Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.9% → 91.1%

Time: 9.5s

Alternatives: 9

Speedup: 11.6×

Specification

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

Math

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

FPCore

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

Initial Program: 57.9% 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: 91.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ t_1 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ t_2 := u2 \cdot \mathsf{PI}\left(\right)\\ t_3 := \cos t\_2\\ \mathbf{if}\;1 - u1 \leq 0.9998400211334229:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \frac{\left(t\_0 + 1\right) - \left(1 - t\_0\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot \left(t\_3 \cdot t\_3 - \sin \left({t\_1}^{2} \cdot \left(t\_1 \cdot u2\right)\right) \cdot \sin t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* (PI) 2.0))))
        (t_1 (cbrt (PI)))
        (t_2 (* u2 (PI)))
        (t_3 (cos t_2)))
   (if (<= (- 1.0 u1) 0.9998400211334229)
     (* (sqrt (- (log (- 1.0 u1)))) (/ (- (+ t_0 1.0) (- 1.0 t_0)) 2.0))
     (*
      (sqrt (- (- u1)))
      (- (* t_3 t_3) (* (sin (* (pow t_1 2.0) (* t_1 u2))) (sin t_2)))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999840021

   1. Initial program 90.3%

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

      1. lift-cos.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. cos-2 N/A

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

      6. cos-mult N/A

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

      7. sin-mult N/A

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

      8. sub-div N/A

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

      9. lower-/.f32 N/A

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

   4. Applied rewrites 90.4%

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

   if 0.999840021 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 36.8%

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

   3. 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 91.9

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

   5. Applied rewrites 91.9%

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

      1. lift-cos.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. associate-*l* N/A

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

      7. cos-2 N/A

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

      8. lower--.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-cos.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-*.f32 N/A

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

   7. Applied rewrites 91.9%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f32 N/A

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

      4. add-cube-cbrt N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f32 N/A

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

      7. pow2 N/A

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

      8. lower-pow.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lower-cbrt.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lower-cbrt.f32 91.9

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

   9. Applied rewrites 91.9%

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

Alternative 2: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.023499999195337296:\\ \;\;\;\;\sqrt{\frac{\left(\left(\left(-u1\right) - 1\right) \cdot u1\right) \cdot u1}{-u1}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.023499999195337296)
     (* (sqrt (/ (* (* (- (- u1) 1.0) u1) u1) (- u1))) t_1)
     (* t_0 1.0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0234999992

   1. Initial program 40.8%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 89.1

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

   5. Applied rewrites 89.1%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

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

      3. flip-- N/A

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

      4. lower-/.f32 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-+.f32 89.1

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

   7. Applied rewrites 89.1%

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

   8. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f32 89.2

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

   10. Applied rewrites 89.2%

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

   11. Taylor expanded in u1 around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f32 89.2

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

   13. Applied rewrites 89.2%

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

   if 0.0234999992 < (*.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 92.2%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 83.7%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.023499999195337296:\\ \;\;\;\;\sqrt{\frac{\left(\left(\left(-u1\right) - 1\right) \cdot u1\right) \cdot u1}{-u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \mathbf{if}\;1 - u1 \leq 0.9998400211334229:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \frac{\left(t\_0 + 1\right) - \left(1 - t\_0\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.5, u1, 1\right) \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* (PI) 2.0)))))
   (if (<= (- 1.0 u1) 0.9998400211334229)
     (* (sqrt (- (log (- 1.0 u1)))) (/ (- (+ t_0 1.0) (- 1.0 t_0)) 2.0))
     (*
      (sqrt (- (* (fma -0.5 u1 1.0) u1) (log1p (* (- u1) u1))))
      (cos (* (* 2.0 (PI)) u2))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999840021

   1. Initial program 90.3%

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

      1. lift-cos.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. cos-2 N/A

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

      6. cos-mult N/A

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

      7. sin-mult N/A

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

      8. sub-div N/A

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

      9. lower-/.f32 N/A

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

   4. Applied rewrites 90.4%

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

   if 0.999840021 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 36.8%

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

   4. Applied rewrites 52.2%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + \frac{-1}{2} \cdot u1\right)} - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 51.7

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-0.5, u1, 1\right)} \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Applied rewrites 51.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-0.5, u1, 1\right) \cdot u1} - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.023499999195337296:\\ \;\;\;\;\sqrt{u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.023499999195337296) (* (sqrt u1) t_1) (* t_0 1.0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0234999992

   1. Initial program 40.8%

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 89.1

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

   7. Applied rewrites 89.1%

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

   if 0.0234999992 < (*.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 92.2%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 83.7%

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

Alternative 5: 74.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.012500000186264515:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 (PI)) u2))) 0.012500000186264515)
     (* (sqrt (- (- u1))) 1.0)
     (* t_0 1.0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0125000002

   1. Initial program 38.6%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 90.5

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

   5. Applied rewrites 90.5%

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

   6. Taylor expanded in u2 around 0

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

   8. Applied rewrites 75.7%

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

   if 0.0125000002 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 90.6%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 80.8%

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

Alternative 6: 91.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq 0.00015999999595806003:\\ \;\;\;\;\sqrt{\frac{\left(\left(\left(-u1\right) - 1\right) \cdot u1\right) \cdot u1}{-u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))))
   (if (<= t_0 0.00015999999595806003)
     (*
      (sqrt (/ (* (* (- (- u1) 1.0) u1) u1) (- 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))) < 1.59999996e-4

   1. Initial program 37.0%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 91.7

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

   5. Applied rewrites 91.7%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

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

      3. flip-- N/A

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

      4. lower-/.f32 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-+.f32 91.7

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

   7. Applied rewrites 91.7%

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

   8. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f32 91.8

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

   10. Applied rewrites 91.8%

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

   11. Taylor expanded in u1 around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f32 91.8

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

   13. Applied rewrites 91.8%

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

   if 1.59999996e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 90.6%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f32 N/A

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

      6. add-sqr-sqrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-sqrt.f32 90.4

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

   4. Applied rewrites 90.4%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-sqrt.f32 N/A

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

      7. lift-sqrt.f32 N/A

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

      8. rem-square-sqrt N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f32 N/A

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

      11. count-2 N/A

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

      12. lift-*.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-*.f32 N/A

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

      16. lower-+.f32 90.6

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

   6. Applied rewrites 90.6%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.00015999999595806003:\\ \;\;\;\;\sqrt{\frac{\left(\left(\left(-u1\right) - 1\right) \cdot u1\right) \cdot u1}{-u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998400211334229:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.5, u1, 1\right) \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9998400211334229)
   (* (sqrt (- (log (- 1.0 u1)))) (cos (* (PI) (+ u2 u2))))
   (*
    (sqrt (- (* (fma -0.5 u1 1.0) u1) (log1p (* (- u1) u1))))
    (cos (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999840021

   1. Initial program 90.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f32 N/A

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

      6. add-sqr-sqrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-sqrt.f32 90.1

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

   4. Applied rewrites 90.1%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-sqrt.f32 N/A

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

      7. lift-sqrt.f32 N/A

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

      8. rem-square-sqrt N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f32 N/A

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

      11. count-2 N/A

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

      12. lift-*.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-*.f32 N/A

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

      16. lower-+.f32 90.3

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

   6. Applied rewrites 90.3%

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

   if 0.999840021 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 36.8%

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + \frac{-1}{2} \cdot u1\right)} - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 51.5

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-0.5, u1, 1\right)} \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Applied rewrites 50.5%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-0.5, u1, 1\right) \cdot u1} - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 64.5% 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 57.7%

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

3. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 75.7

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

5. Applied rewrites 75.7%

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

6. Taylor expanded in u2 around 0

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

8. Applied rewrites 65.6%

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

Alternative 9: 5.0% 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 57.7%

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

3. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 75.7

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

5. Applied rewrites 75.7%

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

6. Taylor expanded in u2 around 0

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

8. Applied rewrites 65.6%

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

9. Taylor expanded in u1 around 0

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

    1. *-commutative N/A

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

    2. unpow2 N/A

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

    3. rem-square-sqrt N/A

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

    4. lower-*.f32 N/A

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

    5. lower-sqrt.f32 4.5

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

11. Applied rewrites 4.5%

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

13. Applied rewrites 4.5%

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

Reproduce

herbie shell --seed 2024307 
(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))))