Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 11.6s

Alternatives: 16

Speedup: 1.1×

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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.125, \frac{\frac{1}{e^{\frac{\frac{-r}{-3}}{s}}}}{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  0.125
  (/ (/ 1.0 (exp (/ (/ (- r) -3.0) s))) (* (PI) (* s r)))
  (* 0.125 (/ (exp (/ (- r) s)) (* (* (PI) s) r)))))

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. times-frac N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

4. Applied rewrites 99.4%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. frac-2neg N/A

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

   5. lift-neg.f32 N/A

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

   6. metadata-eval N/A

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

   7. associate-/r* N/A

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

   8. lift-neg.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. sinh-+-cosh-rev N/A

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

   13. flip-+ N/A

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

   14. sinh-cosh N/A

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

   15. sinh---cosh-rev N/A

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

   16. lower-/.f32 N/A

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

   17. lower-exp.f32 N/A

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

   18. lower-neg.f32 99.4

       \[\leadsto \mathsf{fma}\left(0.125, \frac{\frac{1}{e^{\color{blue}{-\frac{-r}{3 \cdot s}}}}}{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

6. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(0.125, \frac{\color{blue}{\frac{1}{e^{-\frac{\frac{r}{-3}}{s}}}}}{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

7. Final simplification 99.4%

   \[\leadsto \mathsf{fma}\left(0.125, \frac{\frac{1}{e^{\frac{\frac{-r}{-3}}{s}}}}{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]
8. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.125, \frac{\frac{1}{e^{\frac{r}{3 \cdot s}}}}{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  0.125
  (/ (/ 1.0 (exp (/ r (* 3.0 s)))) (* (PI) (* s r)))
  (* 0.125 (/ (exp (/ (- r) s)) (* (* (PI) s) r)))))

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. times-frac N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

4. Applied rewrites 99.4%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. frac-2neg N/A

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

   5. lift-neg.f32 N/A

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

   6. metadata-eval N/A

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

   7. associate-/r* N/A

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

   8. lift-neg.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. sinh-+-cosh-rev N/A

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

   13. flip-+ N/A

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

   14. sinh-cosh N/A

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

   15. sinh---cosh-rev N/A

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

   16. lower-/.f32 N/A

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

   17. lower-exp.f32 N/A

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

   18. lower-neg.f32 99.4

       \[\leadsto \mathsf{fma}\left(0.125, \frac{\frac{1}{e^{\color{blue}{-\frac{-r}{3 \cdot s}}}}}{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

6. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(0.125, \frac{\color{blue}{\frac{1}{e^{-\frac{\frac{r}{-3}}{s}}}}}{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]
7. Step-by-step derivation

   1. lift-neg.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. distribute-neg-frac N/A

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

   4. lift-/.f32 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. metadata-eval N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f32 N/A

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

   9. lower-*.f32 99.4

      \[\leadsto \mathsf{fma}\left(0.125, \frac{\frac{1}{e^{\frac{r}{\color{blue}{3 \cdot s}}}}}{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

8. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(0.125, \frac{\frac{1}{e^{\color{blue}{\frac{r}{3 \cdot s}}}}}{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]
9. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. times-frac N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

4. Applied rewrites 99.4%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/r* N/A

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

   8. lift-/.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*l* N/A

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

   12. associate-/r* N/A

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

4. Applied rewrites 99.4%

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

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. times-frac N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.3

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

7. Applied rewrites 99.3%

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

Alternative 6: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. times-frac N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

4. Applied rewrites 99.4%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-lft-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

6. Applied rewrites 99.3%

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

Alternative 7: 59.2% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. times-frac N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

4. Applied rewrites 99.4%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. frac-2neg N/A

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

   5. lift-neg.f32 N/A

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

   6. metadata-eval N/A

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

   7. associate-/r* N/A

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

   8. lift-neg.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. sinh-+-cosh-rev N/A

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

   13. flip-+ N/A

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

   14. sinh-cosh N/A

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

   15. sinh---cosh-rev N/A

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

   16. lower-/.f32 N/A

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

   17. lower-exp.f32 N/A

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

   18. lower-neg.f32 99.4

       \[\leadsto \mathsf{fma}\left(0.125, \frac{\frac{1}{e^{\color{blue}{-\frac{-r}{3 \cdot s}}}}}{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

6. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(0.125, \frac{\color{blue}{\frac{1}{e^{-\frac{\frac{r}{-3}}{s}}}}}{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

7. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 60.6

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

9. Applied rewrites 60.6%

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

Alternative 8: 16.0% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. times-frac N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

4. Applied rewrites 99.4%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. frac-2neg N/A

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

   5. lift-neg.f32 N/A

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

   6. metadata-eval N/A

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

   7. associate-/r* N/A

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

   8. lift-neg.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. sinh-+-cosh-rev N/A

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

   13. flip-+ N/A

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

   14. sinh-cosh N/A

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

   15. sinh---cosh-rev N/A

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

   16. lower-/.f32 N/A

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

   17. lower-exp.f32 N/A

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

   18. lower-neg.f32 99.4

       \[\leadsto \mathsf{fma}\left(0.125, \frac{\frac{1}{e^{\color{blue}{-\frac{-r}{3 \cdot s}}}}}{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

6. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(0.125, \frac{\color{blue}{\frac{1}{e^{-\frac{\frac{r}{-3}}{s}}}}}{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

7. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 14.9

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

9. Applied rewrites 14.9%

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

Alternative 9: 10.3% accurate, 3.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 10.0%

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

Alternative 10: 10.3% accurate, 4.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 8.8

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

5. Applied rewrites 8.8%

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

7. Applied rewrites 8.8%

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

8. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

10. Applied rewrites 10.0%

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

Alternative 11: 9.4% accurate, 5.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f32 N/A

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

5. Applied rewrites 9.2%

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

Alternative 12: 9.4% accurate, 5.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-/r* N/A

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

   6. associate-*r/ N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. associate-/r* N/A

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

   10. div-add-rev N/A

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

   11. lower-/.f32 N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. lower-PI.f32 9.2

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

5. Applied rewrites 9.2%

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

Alternative 13: 9.4% accurate, 6.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. times-frac N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

4. Applied rewrites 99.4%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. frac-2neg N/A

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

   5. lift-neg.f32 N/A

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

   6. metadata-eval N/A

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

   7. associate-/r* N/A

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

   8. lift-neg.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. sinh-+-cosh-rev N/A

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

   13. flip-+ N/A

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

   14. sinh-cosh N/A

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

   15. sinh---cosh-rev N/A

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

   16. lower-/.f32 N/A

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

   17. lower-exp.f32 N/A

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

   18. lower-neg.f32 99.4

       \[\leadsto \mathsf{fma}\left(0.125, \frac{\frac{1}{e^{\color{blue}{-\frac{-r}{3 \cdot s}}}}}{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

6. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(0.125, \frac{\color{blue}{\frac{1}{e^{-\frac{\frac{r}{-3}}{s}}}}}{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}, 0.125 \cdot \frac{e^{\frac{-r}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right) \]

7. Taylor expanded in s around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 9.2

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

9. Applied rewrites 9.2%

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

Alternative 14: 9.1% accurate, 10.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 8.8

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

5. Applied rewrites 8.8%

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

7. Applied rewrites 8.8%

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

9. Applied rewrites 8.8%

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

Alternative 15: 9.1% accurate, 12.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 8.8

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

5. Applied rewrites 8.8%

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

7. Applied rewrites 8.8%

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

9. Applied rewrites 8.8%

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

Alternative 16: 9.1% 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 99.3%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 8.8

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

5. Applied rewrites 8.8%

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

7. Applied rewrites 8.8%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 8.8%

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

Reproduce

herbie shell --seed 2024351 
(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))))