Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.5% → 93.6%

Time: 9.5s

Alternatives: 10

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: 57.5% 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: 93.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.0949999988079071:\\ \;\;\;\;\sqrt{-\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\log \left(\sqrt{\frac{1}{1 - u1}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 (sqrt (- (log (- 1.0 u1))))) 0.0949999988079071)
     (* (sqrt (- (- (* (* -0.5 u1) u1) u1))) t_0)
     (* (sqrt 2.0) (sqrt (log (sqrt (/ 1.0 (- 1.0 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))) < 0.0949999988

   1. Initial program 52.2%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 62.0

         \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(-0.5, u1, -1\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Applied rewrites 61.1%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(-0.5, u1, -1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 96.6%

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

   if 0.0949999988 < (*.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 95.9%

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

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

      5. sqr-pow N/A

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

      6. log-prod N/A

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

      7. lower-+.f32 N/A

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

      8. lower-log.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. metadata-eval N/A

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

      11. lower-log.f32 N/A

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

      12. lower-pow.f32 N/A

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

      13. metadata-eval 93.2

          \[\leadsto \sqrt{\log \left({\left(1 - u1\right)}^{-0.5}\right) + \log \left({\left(1 - u1\right)}^{\color{blue}{-0.5}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 93.2%

      \[\leadsto \sqrt{\color{blue}{\log \left({\left(1 - u1\right)}^{-0.5}\right) + \log \left({\left(1 - u1\right)}^{-0.5}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-sqrt.f32 82.1

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

   7. Applied rewrites 82.1%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.0949999988079071:\\ \;\;\;\;\sqrt{-\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\log \left(\sqrt{\frac{1}{1 - u1}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;1 - u1 \leq 0.9937999844551086:\\ \;\;\;\;\cos \left(\left(\left(u2 \cdot t\_0\right) \cdot 2\right) \cdot {t\_0}^{2}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\frac{\mathsf{fma}\left(\left(u1 \cdot u1\right) \cdot 0.25, u1, -u1\right)}{-0.5 \cdot u1 - -1} \cdot \frac{u1}{u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cbrt (PI))))
   (if (<= (- 1.0 u1) 0.9937999844551086)
     (* (cos (* (* (* u2 t_0) 2.0) (pow t_0 2.0))) (sqrt (- (log (- 1.0 u1)))))
     (*
      (cos (* (* 2.0 (PI)) u2))
      (sqrt
       (-
        (*
         (/ (fma (* (* u1 u1) 0.25) u1 (- u1)) (- (* -0.5 u1) -1.0))
         (/ u1 u1))))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.993799984

   1. Initial program 95.2%

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

      1. lift-*.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. *-commutative N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f32 N/A

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

      10. pow2 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}} \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2\right) \cdot 2\right)\right) \]

      11. lower-pow.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}} \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2\right) \cdot 2\right)\right) \]

      12. lift-PI.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)}^{2} \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2\right) \cdot 2\right)\right) \]

      13. lower-cbrt.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}}^{2} \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2\right) \cdot 2\right)\right) \]

      14. lower-*.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2\right) \cdot 2\right)}\right) \]

      15. lower-*.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2\right)} \cdot 2\right)\right) \]

      16. lift-PI.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}} \cdot u2\right) \cdot 2\right)\right) \]

      17. lower-cbrt.f32 95.4

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left(\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot u2\right) \cdot 2\right)\right) \]

   4. Applied rewrites 95.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \color{blue}{\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2\right) \cdot 2\right)\right)} \]

   if 0.993799984 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 50.7%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 63.5

         \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(-0.5, u1, -1\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(-0.5, u1, -1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 97.6%

      \[\leadsto \sqrt{-\frac{\left(\left(-0.5 \cdot u1\right) \cdot u1\right) \cdot \left(\left(-0.5 \cdot u1\right) \cdot u1\right) - u1 \cdot u1}{\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1 - \left(-u1\right)}}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 72.8%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9937999844551086:\\ \;\;\;\;\cos \left(\left(\left(u2 \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot 2\right) \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\frac{\mathsf{fma}\left(\left(u1 \cdot u1\right) \cdot 0.25, u1, -u1\right)}{-0.5 \cdot u1 - -1} \cdot \frac{u1}{u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.17000000178813934:\\ \;\;\;\;t\_0 \cdot \sqrt{-\frac{\mathsf{fma}\left(\left(u1 \cdot u1\right) \cdot 0.25, u1, -u1\right)}{-0.5 \cdot u1 - -1} \cdot \frac{u1}{u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\log \left(\sqrt{\frac{1}{1 - u1}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 (sqrt (- (log (- 1.0 u1))))) 0.17000000178813934)
     (*
      t_0
      (sqrt
       (-
        (*
         (/ (fma (* (* u1 u1) 0.25) u1 (- u1)) (- (* -0.5 u1) -1.0))
         (/ u1 u1)))))
     (* (sqrt 2.0) (sqrt (log (sqrt (/ 1.0 (- 1.0 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))) < 0.170000002

   1. Initial program 54.6%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 60.4

         \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(-0.5, u1, -1\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Applied rewrites 59.5%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(-0.5, u1, -1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 95.1%

      \[\leadsto \sqrt{-\frac{\left(\left(-0.5 \cdot u1\right) \cdot u1\right) \cdot \left(\left(-0.5 \cdot u1\right) \cdot u1\right) - u1 \cdot u1}{\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1 - \left(-u1\right)}}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 71.0%

      \[\leadsto \sqrt{-\frac{u1}{u1} \cdot \color{blue}{\frac{\mathsf{fma}\left(0.25 \cdot \left(u1 \cdot u1\right), u1, -u1\right)}{u1 \cdot -0.5 - -1}}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   if 0.170000002 < (*.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 97.3%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing
   3. 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. inv-pow N/A

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

      5. sqr-pow N/A

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

      6. log-prod N/A

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

      7. lower-+.f32 N/A

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

      8. lower-log.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. metadata-eval N/A

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

      11. lower-log.f32 N/A

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

      12. lower-pow.f32 N/A

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

      13. metadata-eval 95.1

          \[\leadsto \sqrt{\log \left({\left(1 - u1\right)}^{-0.5}\right) + \log \left({\left(1 - u1\right)}^{\color{blue}{-0.5}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 95.1%

      \[\leadsto \sqrt{\color{blue}{\log \left({\left(1 - u1\right)}^{-0.5}\right) + \log \left({\left(1 - u1\right)}^{-0.5}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-sqrt.f32 84.6

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

   7. Applied rewrites 84.6%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.17000000178813934:\\ \;\;\;\;\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\frac{\mathsf{fma}\left(\left(u1 \cdot u1\right) \cdot 0.25, u1, -u1\right)}{-0.5 \cdot u1 - -1} \cdot \frac{u1}{u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\log \left(\sqrt{\frac{1}{1 - u1}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.17000000178813934:\\ \;\;\;\;\sqrt{-\frac{\mathsf{fma}\left(0.25, u1 \cdot u1, -1\right)}{-0.5 \cdot u1 - -1} \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\log \left(\sqrt{\frac{1}{1 - u1}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 (sqrt (- (log (- 1.0 u1))))) 0.17000000178813934)
     (*
      (sqrt (- (* (/ (fma 0.25 (* u1 u1) -1.0) (- (* -0.5 u1) -1.0)) u1)))
      t_0)
     (* (sqrt 2.0) (sqrt (log (sqrt (/ 1.0 (- 1.0 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))) < 0.170000002

   1. Initial program 54.6%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 59.8

         \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(-0.5, u1, -1\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Applied rewrites 59.8%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(-0.5, u1, -1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 95.0%

      \[\leadsto \sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 70.2%

      \[\leadsto \sqrt{-\frac{\mathsf{fma}\left(0.25, u1 \cdot u1, -1\right)}{u1 \cdot -0.5 - -1} \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   if 0.170000002 < (*.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 97.3%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing
   3. 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. inv-pow N/A

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

      5. sqr-pow N/A

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

      6. log-prod N/A

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

      7. lower-+.f32 N/A

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

      8. lower-log.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. metadata-eval N/A

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

      11. lower-log.f32 N/A

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

      12. lower-pow.f32 N/A

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

      13. metadata-eval 95.1

          \[\leadsto \sqrt{\log \left({\left(1 - u1\right)}^{-0.5}\right) + \log \left({\left(1 - u1\right)}^{\color{blue}{-0.5}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 95.1%

      \[\leadsto \sqrt{\color{blue}{\log \left({\left(1 - u1\right)}^{-0.5}\right) + \log \left({\left(1 - u1\right)}^{-0.5}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-sqrt.f32 84.6

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

   7. Applied rewrites 84.6%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.17000000178813934:\\ \;\;\;\;\sqrt{-\frac{\mathsf{fma}\left(0.25, u1 \cdot u1, -1\right)}{-0.5 \cdot u1 - -1} \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\log \left(\sqrt{\frac{1}{1 - u1}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9937999844551086:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{-\frac{\mathsf{fma}\left(\left(u1 \cdot u1\right) \cdot 0.25, u1, -u1\right)}{-0.5 \cdot u1 - -1} \cdot \frac{u1}{u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (- 1.0 u1) 0.9937999844551086)
     (* t_0 (sqrt (- (log (- 1.0 u1)))))
     (*
      t_0
      (sqrt
       (-
        (*
         (/ (fma (* (* u1 u1) 0.25) u1 (- u1)) (- (* -0.5 u1) -1.0))
         (/ u1 u1))))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.993799984

   1. Initial program 95.2%

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

   if 0.993799984 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 50.7%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 62.2

         \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(-0.5, u1, -1\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Applied rewrites 62.0%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(-0.5, u1, -1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 97.6%

      \[\leadsto \sqrt{-\frac{\left(\left(-0.5 \cdot u1\right) \cdot u1\right) \cdot \left(\left(-0.5 \cdot u1\right) \cdot u1\right) - u1 \cdot u1}{\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1 - \left(-u1\right)}}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 74.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9937999844551086:\\ \;\;\;\;\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\frac{\mathsf{fma}\left(\left(u1 \cdot u1\right) \cdot 0.25, u1, -u1\right)}{-0.5 \cdot u1 - -1} \cdot \frac{u1}{u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (- (* (* -0.5 u1) u1) u1))) (cos (* (* 2.0 (PI)) u2))))

Derivation

1. Initial program 62.3%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f32 56.5

      \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(-0.5, u1, -1\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Applied rewrites 56.2%

   \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(-0.5, u1, -1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
6. Step-by-step derivation

7. Applied rewrites 87.1%

   \[\leadsto \sqrt{-\left(\left(-0.5 \cdot u1\right) \cdot u1 - \color{blue}{u1}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
8. Add Preprocessing

Alternative 7: 88.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \cos \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (cos (* (+ u2 u2) (PI))) (sqrt (- (* (- (* -0.5 u1) 1.0) u1)))))

Derivation

1. Initial program 62.3%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f32 56.7

      \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(-0.5, u1, -1\right)} \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Applied rewrites 57.0%

   \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(-0.5, u1, -1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
6. Step-by-step derivation

7. Applied rewrites 87.1%

   \[\leadsto \sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lift-PI.f32 N/A

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

   6. count-2 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-*.f32 N/A

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

   11. lower-+.f32 87.1

       \[\leadsto \sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(u2 + u2\right)}\right) \]

9. Applied rewrites 87.1%

   \[\leadsto \sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

10. Final simplification 87.1%

    \[\leadsto \cos \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \]
11. Add Preprocessing

Alternative 8: 76.6% 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 62.3%

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

4. Applied rewrites 42.4%

   \[\leadsto \color{blue}{e^{\log \left(\mathsf{log1p}\left(u1\right)\right) \cdot 0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

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 74.2

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

7. Applied rewrites 74.2%

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

Alternative 9: 65.2% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 1.0e0 * sqrt(-(-u1))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 62.3%

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

3. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 74.2

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

5. Applied rewrites 74.2%

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

6. Taylor expanded in u2 around 0

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

8. Applied rewrites 64.0%

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

9. Final simplification 64.0%

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

Alternative 10: 4.8% accurate, 12.8× speedup

Math

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

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(Float32(-sqrt(u1)) * Float32(1.0))
end

MATLAB

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

Derivation

1. Initial program 62.3%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 N/A

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

   6. lower-sqrt.f32 3.4

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

5. Applied rewrites 3.4%

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

6. Taylor expanded in u2 around 0

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

8. Applied rewrites 4.4%

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

Reproduce

herbie shell --seed 2024268 
(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))))