Beckmann Sample, near normal, slope_x

Percentage Accurate: 58.0% → 91.4%

Time: 8.0s

Alternatives: 9

Speedup: 11.6×

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: 58.0% 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: 91.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \mathsf{PI}\left(\right)\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ t_3 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_3 \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot \left(0.5 + \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot 0.5 - {\sin \left(\mathsf{PI}\left(\right) \cdot u2\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_3} \cdot \left(\left(t\_1 + t\_2\right) \cdot \left(t\_1 - t\_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (PI)))
        (t_1 (cos t_0))
        (t_2 (sin t_0))
        (t_3 (- (log (- 1.0 u1)))))
   (if (<= t_3 0.00019999999494757503)
     (*
      (sqrt (- (- u1)))
      (+
       0.5
       (- (* (cos (* (* 2.0 (PI)) u2)) 0.5) (pow (sin (* (PI) u2)) 2.0))))
     (* (sqrt t_3) (* (+ t_1 t_2) (- t_1 t_2))))))

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%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 92.5

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

   5. Applied rewrites 92.5%

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

      1. lift-cos.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. associate-*l* N/A

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

      7. cos-2 N/A

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

      8. lower--.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-cos.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-*.f32 N/A

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

   7. Applied rewrites 92.6%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-cos.f32 N/A

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

      4. lift-cos.f32 N/A

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

      5. sqr-cos-a N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f32 N/A

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

      8. lower--.f32 N/A

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

   9. Applied rewrites 92.6%

      \[\leadsto \sqrt{-\left(-u1\right)} \cdot \color{blue}{\left(0.5 + \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot 0.5 - {\sin \left(\mathsf{PI}\left(\right) \cdot u2\right)}^{2}\right)\right)} \]

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

   1. Initial program 90.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. difference-of-squares N/A

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

      7. lower-*.f32 N/A

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

      8. lower-+.f32 N/A

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

      9. lower-cos.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-sin.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lower--.f32 N/A

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

      16. lower-cos.f32 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f32 N/A

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

   4. Applied rewrites 90.4%

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

Alternative 2: 91.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \mathsf{PI}\left(\right)\\ t_1 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_1 \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot \left(0.5 + \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot 0.5 - {\sin \left(\mathsf{PI}\left(\right) \cdot u2\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot \left({\cos t\_0}^{2} - {\sin t\_0}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (PI))) (t_1 (- (log (- 1.0 u1)))))
   (if (<= t_1 0.00019999999494757503)
     (*
      (sqrt (- (- u1)))
      (+
       0.5
       (- (* (cos (* (* 2.0 (PI)) u2)) 0.5) (pow (sin (* (PI) u2)) 2.0))))
     (* (sqrt t_1) (- (pow (cos t_0) 2.0) (pow (sin t_0) 2.0))))))

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%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 92.5

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

   5. Applied rewrites 92.5%

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

      1. lift-cos.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. associate-*l* N/A

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

      7. cos-2 N/A

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

      8. lower--.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-cos.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-*.f32 N/A

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

   7. Applied rewrites 92.6%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-cos.f32 N/A

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

      4. lift-cos.f32 N/A

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

      5. sqr-cos-a N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f32 N/A

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

      8. lower--.f32 N/A

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

   9. Applied rewrites 92.6%

      \[\leadsto \sqrt{-\left(-u1\right)} \cdot \color{blue}{\left(0.5 + \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot 0.5 - {\sin \left(\mathsf{PI}\left(\right) \cdot u2\right)}^{2}\right)\right)} \]

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

   1. Initial program 90.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. lower--.f32 N/A

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

      7. pow2 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-cos.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f32 N/A

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

      14. lower-sin.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 90.3

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

   4. Applied rewrites 90.3%

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

Alternative 3: 91.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ t_1 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_1 \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot \left(0.5 + \left(t\_0 \cdot 0.5 - {\sin \left(\mathsf{PI}\left(\right) \cdot u2\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot t\_0\\ \end{array} \end{array} \]

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 0.00019999999494757503)
     (* (sqrt (- (- u1))) (+ 0.5 (- (* t_0 0.5) (pow (sin (* (PI) u2)) 2.0))))
     (* (sqrt t_1) t_0))))

Derivation

1. Split input into 2 regimes

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

   1. Initial program 36.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 92.5

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

   5. Applied rewrites 92.5%

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

      1. lift-cos.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. associate-*l* N/A

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

      7. cos-2 N/A

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

      8. lower--.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-cos.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-*.f32 N/A

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

   7. Applied rewrites 92.6%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-cos.f32 N/A

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

      4. lift-cos.f32 N/A

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

      5. sqr-cos-a N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f32 N/A

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

      8. lower--.f32 N/A

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

   9. Applied rewrites 92.6%

      \[\leadsto \sqrt{-\left(-u1\right)} \cdot \color{blue}{\left(0.5 + \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot 0.5 - {\sin \left(\mathsf{PI}\left(\right) \cdot u2\right)}^{2}\right)\right)} \]

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

   1. Initial program 90.2%

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

Alternative 4: 86.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;\sqrt{-t\_0} \cdot t\_1 \leq 0.014999999664723873:\\ \;\;\;\;\sqrt{u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{-0.5 \cdot t\_0}\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* (sqrt (- t_0)) t_1) 0.014999999664723873)
     (* (sqrt u1) t_1)
     (* (* (sqrt 2.0) (sqrt (* -0.5 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))) < 0.0149999997

   1. Initial program 39.6%

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

   4. Applied rewrites 53.5%

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

   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 90.4

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

   7. Applied rewrites 90.4%

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

   if 0.0149999997 < (*.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 90.5%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 80.3%

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

      1. lift-sqrt.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. neg-mul-1 N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. lift-log.f32 N/A

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

      7. log-pow N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\log \left({\left(1 - u1\right)}^{\frac{-1}{2}}\right)}} \cdot 1 \]

      8. lift-pow.f32 N/A

         \[\leadsto \sqrt{2 \cdot \log \color{blue}{\left({\left(1 - u1\right)}^{\frac{-1}{2}}\right)}} \cdot 1 \]

      9. lift-log.f32 N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\log \left({\left(1 - u1\right)}^{\frac{-1}{2}}\right)}} \cdot 1 \]

      10. sqrt-prod N/A

          \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{\log \left({\left(1 - u1\right)}^{\frac{-1}{2}}\right)}\right)} \cdot 1 \]

      11. lower-*.f32 N/A

          \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{\log \left({\left(1 - u1\right)}^{\frac{-1}{2}}\right)}\right)} \cdot 1 \]

      12. lower-sqrt.f32 N/A

          \[\leadsto \left(\color{blue}{\sqrt{2}} \cdot \sqrt{\log \left({\left(1 - u1\right)}^{\frac{-1}{2}}\right)}\right) \cdot 1 \]

      13. lower-sqrt.f32 76.9

          \[\leadsto \left(\sqrt{2} \cdot \color{blue}{\sqrt{\log \left({\left(1 - u1\right)}^{-0.5}\right)}}\right) \cdot 1 \]

      14. lift-log.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{\color{blue}{\log \left({\left(1 - u1\right)}^{\frac{-1}{2}}\right)}}\right) \cdot 1 \]

      15. lift-pow.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{\log \color{blue}{\left({\left(1 - u1\right)}^{\frac{-1}{2}}\right)}}\right) \cdot 1 \]

      16. log-pow N/A

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

      17. lift-log.f32 N/A

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

      18. lower-*.f32 80.5

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \log \left(1 - u1\right)}}\right) \cdot 1 \]

      19. lift-log.f32 N/A

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

      20. lift--.f32 N/A

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

      21. sub-neg N/A

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

      22. lift-neg.f32 N/A

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

      23. lower-log1p.f32 49.9

          \[\leadsto \left(\sqrt{2} \cdot \sqrt{-0.5 \cdot \color{blue}{\mathsf{log1p}\left(-u1\right)}}\right) \cdot 1 \]

   7. Applied rewrites 49.9%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{-0.5 \cdot \mathsf{log1p}\left(-u1\right)}\right)} \cdot 1 \]
   8. Step-by-step derivation

      1. lift-log1p.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. sub-neg N/A

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

      4. lower-log.f32 N/A

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

      5. lift--.f32 80.5

         \[\leadsto \left(\sqrt{2} \cdot \sqrt{-0.5 \cdot \log \color{blue}{\left(1 - u1\right)}}\right) \cdot 1 \]

   9. Applied rewrites 80.5%

      \[\leadsto \left(\sqrt{2} \cdot \sqrt{-0.5 \cdot \color{blue}{\log \left(1 - u1\right)}}\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 86.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.014999999664723873:\\ \;\;\;\;\sqrt{u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

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) 0.014999999664723873) (* (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))) < 0.0149999997

   1. Initial program 39.6%

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

   4. Applied rewrites 51.4%

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

   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 90.4

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

   7. Applied rewrites 90.4%

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

   if 0.0149999997 < (*.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 90.5%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 80.3%

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

Alternative 6: 75.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.014000000432133675:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 (PI)) u2))) 0.014000000432133675)
     (* (sqrt (- (- u1))) 1.0)
     (* 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))) < 0.0140000004

   1. Initial program 39.0%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 33.3%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{-\color{blue}{-1 \cdot u1}} \cdot 1 \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}} \cdot 1 \]

      2. lower-neg.f32 73.4

         \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot 1 \]

   8. Applied rewrites 73.4%

      \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot 1 \]

   if 0.0140000004 < (*.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 90.3%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 79.7%

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

Alternative 7: 91.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9998000264167786:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (- 1.0 u1) 0.9998000264167786)
     (* (sqrt (- (log (- 1.0 u1)))) t_0)
     (* (sqrt u1) t_0))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999800026

   1. Initial program 90.2%

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

   if 0.999800026 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 36.7%

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

   4. Applied rewrites 54.9%

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

   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 92.5

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

   7. Applied rewrites 92.5%

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

Alternative 8: 64.8% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\left(-u1\right)} \cdot 1 \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-(-u1)) * 1.0f;
}

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)) * 1.0e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-Float32(-u1))) * Float32(1.0))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-(-u1)) * single(1.0);
end

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in u2 around 0

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

5. Applied rewrites 51.6%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{-\color{blue}{-1 \cdot u1}} \cdot 1 \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}} \cdot 1 \]

   2. lower-neg.f32 64.1

      \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot 1 \]

8. Applied rewrites 64.1%

   \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot 1 \]
9. Add Preprocessing

Alternative 9: -0.0% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \frac{0}{0} \cdot 1 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* (/ 0.0 0.0) 1.0))

C

float code(float cosTheta_i, float u1, float u2) {
	return (0.0f / 0.0f) * 1.0f;
}

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 = (0.0e0 / 0.0e0) * 1.0e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(0.0) / Float32(0.0)) * Float32(1.0))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(0.0) / single(0.0)) * single(1.0);
end

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in u2 around 0

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

5. Applied rewrites 51.6%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. neg-mul-1 N/A

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

   4. metadata-eval N/A

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

   5. associate-*r* N/A

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

   6. lift-log.f32 N/A

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

   7. log-pow N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\log \left({\left(1 - u1\right)}^{\frac{-1}{2}}\right)}} \cdot 1 \]

   8. lift-pow.f32 N/A

      \[\leadsto \sqrt{2 \cdot \log \color{blue}{\left({\left(1 - u1\right)}^{\frac{-1}{2}}\right)}} \cdot 1 \]

   9. lift-log.f32 N/A

      \[\leadsto \sqrt{2 \cdot \color{blue}{\log \left({\left(1 - u1\right)}^{\frac{-1}{2}}\right)}} \cdot 1 \]

   10. sqrt-prod N/A

       \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{\log \left({\left(1 - u1\right)}^{\frac{-1}{2}}\right)}\right)} \cdot 1 \]

   11. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{\log \left({\left(1 - u1\right)}^{\frac{-1}{2}}\right)}\right)} \cdot 1 \]

   12. lower-sqrt.f32 N/A

       \[\leadsto \left(\color{blue}{\sqrt{2}} \cdot \sqrt{\log \left({\left(1 - u1\right)}^{\frac{-1}{2}}\right)}\right) \cdot 1 \]

   13. lower-sqrt.f32 47.3

       \[\leadsto \left(\sqrt{2} \cdot \color{blue}{\sqrt{\log \left({\left(1 - u1\right)}^{-0.5}\right)}}\right) \cdot 1 \]

   14. lift-log.f32 N/A

       \[\leadsto \left(\sqrt{2} \cdot \sqrt{\color{blue}{\log \left({\left(1 - u1\right)}^{\frac{-1}{2}}\right)}}\right) \cdot 1 \]

   15. lift-pow.f32 N/A

       \[\leadsto \left(\sqrt{2} \cdot \sqrt{\log \color{blue}{\left({\left(1 - u1\right)}^{\frac{-1}{2}}\right)}}\right) \cdot 1 \]

   16. log-pow N/A

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

   17. lift-log.f32 N/A

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

   18. lower-*.f32 51.7

       \[\leadsto \left(\sqrt{2} \cdot \sqrt{\color{blue}{-0.5 \cdot \log \left(1 - u1\right)}}\right) \cdot 1 \]

   19. lift-log.f32 N/A

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

   20. lift--.f32 N/A

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

   21. sub-neg N/A

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

   22. lift-neg.f32 N/A

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

   23. lower-log1p.f32 63.8

       \[\leadsto \left(\sqrt{2} \cdot \sqrt{-0.5 \cdot \color{blue}{\mathsf{log1p}\left(-u1\right)}}\right) \cdot 1 \]

7. Applied rewrites 63.8%

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{-0.5 \cdot \mathsf{log1p}\left(-u1\right)}\right)} \cdot 1 \]
8. Step-by-step derivation

9. Applied rewrites -0.0%

   \[\leadsto \color{blue}{\frac{0}{0} \cdot 1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024271 
(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))))