Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 16.5s

Alternatives: 16

Speedup: N/A×

Specification

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

Math

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

FPCore

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 94.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r\\ \mathbf{if}\;\frac{e^{\frac{-r}{s}} \cdot 0.25}{t\_0} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{s}, r, 1\right), \frac{0.25}{t\_0}, \frac{{\left(e^{r}\right)}^{\left(\frac{-0.3333333333333333}{s}\right)} \cdot \frac{0.125}{\mathsf{PI}\left(\right) \cdot s}}{r}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (* (* 2.0 (PI)) s) r)))
   (if (<=
        (+
         (/ (* (exp (/ (- r) s)) 0.25) t_0)
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r)))
        9.9999998245167e-15)
     (fma
      (fma (/ -1.0 s) r 1.0)
      (/ 0.25 t_0)
      (/ (* (pow (exp r) (/ -0.3333333333333333 s)) (/ 0.125 (* (PI) s))) r))
     (/
      (-
       (/
        (-
         (/ -0.16666666666666666 (PI))
         (/ (* (/ -0.06944444444444445 (PI)) r) s))
        s)
       (/ -0.25 (* (PI) r)))
      s))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 9.99999982e-15

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 4.6%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{r}{s}}{s}, \mathsf{fma}\left(-0.16666666666666666, \frac{r}{s}, 0.5\right), \frac{-1}{s}\right), r, 1\right)}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]
   6. Step-by-step derivation

   7. Applied rewrites 4.6%

      \[\leadsto \frac{0.25 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{r}{s}}{s}, \frac{r}{s} \cdot -0.16666666666666666 + 0.5, \frac{-1}{s}\right), r, 1\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   8. Taylor expanded in s around inf

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

   10. Applied rewrites 4.6%

       \[\leadsto \frac{0.25 \cdot \mathsf{fma}\left(\frac{-1}{s}, r, 1\right)}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   12. Applied rewrites 98.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{s}, r, 1\right), \frac{0.25}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot s\right) \cdot r}, \frac{\frac{0.125}{\mathsf{PI}\left(\right) \cdot s} \cdot {\left(e^{r}\right)}^{\left(\frac{-0.3333333333333333}{s}\right)}}{r}\right)} \]

   if 9.99999982e-15 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 98.2%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 65.1%

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

   6. Taylor expanded in s around inf

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

   8. Applied rewrites 66.7%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{s}, r, 1\right), \frac{0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}, \frac{{\left(e^{r}\right)}^{\left(\frac{-0.3333333333333333}{s}\right)} \cdot \frac{0.125}{\mathsf{PI}\left(\right) \cdot s}}{r}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 14.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\\ t_1 := \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}\\ \mathbf{if}\;\frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{s}, \frac{t\_0 - \frac{r}{s} \cdot {\left(\frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right)}\right)}^{-1}}{s}, \frac{t\_1}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - t\_1}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ -0.16666666666666666 (PI))) (t_1 (/ -0.25 (* (PI) r))))
   (if (<=
        (+
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* 2.0 (PI)) s) r))
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r)))
        0.0)
     (fma
      (/ 1.0 s)
      (/
       (-
        t_0
        (*
         (/ r s)
         (pow
          (/ (PI) (fma (/ 0.021604938271604937 s) r -0.06944444444444445))
          -1.0)))
       s)
      (/ t_1 (- s)))
     (/ (- (/ (- t_0 (/ (* (/ -0.06944444444444445 (PI)) r) s)) s) t_1) s))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 3.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 5.8%

      \[\leadsto \mathsf{fma}\left(\frac{1}{s}, \color{blue}{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 7.5%

      \[\leadsto \mathsf{fma}\left(\frac{1}{s}, \frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - {\left(\frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right)}\right)}^{-1} \cdot \frac{r}{s}}{s}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]

   if 0.0 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 96.9%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 57.2%

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

   6. Taylor expanded in s around inf

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

   8. Applied rewrites 59.0%

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

4. Final simplification 15.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{s}, \frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r}{s} \cdot {\left(\frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\frac{0.021604938271604937}{s}, r, -0.06944444444444445\right)}\right)}^{-1}}{s}, \frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 13.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot r\\ t_1 := \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;\frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{s}, \frac{t\_1 - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \frac{{\left(-4 \cdot t\_0\right)}^{-1}}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1 - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{t\_0}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (PI) r)) (t_1 (/ -0.16666666666666666 (PI))))
   (if (<=
        (+
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* 2.0 (PI)) s) r))
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r)))
        0.0)
     (fma
      (/ 1.0 s)
      (/
       (-
        t_1
        (*
         (/ (- -0.06944444444444445 (* (/ -0.021604938271604937 s) r)) (PI))
         (/ r s)))
       s)
      (/ (pow (* -4.0 t_0) -1.0) (- s)))
     (/
      (- (/ (- t_1 (/ (* (/ -0.06944444444444445 (PI)) r) s)) s) (/ -0.25 t_0))
      s))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 3.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 5.4%

      \[\leadsto \mathsf{fma}\left(\frac{1}{s}, \color{blue}{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 8.2%

      \[\leadsto \mathsf{fma}\left(\frac{1}{s}, \frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, -\frac{{\left(\left(\mathsf{PI}\left(\right) \cdot r\right) \cdot -4\right)}^{-1}}{s}\right) \]

   if 0.0 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 96.9%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 57.2%

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

   6. Taylor expanded in s around inf

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

   8. Applied rewrites 59.0%

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

4. Final simplification 14.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{s}, \frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \frac{{\left(-4 \cdot \left(\mathsf{PI}\left(\right) \cdot r\right)\right)}^{-1}}{-s}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 13.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;\frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{s}, \frac{t\_0 - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, {\left(4 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r\right)\right)}^{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ -0.16666666666666666 (PI))))
   (if (<=
        (+
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* 2.0 (PI)) s) r))
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r)))
        0.0)
     (fma
      (/ 1.0 s)
      (/
       (-
        t_0
        (*
         (/ (- -0.06944444444444445 (* (/ -0.021604938271604937 s) r)) (PI))
         (/ r s)))
       s)
      (pow (* 4.0 (* (* (PI) s) r)) -1.0))
     (/
      (-
       (/ (- t_0 (/ (* (/ -0.06944444444444445 (PI)) r) s)) s)
       (/ -0.25 (* (PI) r)))
      s))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 3.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 5.0%

      \[\leadsto \mathsf{fma}\left(\frac{1}{s}, \color{blue}{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 5.4%

      \[\leadsto \mathsf{fma}\left(\frac{1}{s}, \frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r} \cdot \frac{-1}{s}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 8.1%

       \[\leadsto \mathsf{fma}\left(\frac{1}{s}, \frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, {\left(\left(\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r\right) \cdot 4\right)}^{-1}\right) \]

   if 0.0 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 96.9%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 57.2%

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

   6. Taylor expanded in s around inf

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

   8. Applied rewrites 59.0%

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

4. Final simplification 14.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{s}, \frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, {\left(4 \cdot \left(\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r\right)\right)}^{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 13.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot r\\ t_1 := \frac{-0.16666666666666666}{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;\frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{s}, \frac{t\_1 - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, {\left(\left(t\_0 \cdot s\right) \cdot 4\right)}^{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1 - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{t\_0}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (PI) r)) (t_1 (/ -0.16666666666666666 (PI))))
   (if (<=
        (+
         (/ (* (exp (/ (- r) s)) 0.25) (* (* (* 2.0 (PI)) s) r))
         (/ (* (exp (/ (- r) (* 3.0 s))) 0.75) (* (* (* 6.0 (PI)) s) r)))
        0.0)
     (fma
      (/ 1.0 s)
      (/
       (-
        t_1
        (*
         (/ (- -0.06944444444444445 (* (/ -0.021604938271604937 s) r)) (PI))
         (/ r s)))
       s)
      (pow (* (* t_0 s) 4.0) -1.0))
     (/
      (- (/ (- t_1 (/ (* (/ -0.06944444444444445 (PI)) r) s)) s) (/ -0.25 t_0))
      s))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 3.4%

      \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{r \cdot \left(\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} - \frac{r}{\mathsf{PI}\left(\right)} \cdot \frac{-0.021604938271604937}{s}\right)}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 5.4%

      \[\leadsto \mathsf{fma}\left(\frac{1}{s}, \color{blue}{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}}, -\frac{\frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 7.9%

      \[\leadsto \mathsf{fma}\left(\frac{1}{s}, \frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, {\left(\left(\left(\mathsf{PI}\left(\right) \cdot r\right) \cdot s\right) \cdot 4\right)}^{-1}\right) \]

   if 0.0 < (+.f32 (/.f32 (*.f32 #s(literal 1/4 binary32) (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) s) r)) (/.f32 (*.f32 #s(literal 3/4 binary32) (exp.f32 (/.f32 (neg.f32 r) (*.f32 #s(literal 3 binary32) s)))) (*.f32 (*.f32 (*.f32 #s(literal 6 binary32) (PI.f32)) s) r)))

   1. Initial program 96.9%

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

   3. Taylor expanded in s around -inf

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

   5. Applied rewrites 57.2%

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

   6. Taylor expanded in s around inf

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

   8. Applied rewrites 59.0%

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

4. Final simplification 14.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-r}{s}} \cdot 0.25}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-r}{3 \cdot s}} \cdot 0.75}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{s}, \frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{-0.06944444444444445 - \frac{-0.021604938271604937}{s} \cdot r}{\mathsf{PI}\left(\right)} \cdot \frac{r}{s}}{s}, {\left(\left(\left(\mathsf{PI}\left(\right) \cdot r\right) \cdot s\right) \cdot 4\right)}^{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-0.16666666666666666}{\mathsf{PI}\left(\right)} - \frac{\frac{-0.06944444444444445}{\mathsf{PI}\left(\right)} \cdot r}{s}}{s} - \frac{-0.25}{\mathsf{PI}\left(\right) \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-*.f32 99.6

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 8: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 9: 10.0% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 9.9%

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

6. Taylor expanded in s around inf

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

8. Applied rewrites 10.2%

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

9. Final simplification 10.2%

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

Alternative 10: 9.0% accurate, 5.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.1

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

5. Applied rewrites 9.1%

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

6. Taylor expanded in s around inf

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

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

8. Applied rewrites 9.3%

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

Alternative 11: 9.0% accurate, 6.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 9.3%

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

Alternative 12: 9.0% accurate, 6.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.1

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

5. Applied rewrites 9.1%

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

7. Applied rewrites 9.1%

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

Alternative 13: 9.0% accurate, 10.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.1

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

5. Applied rewrites 9.1%

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

Alternative 14: 9.0% accurate, 10.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.1

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

5. Applied rewrites 9.1%

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

7. Applied rewrites 9.1%

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

9. Applied rewrites 9.1%

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

Alternative 15: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.1

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

5. Applied rewrites 9.1%

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

7. Applied rewrites 9.1%

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

9. Applied rewrites 9.1%

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

Alternative 16: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.1

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

5. Applied rewrites 9.1%

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

7. Applied rewrites 9.1%

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

9. Applied rewrites 9.1%

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

Reproduce

herbie shell --seed 2024271 
(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))))