Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.8% → 96.9%

Time: 11.6s

Alternatives: 9

Speedup: 7.7×

Specification

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 (PI)) u2))))

Initial Program: 57.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 (PI)) u2))))

Alternative 1: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{u1}}\\ t_1 := \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \mathbf{if}\;1 - u1 \leq 0.996999979019165:\\ \;\;\;\;t\_1 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{u1} \cdot 0.5, \mathsf{fma}\left(\frac{-1}{u1}, 0.0625, 0.25\right), t\_0 \cdot 0.16666666666666666\right), u1, t\_0 \cdot 0.25\right) + \sqrt{u1}\right)}^{2}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 u1))) (t_1 (sin (* u2 (* (PI) 2.0)))))
   (if (<= (- 1.0 u1) 0.996999979019165)
     (* t_1 (sqrt (- (log (- 1.0 u1)))))
     (*
      (sqrt
       (pow
        (+
         (*
          (* u1 u1)
          (fma
           (fma
            (* (sqrt u1) 0.5)
            (fma (/ -1.0 u1) 0.0625 0.25)
            (* t_0 0.16666666666666666))
           u1
           (* t_0 0.25)))
         (sqrt u1))
        2.0))
      t_1))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.996999979

   1. Initial program 94.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing

   if 0.996999979 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 44.0%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{-\color{blue}{-1 \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-neg.f32 87.5

         \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Applied rewrites 87.5%

      \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. +-lft-identity N/A

         \[\leadsto \sqrt{\color{blue}{0 + \left(-\left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. flip3-+ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{0}^{3} + {\left(-\left(-u1\right)\right)}^{3}}{0 \cdot 0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. lower-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{0}^{3} + {\left(-\left(-u1\right)\right)}^{3}}{0 \cdot 0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\color{blue}{0} + {\left(-\left(-u1\right)\right)}^{3}}{0 \cdot 0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-+.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{0 + {\left(-\left(-u1\right)\right)}^{3}}}{0 \cdot 0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \sqrt{\frac{0 + \color{blue}{{\left(-\left(-u1\right)\right)}^{3}}}{0 \cdot 0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\frac{0 + {\left(-\left(-u1\right)\right)}^{3}}{\color{blue}{0} + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-+.f32 N/A

         \[\leadsto \sqrt{\frac{0 + {\left(-\left(-u1\right)\right)}^{3}}{\color{blue}{0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Applied rewrites 87.5%

      \[\leadsto \sqrt{\color{blue}{\frac{0 + {\left(-\left(-u1\right)\right)}^{3}}{0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{0 + {\left(-\left(-u1\right)\right)}^{3}}{0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lift-+.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{0 + {\left(-\left(-u1\right)\right)}^{3}}}{0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. +-lft-identity N/A

         \[\leadsto \sqrt{\frac{\color{blue}{{\left(-\left(-u1\right)\right)}^{3}}}{0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. lift-pow.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{{\left(-\left(-u1\right)\right)}^{3}}}{0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. sqr-pow N/A

         \[\leadsto \sqrt{\frac{\color{blue}{{\left(-\left(-u1\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(-\left(-u1\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lift-+.f32 N/A

         \[\leadsto \sqrt{\frac{{\left(-\left(-u1\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(-\left(-u1\right)\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. +-lft-identity N/A

         \[\leadsto \sqrt{\frac{{\left(-\left(-u1\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(-\left(-u1\right)\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lift--.f32 N/A

         \[\leadsto \sqrt{\frac{{\left(-\left(-u1\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(-\left(-u1\right)\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   9. Applied rewrites 87.3%

      \[\leadsto \sqrt{\color{blue}{{\left(\frac{{\left(-\left(-u1\right)\right)}^{1.5}}{-\left(-u1\right)}\right)}^{2}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. Taylor expanded in u1 around 0

       \[\leadsto \sqrt{{\color{blue}{\left(-1 \cdot \left(\sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right) + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} + \frac{1}{16} \cdot \frac{1}{u1 \cdot {\left(\sqrt{-1}\right)}^{2}}\right)\right)\right)\right)\right)}}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \sqrt{{\color{blue}{\left({u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} + \frac{1}{16} \cdot \frac{1}{u1 \cdot {\left(\sqrt{-1}\right)}^{2}}\right)\right)\right)\right) + -1 \cdot \left(\sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

       2. mul-1-neg N/A

          \[\leadsto \sqrt{{\left({u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} + \frac{1}{16} \cdot \frac{1}{u1 \cdot {\left(\sqrt{-1}\right)}^{2}}\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}\right)}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

       3. unsub-neg N/A

          \[\leadsto \sqrt{{\color{blue}{\left({u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} + \frac{1}{16} \cdot \frac{1}{u1 \cdot {\left(\sqrt{-1}\right)}^{2}}\right)\right)\right)\right) - \sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

       4. lower--.f32 N/A

          \[\leadsto \sqrt{{\color{blue}{\left({u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} + \frac{1}{16} \cdot \frac{1}{u1 \cdot {\left(\sqrt{-1}\right)}^{2}}\right)\right)\right)\right) - \sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   12. Applied rewrites 97.8%

       \[\leadsto \sqrt{{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \mathsf{fma}\left(\frac{-1}{u1}, 0.0625, 0.25\right), 0.16666666666666666 \cdot \sqrt{\frac{1}{u1}}\right), u1, \sqrt{\frac{1}{u1}} \cdot 0.25\right) \cdot \left(u1 \cdot u1\right) - \left(-\sqrt{u1}\right)\right)}}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.996999979019165:\\ \;\;\;\;\sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{u1} \cdot 0.5, \mathsf{fma}\left(\frac{-1}{u1}, 0.0625, 0.25\right), \sqrt{\frac{1}{u1}} \cdot 0.16666666666666666\right), u1, \sqrt{\frac{1}{u1}} \cdot 0.25\right) + \sqrt{u1}\right)}^{2}} \cdot \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ t_1 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_1 \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{\frac{{\left(-\left(-u1\right)\right)}^{3}}{u1 \cdot u1}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* u2 (* (PI) 2.0)))) (t_1 (- (log (- 1.0 u1)))))
   (if (<= t_1 0.00019999999494757503)
     (* (sqrt (/ (pow (- (- u1)) 3.0) (* u1 u1))) t_0)
     (* t_0 (sqrt t_1)))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.99999995e-4

   1. Initial program 36.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{-\color{blue}{-1 \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-neg.f32 92.0

         \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Applied rewrites 92.0%

      \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. +-lft-identity N/A

         \[\leadsto \sqrt{\color{blue}{0 + \left(-\left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. flip3-+ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{0}^{3} + {\left(-\left(-u1\right)\right)}^{3}}{0 \cdot 0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. lower-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{0}^{3} + {\left(-\left(-u1\right)\right)}^{3}}{0 \cdot 0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\color{blue}{0} + {\left(-\left(-u1\right)\right)}^{3}}{0 \cdot 0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-+.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{0 + {\left(-\left(-u1\right)\right)}^{3}}}{0 \cdot 0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \sqrt{\frac{0 + \color{blue}{{\left(-\left(-u1\right)\right)}^{3}}}{0 \cdot 0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\frac{0 + {\left(-\left(-u1\right)\right)}^{3}}{\color{blue}{0} + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-+.f32 N/A

         \[\leadsto \sqrt{\frac{0 + {\left(-\left(-u1\right)\right)}^{3}}{\color{blue}{0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Applied rewrites 92.0%

      \[\leadsto \sqrt{\color{blue}{\frac{0 + {\left(-\left(-u1\right)\right)}^{3}}{0 + \left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right) - 0 \cdot \left(-\left(-u1\right)\right)\right)}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   9. Applied rewrites 26.6%

      \[\leadsto \sqrt{\frac{0 + {\left(-\left(-u1\right)\right)}^{3}}{0 + \left(\color{blue}{\frac{{\left(-\left(-u1\right)\right)}^{8} - 0}{{\left(-u1\right)}^{6}}} - 0 \cdot \left(-\left(-u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. Taylor expanded in u1 around 0

       \[\leadsto \sqrt{\frac{0 + {\left(-\left(-u1\right)\right)}^{3}}{\color{blue}{{u1}^{2}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \sqrt{\frac{0 + {\left(-\left(-u1\right)\right)}^{3}}{\color{blue}{u1 \cdot u1}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

       2. lower-*.f32 92.0

          \[\leadsto \sqrt{\frac{0 + {\left(-\left(-u1\right)\right)}^{3}}{\color{blue}{u1 \cdot u1}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   12. Applied rewrites 92.0%

       \[\leadsto \sqrt{\frac{0 + {\left(-\left(-u1\right)\right)}^{3}}{\color{blue}{u1 \cdot u1}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   if 1.99999995e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 89.4%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{\frac{{\left(-\left(-u1\right)\right)}^{3}}{u1 \cdot u1}} \cdot \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ t_1 := \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \mathbf{if}\;t\_0 \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{\frac{1}{\frac{1}{u1}}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))) (t_1 (sin (* u2 (* (PI) 2.0)))))
   (if (<= t_0 0.00019999999494757503)
     (* (sqrt (/ 1.0 (/ 1.0 u1))) t_1)
     (* t_1 (sqrt t_0)))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.99999995e-4

   1. Initial program 36.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing

   4. Applied rewrites 38.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. flip-- N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{log1p}\left(u1\right) \cdot \mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) \cdot \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}{\mathsf{log1p}\left(u1\right) + \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. clear-num N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\mathsf{log1p}\left(u1\right) + \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}{\mathsf{log1p}\left(u1\right) \cdot \mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) \cdot \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. lower-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\mathsf{log1p}\left(u1\right) + \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}{\mathsf{log1p}\left(u1\right) \cdot \mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) \cdot \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. clear-num N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{\frac{\mathsf{log1p}\left(u1\right) \cdot \mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) \cdot \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}{\mathsf{log1p}\left(u1\right) + \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. flip-- N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{\color{blue}{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lift--.f32 N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{\color{blue}{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-/.f32 37.4

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. lift--.f32 N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{\color{blue}{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lift-log1p.f32 N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{\color{blue}{\log \left(1 + u1\right)} - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. lift-log1p.f32 N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{\log \left(1 + u1\right) - \color{blue}{\log \left(1 + \left(-u1\right) \cdot u1\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      12. diff-log N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{\color{blue}{\log \left(\frac{1 + u1}{1 + \left(-u1\right) \cdot u1}\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      13. clear-num N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{\log \color{blue}{\left(\frac{1}{\frac{1 + \left(-u1\right) \cdot u1}{1 + u1}}\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. Applied rewrites 40.2%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1}{-\mathsf{log1p}\left(-u1\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{u1}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

      1. lower-/.f32 92.0

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{u1}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   9. Applied rewrites 92.0%

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{u1}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   if 1.99999995e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 89.4%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{\frac{1}{\frac{1}{u1}}} \cdot \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9977999925613403:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\frac{1}{u1}}} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* (PI) 2.0))))
   (if (<= (- 1.0 u1) 0.9977999925613403)
     (* t_0 (sqrt (- (log (- 1.0 u1)))))
     (* (sqrt (/ 1.0 (/ 1.0 u1))) (sin t_0)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.997799993

   1. Initial program 93.6%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot u2\right) \]

      6. lower-PI.f32 80.7

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot u2\right) \]

   5. Applied rewrites 80.7%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)} \]

   if 0.997799993 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 42.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing

   4. Applied rewrites 33.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. flip-- N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{log1p}\left(u1\right) \cdot \mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) \cdot \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}{\mathsf{log1p}\left(u1\right) + \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. clear-num N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\mathsf{log1p}\left(u1\right) + \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}{\mathsf{log1p}\left(u1\right) \cdot \mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) \cdot \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. lower-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\mathsf{log1p}\left(u1\right) + \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}{\mathsf{log1p}\left(u1\right) \cdot \mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) \cdot \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. clear-num N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{\frac{\mathsf{log1p}\left(u1\right) \cdot \mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) \cdot \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}{\mathsf{log1p}\left(u1\right) + \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. flip-- N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{\color{blue}{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lift--.f32 N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{\color{blue}{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-/.f32 34.0

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. lift--.f32 N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{\color{blue}{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lift-log1p.f32 N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{\color{blue}{\log \left(1 + u1\right)} - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. lift-log1p.f32 N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{\log \left(1 + u1\right) - \color{blue}{\log \left(1 + \left(-u1\right) \cdot u1\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      12. diff-log N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{\color{blue}{\log \left(\frac{1 + u1}{1 + \left(-u1\right) \cdot u1}\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      13. clear-num N/A

          \[\leadsto \sqrt{\frac{1}{\frac{1}{\log \color{blue}{\left(\frac{1}{\frac{1 + \left(-u1\right) \cdot u1}{1 + u1}}\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. Applied rewrites 37.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1}{-\mathsf{log1p}\left(-u1\right)}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{u1}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

      1. lower-/.f32 88.3

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{u1}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   9. Applied rewrites 88.3%

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\frac{1}{u1}}}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9977999925613403:\\ \;\;\;\;\left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\frac{1}{u1}}} \cdot \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9977999925613403:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* (PI) 2.0))))
   (if (<= (- 1.0 u1) 0.9977999925613403)
     (* t_0 (sqrt (- (log (- 1.0 u1)))))
     (* (sqrt u1) (sin t_0)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.997799993

   1. Initial program 93.6%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot u2\right) \]

      6. lower-PI.f32 80.7

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot u2\right) \]

   5. Applied rewrites 80.7%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)} \]

   if 0.997799993 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 42.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing

   4. Applied rewrites 16.4%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. lower-sqrt.f32 88.2

         \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Applied rewrites 88.2%

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9977999925613403:\\ \;\;\;\;\left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \cdot \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt u1) (sin (* u2 (* (PI) 2.0)))))

Derivation

1. Initial program 57.7%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing

4. Applied rewrites 13.6%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
6. Step-by-step derivation

   1. lower-sqrt.f32 76.1

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

7. Applied rewrites 76.1%

   \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

8. Final simplification 76.1%

   \[\leadsto \sqrt{u1} \cdot \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \]
9. Add Preprocessing

Alternative 7: 65.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\frac{-1}{-u1}}} \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ 1.0 (/ -1.0 (- u1)))) (* u2 (* (PI) 2.0))))

Derivation

1. Initial program 57.7%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{-\color{blue}{-1 \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. lower-neg.f32 76.1

      \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Applied rewrites 76.1%

   \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

6. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\left(-u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{-\left(-u1\right)} \cdot \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2\right)}\right) \]

   2. associate-*r* N/A

      \[\leadsto \sqrt{-\left(-u1\right)} \cdot \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

   3. lower-*.f32 N/A

      \[\leadsto \sqrt{-\left(-u1\right)} \cdot \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

   4. *-commutative N/A

      \[\leadsto \sqrt{-\left(-u1\right)} \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot u2\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{-\left(-u1\right)} \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot u2\right) \]

   6. lower-PI.f32 63.9

      \[\leadsto \sqrt{-\left(-u1\right)} \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot u2\right) \]

8. Applied rewrites 63.9%

   \[\leadsto \sqrt{-\left(-u1\right)} \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)} \]

10. Applied rewrites 63.9%

    \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{-1}{-u1}}}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

11. Final simplification 63.9%

    \[\leadsto \sqrt{\frac{1}{\frac{-1}{-u1}}} \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \]
12. Add Preprocessing

Alternative 8: 65.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\left(-u1\right)} \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (- u1))) (* u2 (* (PI) 2.0))))

Derivation

1. Initial program 57.7%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{-\color{blue}{-1 \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. lower-neg.f32 76.1

      \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Applied rewrites 76.1%

   \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

6. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\left(-u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{-\left(-u1\right)} \cdot \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2\right)}\right) \]

   2. associate-*r* N/A

      \[\leadsto \sqrt{-\left(-u1\right)} \cdot \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

   3. lower-*.f32 N/A

      \[\leadsto \sqrt{-\left(-u1\right)} \cdot \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

   4. *-commutative N/A

      \[\leadsto \sqrt{-\left(-u1\right)} \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot u2\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{-\left(-u1\right)} \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot u2\right) \]

   6. lower-PI.f32 63.9

      \[\leadsto \sqrt{-\left(-u1\right)} \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot u2\right) \]

8. Applied rewrites 63.9%

   \[\leadsto \sqrt{-\left(-u1\right)} \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)} \]

9. Final simplification 63.9%

   \[\leadsto \sqrt{-\left(-u1\right)} \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \]
10. Add Preprocessing

Alternative 9: 4.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \left(-\sqrt{u1}\right) \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (- (sqrt u1)) (* u2 (* (PI) 2.0))))

Derivation

1. Initial program 57.7%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\left(\sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{u1}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. unpow2 N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{u1}\right) \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. rem-square-sqrt N/A

      \[\leadsto \left(\color{blue}{-1} \cdot \sqrt{u1}\right) \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. lower-neg.f32 N/A

      \[\leadsto \color{blue}{\left(-\sqrt{u1}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. lower-sqrt.f32 4.0

      \[\leadsto \left(-\color{blue}{\sqrt{u1}}\right) \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Applied rewrites 4.0%

   \[\leadsto \color{blue}{\left(-\sqrt{u1}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

6. Taylor expanded in u2 around 0

   \[\leadsto \left(-\sqrt{u1}\right) \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(-\sqrt{u1}\right) \cdot \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2\right)}\right) \]

   2. associate-*r* N/A

      \[\leadsto \left(-\sqrt{u1}\right) \cdot \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

   3. lower-*.f32 N/A

      \[\leadsto \left(-\sqrt{u1}\right) \cdot \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

   4. *-commutative N/A

      \[\leadsto \left(-\sqrt{u1}\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot u2\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \left(-\sqrt{u1}\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot u2\right) \]

   6. lower-PI.f32 4.8

      \[\leadsto \left(-\sqrt{u1}\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot u2\right) \]

8. Applied rewrites 4.8%

   \[\leadsto \left(-\sqrt{u1}\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)} \]

9. Final simplification 4.8%

   \[\leadsto \left(-\sqrt{u1}\right) \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024276 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 (PI)) u2))))