Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.3% → 91.4%

Time: 7.5s

Alternatives: 7

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: 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: 91.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ t_1 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_1 \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot \left({\cos \left({t\_0}^{2} \cdot \left(t\_0 \cdot u2\right)\right)}^{2} - {\sin \left(u2 \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cbrt (PI))) (t_1 (- (log (- 1.0 u1)))))
   (if (<= t_1 0.0001500000071246177)
     (*
      (sqrt (- (- u1)))
      (-
       (pow (cos (* (pow t_0 2.0) (* t_0 u2))) 2.0)
       (pow (sin (* u2 (PI))) 2.0)))
     (* (sqrt t_1) (cos (* (PI) (+ u2 u2)))))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.50000007e-4

   1. Initial program 38.4%

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

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

      5. count-2-rev N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. cos-sum N/A

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

      9. lower--.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. lower-pow.f32 N/A

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

      14. lower-cos.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. lift-*.f32 N/A

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

      18. *-commutative N/A

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

   7. Applied rewrites 92.4%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f32 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f32 N/A

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

      7. pow2 N/A

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

      8. lower-pow.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lower-cbrt.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lower-cbrt.f32 92.6

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

   9. Applied rewrites 92.6%

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

   if 1.50000007e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 90.1%

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

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

      3. associate-*l* N/A

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

      4. count-2-rev N/A

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

      5. lower-fma.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 73.6

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

   4. Applied rewrites 73.6%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 90.1

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

   6. Applied rewrites 90.1%

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

Alternative 2: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0 \leq 0.020999999716877937:\\ \;\;\;\;\sqrt{u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* (sqrt (- (log (- 1.0 u1)))) t_0) 0.020999999716877937)
     (* (sqrt u1) t_0)
     (*
      (sqrt (log (/ 1.0 (- 1.0 u1))))
      (fma (* -2.0 (* u2 u2)) (* (PI) (PI)) 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))) < 0.0209999997

   1. Initial program 43.2%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 40.4

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

   4. Applied rewrites 40.4%

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

   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 89.4

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

   7. Applied rewrites 89.4%

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

   if 0.0209999997 < (*.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 92.8%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 56.5

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

   7. Applied rewrites 56.0%

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

Alternative 3: 86.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0 \leq 0.020999999716877937:\\ \;\;\;\;\sqrt{u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\frac{u1 + 1}{1 - u1 \cdot u1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* (sqrt (- (log (- 1.0 u1)))) t_0) 0.020999999716877937)
     (* (sqrt u1) t_0)
     (sqrt (log (/ (+ u1 1.0) (- 1.0 (* u1 u1))))))))

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))) < 0.0209999997

   1. Initial program 43.2%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 40.4

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

   4. Applied rewrites 40.4%

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

   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 89.4

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

   7. Applied rewrites 89.4%

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

   if 0.0209999997 < (*.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 92.8%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 90.4

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

   4. Applied rewrites 90.4%

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

      1. lift-/.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-neg N/A

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

      4. lift-neg.f32 N/A

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

      5. flip-+ N/A

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

      6. clear-num-rev N/A

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

      7. lift-neg.f32 N/A

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

      8. lift-neg.f32 N/A

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

      9. sqr-neg-rev N/A

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

      10. lower-/.f32 N/A

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

      11. lift-neg.f32 N/A

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

      12. neg-sub0 N/A

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

      13. associate--r- N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. lower-+.f32 N/A

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

      17. metadata-eval N/A

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

      18. lower--.f32 N/A

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

      19. lower-*.f32 89.7

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

   6. Applied rewrites 89.7%

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

   7. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\log \left(\frac{1 + u1}{1 - {u1}^{2}}\right)}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f32 N/A

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

      2. lower-log.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f32 N/A

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

      6. lower--.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 75.4

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

   9. Applied rewrites 75.4%

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

Alternative 4: 91.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))))
   (if (<= t_0 0.0001500000071246177)
     (* (sqrt u1) (cos (* (* 2.0 (PI)) u2)))
     (* (sqrt t_0) (cos (* (PI) (+ u2 u2)))))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.50000007e-4

   1. Initial program 38.4%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 35.8

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

   4. Applied rewrites 35.8%

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

   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 92.6

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

   7. Applied rewrites 92.6%

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

   if 1.50000007e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 90.1%

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

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

      3. associate-*l* N/A

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

      4. count-2-rev N/A

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

      5. lower-fma.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 73.6

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

   4. Applied rewrites 73.6%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. distribute-lft-out N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 90.1

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

   6. Applied rewrites 90.1%

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

Alternative 5: 76.8% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 60.6%

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

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lower-log.f32 N/A

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

   5. lower-/.f32 57.9

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

4. Applied rewrites 57.9%

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

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 75.1

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

7. Applied rewrites 75.1%

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

Alternative 6: 69.7% accurate, 5.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 60.6%

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

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lower-log.f32 N/A

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

   5. lower-/.f32 57.9

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

4. Applied rewrites 57.9%

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

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 75.1

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

7. Applied rewrites 75.1%

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

8. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 65.5

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

10. Applied rewrites 65.2%

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

12. Applied rewrites 69.9%

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

13. Final simplification 69.9%

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

Alternative 7: 64.9% 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 60.6%

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

3. Taylor expanded in 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 75.1

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

5. Applied rewrites 75.1%

   \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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. associate-*l* N/A

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

   5. count-2-rev N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. cos-sum N/A

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

   9. lower--.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. pow2 N/A

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

   13. lower-pow.f32 N/A

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

   14. lower-cos.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

7. Applied rewrites 75.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-cube-cbrt N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. pow2 N/A

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

   8. lower-pow.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-cbrt.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lower-cbrt.f32 75.1

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

9. Applied rewrites 75.1%

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

10. Taylor expanded in u2 around 0

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

12. Applied rewrites 65.5%

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

Reproduce

herbie shell --seed 2024305 
(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))))