Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.3% → 91.3%

Time: 9.6s

Alternatives: 9

Speedup: 21.0×

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

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

FPCore

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

Initial Program: 57.3% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 91.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ t_1 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;t\_0 \leq 0.0001221999991685152:\\ \;\;\;\;\sqrt{\frac{\left(u1 + 1\right) \cdot \left(\left(-u1\right) \cdot u1\right)}{-u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0} \cdot \cos \left(\left(\left(u2 \cdot 2\right) \cdot t\_1\right) \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))) (t_1 (sqrt (PI))))
   (if (<= t_0 0.0001221999991685152)
     (*
      (sqrt (/ (* (+ u1 1.0) (* (- u1) u1)) (- u1)))
      (cos (* (* 2.0 (PI)) u2)))
     (* (sqrt t_0) (cos (* (* (* u2 2.0) t_1) t_1))))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.222e-4

   1. Initial program 36.3%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 92.6

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

   5. Applied rewrites 92.6%

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

      1. +-lft-identity N/A

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

      2. flip-+ N/A

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

      3. lower-/.f32 N/A

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

      4. metadata-eval N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 92.6

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\frac{\color{blue}{{u1}^{2} \cdot \left(-1 \cdot u1 - 1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-1 \cdot u1 - 1\right) \cdot {u1}^{2}}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-1 \cdot u1 - 1\right) \cdot {u1}^{2}}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. lower--.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-1 \cdot u1 - 1\right)} \cdot {u1}^{2}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right)} - 1\right) \cdot {u1}^{2}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-neg.f32 N/A

         \[\leadsto \sqrt{\frac{\left(\color{blue}{\left(-u1\right)} - 1\right) \cdot {u1}^{2}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{\left(\left(-u1\right) - 1\right) \cdot \color{blue}{\left(u1 \cdot u1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lower-*.f32 92.6

         \[\leadsto \sqrt{\frac{\left(\left(-u1\right) - 1\right) \cdot \color{blue}{\left(u1 \cdot u1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. Applied rewrites 92.6%

       \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(-u1\right) - 1\right) \cdot \left(u1 \cdot u1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   11. Taylor expanded in u1 around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f32 92.6

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

   13. Applied rewrites 92.6%

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

   if 1.222e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 88.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f32 N/A

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

      6. add-sqr-sqrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-sqrt.f32 88.1

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

   4. Applied rewrites 88.1%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.0001221999991685152:\\ \;\;\;\;\sqrt{\frac{\left(u1 + 1\right) \cdot \left(\left(-u1\right) \cdot u1\right)}{-u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(\left(u2 \cdot 2\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.015799999237060547:\\ \;\;\;\;\sqrt{\frac{\left(u1 + 1\right) \cdot \left(\left(-u1\right) \cdot u1\right)}{-u1}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.015799999237060547)
     (* (sqrt (/ (* (+ u1 1.0) (* (- u1) u1)) (- u1))) t_1)
     (* t_0 1.0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0158

   1. Initial program 42.0%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 89.0

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

   5. Applied rewrites 89.0%

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

      1. +-lft-identity N/A

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

      2. flip-+ N/A

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

      3. lower-/.f32 N/A

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

      4. metadata-eval N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 89.0

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\frac{\color{blue}{{u1}^{2} \cdot \left(-1 \cdot u1 - 1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-1 \cdot u1 - 1\right) \cdot {u1}^{2}}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-1 \cdot u1 - 1\right) \cdot {u1}^{2}}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. lower--.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-1 \cdot u1 - 1\right)} \cdot {u1}^{2}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right)} - 1\right) \cdot {u1}^{2}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-neg.f32 N/A

         \[\leadsto \sqrt{\frac{\left(\color{blue}{\left(-u1\right)} - 1\right) \cdot {u1}^{2}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{\left(\left(-u1\right) - 1\right) \cdot \color{blue}{\left(u1 \cdot u1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lower-*.f32 89.0

         \[\leadsto \sqrt{\frac{\left(\left(-u1\right) - 1\right) \cdot \color{blue}{\left(u1 \cdot u1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. Applied rewrites 89.0%

       \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(-u1\right) - 1\right) \cdot \left(u1 \cdot u1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   11. Taylor expanded in u1 around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f32 89.0

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

   13. Applied rewrites 89.0%

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

   if 0.0158 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 91.0%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 79.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.015799999237060547:\\ \;\;\;\;\sqrt{\frac{\left(u1 + 1\right) \cdot \left(\left(-u1\right) \cdot u1\right)}{-u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.015799999237060547:\\ \;\;\;\;\sqrt{\frac{\left(\left(\left(-u1\right) - 1\right) \cdot u1\right) \cdot u1}{-u1}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.015799999237060547)
     (* (sqrt (/ (* (* (- (- u1) 1.0) u1) u1) (- u1))) t_1)
     (* t_0 1.0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0158

   1. Initial program 42.0%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 89.0

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

   5. Applied rewrites 89.0%

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

      1. +-lft-identity N/A

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

      2. flip-+ N/A

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

      3. lower-/.f32 N/A

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

      4. metadata-eval N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 89.0

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\frac{\color{blue}{{u1}^{2} \cdot \left(-1 \cdot u1 - 1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-1 \cdot u1 - 1\right) \cdot {u1}^{2}}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-1 \cdot u1 - 1\right) \cdot {u1}^{2}}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. lower--.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-1 \cdot u1 - 1\right)} \cdot {u1}^{2}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right)} - 1\right) \cdot {u1}^{2}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-neg.f32 N/A

         \[\leadsto \sqrt{\frac{\left(\color{blue}{\left(-u1\right)} - 1\right) \cdot {u1}^{2}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{\left(\left(-u1\right) - 1\right) \cdot \color{blue}{\left(u1 \cdot u1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lower-*.f32 89.0

         \[\leadsto \sqrt{\frac{\left(\left(-u1\right) - 1\right) \cdot \color{blue}{\left(u1 \cdot u1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. Applied rewrites 89.0%

       \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(-u1\right) - 1\right) \cdot \left(u1 \cdot u1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   11. Step-by-step derivation

       1. lift--.f32 N/A

          \[\leadsto \sqrt{\frac{\mathsf{Rewrite=>}\left(lower-*.f32, \left(\left(\left(-u1\right) - 1\right) \cdot u1\right)\right) \cdot u1}{\color{blue}{0 - \left(-\left(-u1\right)\right)}}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

       2. sub0-neg N/A

          \[\leadsto \sqrt{\frac{\mathsf{Rewrite=>}\left(lower-*.f32, \left(\left(\left(-u1\right) - 1\right) \cdot u1\right)\right) \cdot u1}{\color{blue}{\mathsf{neg}\left(\left(-\left(-u1\right)\right)\right)}}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

       3. lift-neg.f32 N/A

          \[\leadsto \sqrt{\frac{\mathsf{Rewrite=>}\left(lower-*.f32, \left(\left(\left(-u1\right) - 1\right) \cdot u1\right)\right) \cdot u1}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(-u1\right)\right)\right)}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

       4. remove-double-neg N/A

          \[\leadsto \sqrt{\frac{\mathsf{Rewrite=>}\left(lower-*.f32, \left(\left(\left(-u1\right) - 1\right) \cdot u1\right)\right) \cdot u1}{\color{blue}{-u1}}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   12. Applied rewrites 89.0%

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

   if 0.0158 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 91.0%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 79.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.015799999237060547:\\ \;\;\;\;\sqrt{\frac{\left(\left(\left(-u1\right) - 1\right) \cdot u1\right) \cdot u1}{-u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.015799999237060547:\\ \;\;\;\;\sqrt{\frac{\left(-u1\right) \cdot u1}{-u1}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.015799999237060547)
     (* (sqrt (/ (* (- u1) u1) (- u1))) t_1)
     (* t_0 1.0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0158

   1. Initial program 42.0%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 89.0

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

   5. Applied rewrites 89.0%

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

      1. +-lft-identity N/A

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

      2. flip-+ N/A

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

      3. lower-/.f32 N/A

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

      4. metadata-eval N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 89.0

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\frac{\color{blue}{-1 \cdot {u1}^{2}}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

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

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

      4. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-1 \cdot u1\right)} \cdot u1}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-1 \cdot u1\right) \cdot u1}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. mul-1-neg N/A

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

      7. lower-neg.f32 89.0

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

   10. Applied rewrites 89.0%

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

   if 0.0158 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 91.0%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 79.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.015799999237060547:\\ \;\;\;\;\sqrt{\frac{\left(-u1\right) \cdot u1}{-u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq 0.0001221999991685152:\\ \;\;\;\;\sqrt{\frac{\left(u1 + 1\right) \cdot \left(\left(-u1\right) \cdot u1\right)}{-u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0} \cdot \frac{1}{\frac{1}{\cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))))
   (if (<= t_0 0.0001221999991685152)
     (*
      (sqrt (/ (* (+ u1 1.0) (* (- u1) u1)) (- u1)))
      (cos (* (* 2.0 (PI)) u2)))
     (* (sqrt t_0) (/ 1.0 (/ 1.0 (cos (* u2 (* (PI) 2.0)))))))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.222e-4

   1. Initial program 36.3%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 92.6

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

   5. Applied rewrites 92.6%

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

      1. +-lft-identity N/A

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

      2. flip-+ N/A

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

      3. lower-/.f32 N/A

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

      4. metadata-eval N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 92.6

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\frac{\color{blue}{{u1}^{2} \cdot \left(-1 \cdot u1 - 1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-1 \cdot u1 - 1\right) \cdot {u1}^{2}}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-1 \cdot u1 - 1\right) \cdot {u1}^{2}}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. lower--.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-1 \cdot u1 - 1\right)} \cdot {u1}^{2}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right)} - 1\right) \cdot {u1}^{2}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-neg.f32 N/A

         \[\leadsto \sqrt{\frac{\left(\color{blue}{\left(-u1\right)} - 1\right) \cdot {u1}^{2}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{\left(\left(-u1\right) - 1\right) \cdot \color{blue}{\left(u1 \cdot u1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lower-*.f32 92.6

         \[\leadsto \sqrt{\frac{\left(\left(-u1\right) - 1\right) \cdot \color{blue}{\left(u1 \cdot u1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. Applied rewrites 92.6%

       \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(-u1\right) - 1\right) \cdot \left(u1 \cdot u1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   11. Taylor expanded in u1 around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f32 92.6

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

   13. Applied rewrites 92.6%

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

   if 1.222e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 88.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. cos-2 N/A

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

      6. flip-- N/A

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

      7. cos-sin-sum N/A

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

      8. lower-/.f32 N/A

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

   4. Applied rewrites 88.0%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lower-/.f32 87.9

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

      5. /-rgt-identity N/A

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

      6. lift--.f32 N/A

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

      7. lift-pow.f32 N/A

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

      8. sqr-pow N/A

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

      9. lift-pow.f32 N/A

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

      10. sqr-pow N/A

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

      11. cos-sin-sum N/A

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

   6. Applied rewrites 88.1%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.0001221999991685152:\\ \;\;\;\;\sqrt{\frac{\left(u1 + 1\right) \cdot \left(\left(-u1\right) \cdot u1\right)}{-u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \frac{1}{\frac{1}{\cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.015799999237060547:\\ \;\;\;\;\sqrt{u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.015799999237060547) (* (sqrt u1) t_1) (* t_0 1.0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0158

   1. Initial program 42.0%

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

   4. Applied rewrites 52.9%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 89.0

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

   7. Applied rewrites 89.0%

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

   if 0.0158 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 91.0%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 79.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 (PI)) u2))) 0.009999999776482582)
     (sqrt u1)
     (* t_0 1.0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.00999999978

   1. Initial program 37.6%

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right)}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right)}} \]

      2. lower-log1p.f32 73.1

         \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)}} \]

   7. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right)}} \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \]
   9. Step-by-step derivation

   10. Applied rewrites 73.1%

       \[\leadsto \sqrt{u1} \]

   if 0.00999999978 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 87.6%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 76.7%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 91.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ t_1 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_1 \leq 0.0001221999991685152:\\ \;\;\;\;\sqrt{\frac{\left(u1 + 1\right) \cdot \left(\left(-u1\right) \cdot u1\right)}{-u1}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 (PI)) u2))) (t_1 (- (log (- 1.0 u1)))))
   (if (<= t_1 0.0001221999991685152)
     (* (sqrt (/ (* (+ u1 1.0) (* (- u1) u1)) (- u1))) t_0)
     (* (sqrt t_1) t_0))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.222e-4

   1. Initial program 36.3%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 92.6

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

   5. Applied rewrites 92.6%

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

      1. +-lft-identity N/A

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

      2. flip-+ N/A

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

      3. lower-/.f32 N/A

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

      4. metadata-eval N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 92.6

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\frac{\color{blue}{{u1}^{2} \cdot \left(-1 \cdot u1 - 1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-1 \cdot u1 - 1\right) \cdot {u1}^{2}}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-1 \cdot u1 - 1\right) \cdot {u1}^{2}}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. lower--.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(-1 \cdot u1 - 1\right)} \cdot {u1}^{2}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right)} - 1\right) \cdot {u1}^{2}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-neg.f32 N/A

         \[\leadsto \sqrt{\frac{\left(\color{blue}{\left(-u1\right)} - 1\right) \cdot {u1}^{2}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{\left(\left(-u1\right) - 1\right) \cdot \color{blue}{\left(u1 \cdot u1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lower-*.f32 92.6

         \[\leadsto \sqrt{\frac{\left(\left(-u1\right) - 1\right) \cdot \color{blue}{\left(u1 \cdot u1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. Applied rewrites 92.6%

       \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(-u1\right) - 1\right) \cdot \left(u1 \cdot u1\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   11. Taylor expanded in u1 around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f32 92.6

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

   13. Applied rewrites 92.6%

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

   if 1.222e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 88.1%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.0001221999991685152:\\ \;\;\;\;\sqrt{\frac{\left(u1 + 1\right) \cdot \left(\left(-u1\right) \cdot u1\right)}{-u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.9% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt u1))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1);
end

Derivation

1. Initial program 57.5%

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

4. Applied rewrites 45.0%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right)}} \]
6. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right)}} \]

   2. lower-log1p.f32 64.8

      \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)}} \]

7. Applied rewrites 64.8%

   \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right)}} \]

8. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1} \]
9. Step-by-step derivation

10. Applied rewrites 64.8%

    \[\leadsto \sqrt{u1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024314 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :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)))) (cos (* (* 2.0 (PI)) u2))))