6666666666666666 \cdot \left(r \cdot r\right)\right) \cdot r}{s}, 0.3333333333333333, -0.6666666666666666 \cdot \left(r \cdot r\right)\right)}{s}\right)}{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \frac{r}{s}, 1\right), r, s\right) \cdot r}}{\frac{0.125}{\mathsf{PI}\left(\right)}} \cdot r} \]

11. Taylor expanded in r around 0

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

    1. *-commutative N/A

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

    2. distribute-lft-out N/A

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

    3. associate-*l* N/A

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

    4. lower-*.f32 N/A

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

    5. lower-*.f32 N/A

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

    6. distribute-rgt-out N/A

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

    7. lower-*.f32 N/A

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

    8. lower-PI.f32 N/A

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

    9. +-commutative N/A

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

    10. lower-+.f32 11.0

        \[\leadsto \frac{r + \left(r - \frac{\mathsf{fma}\left(\frac{\left(-0.6666666666666666 \cdot \left(r \cdot r\right)\right) \cdot r}{s}, 0.3333333333333333, -0.6666666666666666 \cdot \left(r \cdot r\right)\right)}{s}\right)}{\left(8 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(s + r\right)}\right) \cdot r\right)\right) \cdot r} \]

13. Applied rewrites 11.0%

    \[\leadsto \frac{r + \left(r - \frac{\mathsf{fma}\left(\frac{\left(-0.6666666666666666 \cdot \left(r \cdot r\right)\right) \cdot r}{s}, 0.3333333333333333, -0.6666666666666666 \cdot \left(r \cdot r\right)\right)}{s}\right)}{\color{blue}{\left(8 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(s + r\right)\right) \cdot r\right)\right)} \cdot r} \]

14. Final simplification 11.0%

    \[\leadsto \frac{\left(r - \frac{\mathsf{fma}\left(\frac{\left(\left(r \cdot r\right) \cdot -0.6666666666666666\right) \cdot r}{s}, 0.3333333333333333, \left(r \cdot r\right) \cdot -0.6666666666666666\right)}{s}\right) + r}{\left(\left(\left(\left(s + r\right) \cdot \mathsf{PI}\left(\right)\right) \cdot r\right) \cdot 8\right) \cdot r} \]
15. Add Preprocessing

Alternative 6: 9.7% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 10.1%

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

6. Taylor expanded in s around inf

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

8. Applied rewrites 10.5%

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

9. Final simplification 10.5%

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

Alternative 7: 8.8% accurate, 5.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 9.6%

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

7. Applied rewrites 9.6%

   \[\leadsto \frac{1}{s} \cdot \color{blue}{\left(\frac{-0.16666666666666666}{s \cdot \mathsf{PI}\left(\right)} - \frac{-0.25}{r \cdot \mathsf{PI}\left(\right)}\right)} \]
8. Step-by-step derivation

9. Applied rewrites 9.7%

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

10. Final simplification 9.7%

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

Alternative 8: 8.8% accurate, 5.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 9.6%

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

7. Applied rewrites 9.6%

   \[\leadsto \frac{1}{s} \cdot \color{blue}{\left(\frac{-0.16666666666666666}{s \cdot \mathsf{PI}\left(\right)} - \frac{-0.25}{r \cdot \mathsf{PI}\left(\right)}\right)} \]
8. Step-by-step derivation

9. Applied rewrites 9.6%

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

10. Final simplification 9.6%

    \[\leadsto \frac{\frac{-0.25}{r} - \frac{-0.16666666666666666}{s}}{\mathsf{PI}\left(\right)} \cdot \frac{-1}{s} \]
11. Add Preprocessing

Alternative 9: 8.8% accurate, 6.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 9.6%

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

Alternative 10: 8.9% accurate, 9.0× speedup

Math

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

FPCore

(FPCore (s r) :precision binary32 (* (/ 0.25 (* (PI) s)) (/ 1.0 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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.4

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

5. Applied rewrites 9.4%

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

7. Applied rewrites 9.4%

   \[\leadsto \frac{1}{r} \cdot \color{blue}{\frac{0.25}{\mathsf{PI}\left(\right) \cdot s}} \]

8. Final simplification 9.4%

   \[\leadsto \frac{0.25}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{1}{r} \]
9. Add Preprocessing

Alternative 11: 8.9% accurate, 10.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.4

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

5. Applied rewrites 9.4%

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

Alternative 12: 8.9% accurate, 10.6× 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 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.4

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

5. Applied rewrites 9.4%

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

7. Applied rewrites 9.4%

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

9. Applied rewrites 9.4%

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

10. Final simplification 9.4%

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

Alternative 13: 8.9% accurate, 10.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.4

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

5. Applied rewrites 9.4%

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

7. Applied rewrites 9.4%

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

Alternative 14: 8.9% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(4 \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 1.0 (* (* 4.0 (* (PI) s)) r)))

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.4

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

5. Applied rewrites 9.4%

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

7. Applied rewrites 9.4%

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

8. Final simplification 9.4%

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

Alternative 15: 8.9% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.4

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

5. Applied rewrites 9.4%

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

7. Applied rewrites 9.4%

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

9. Applied rewrites 9.4%

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

10. Final simplification 9.4%

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

Alternative 16: 8.9% 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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.4

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

5. Applied rewrites 9.4%

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

7. Applied rewrites 9.4%

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

9. Applied rewrites 9.4%

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

Reproduce

herbie shell --seed 2024296 
(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))))