Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.2% → 97.0%

Time: 9.0s

Alternatives: 7

Speedup: 8.9×

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

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

FPCore

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

Initial Program: 57.2% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9965000152587891:\\ \;\;\;\;\sin \left(\frac{\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{3}}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\left(\frac{-1}{u1} - 0.5\right) \cdot \left(u1 \cdot u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9965000152587891)
   (*
    (sin (/ (* (* 2.0 u2) (pow (PI) 3.0)) (* (PI) (PI))))
    (sqrt (- (log (- 1.0 u1)))))
   (* (sin (* (* 2.0 (PI)) u2)) (sqrt (- (* (- (/ -1.0 u1) 0.5) (* u1 u1)))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.99650002

   1. Initial program 94.5%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. +-lft-identity N/A

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

      6. flip3-+ N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\color{blue}{\frac{{0}^{3} + {\mathsf{PI}\left(\right)}^{3}}{0 \cdot 0 + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right) - 0 \cdot \mathsf{PI}\left(\right)\right)}} \cdot \left(2 \cdot u2\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\frac{\color{blue}{0} + {\mathsf{PI}\left(\right)}^{3}}{0 \cdot 0 + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right) - 0 \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(2 \cdot u2\right)\right) \]

      8. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\frac{0 + {\color{blue}{\mathsf{PI}\left(\right)}}^{3}}{0 \cdot 0 + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right) - 0 \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(2 \cdot u2\right)\right) \]

      9. add-cbrt-cube N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\frac{0 + {\color{blue}{\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right)}}^{3}}{0 \cdot 0 + \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right) - 0 \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(2 \cdot u2\right)\right) \]

      10. rem-cube-cbrt N/A

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

      11. +-lft-identity N/A

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

      12. metadata-eval N/A

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

      13. +-lft-identity N/A

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

      14. distribute-rgt-out-- N/A

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

      15. lift-PI.f32 N/A

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

      16. lift-PI.f32 N/A

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

      17. --rgt-identity-rev N/A

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

      18. associate-*l/ N/A

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

   4. Applied rewrites 94.7%

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

   if 0.99650002 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 43.9%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 41.5

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

   5. Applied rewrites 40.6%

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

   7. Applied rewrites 97.6%

      \[\leadsto \sqrt{-\left(-1 - 0.5 \cdot u1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. Taylor expanded in u1 around inf

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

   10. Applied rewrites 97.7%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9965000152587891:\\ \;\;\;\;\sin \left(\frac{\left(2 \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{3}}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\left(\frac{-1}{u1} - 0.5\right) \cdot \left(u1 \cdot u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;1 - u1 \leq 0.9965000152587891:\\ \;\;\;\;\left(\left(\sin t\_0 \cdot \cos t\_0\right) \cdot 2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\left(\frac{-1}{u1} - 0.5\right) \cdot \left(u1 \cdot u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (PI))))
   (if (<= (- 1.0 u1) 0.9965000152587891)
     (* (* (* (sin t_0) (cos t_0)) 2.0) (sqrt (- (log (- 1.0 u1)))))
     (*
      (sin (* (* 2.0 (PI)) u2))
      (sqrt (- (* (- (/ -1.0 u1) 0.5) (* u1 u1))))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.99650002

   1. Initial program 94.5%

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

      1. lift-sin.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. sin-2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-sin.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 94.6

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

   4. Applied rewrites 94.6%

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

   if 0.99650002 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 43.9%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 40.0

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

   5. Applied rewrites 41.3%

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

   7. Applied rewrites 97.6%

      \[\leadsto \sqrt{-\left(-1 - 0.5 \cdot u1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. Taylor expanded in u1 around inf

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

   10. Applied rewrites 97.7%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9965000152587891:\\ \;\;\;\;\left(\left(\sin \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\left(\frac{-1}{u1} - 0.5\right) \cdot \left(u1 \cdot u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9965000152587891:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{-\left(\frac{-1}{u1} - 0.5\right) \cdot \left(u1 \cdot u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (* 2.0 (PI)) u2))))
   (if (<= (- 1.0 u1) 0.9965000152587891)
     (* t_0 (sqrt (- (log (- 1.0 u1)))))
     (* t_0 (sqrt (- (* (- (/ -1.0 u1) 0.5) (* u1 u1))))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.99650002

   1. Initial program 94.5%

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

   if 0.99650002 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 43.9%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 39.7

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

   5. Applied rewrites 40.4%

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

   7. Applied rewrites 97.6%

      \[\leadsto \sqrt{-\left(-1 - 0.5 \cdot u1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. Taylor expanded in u1 around inf

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

   10. Applied rewrites 97.7%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9965000152587891:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\left(\frac{-1}{u1} - 0.5\right) \cdot \left(u1 \cdot u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\left(\frac{-1}{u1} - 0.5\right) \cdot \left(u1 \cdot u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (* (* 2.0 (PI)) u2)) (sqrt (- (* (- (/ -1.0 u1) 0.5) (* u1 u1))))))

Derivation

1. Initial program 56.0%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f32 37.9

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

5. Applied rewrites 38.6%

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

7. Applied rewrites 88.4%

   \[\leadsto \sqrt{-\left(-1 - 0.5 \cdot u1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

8. Taylor expanded in u1 around inf

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

10. Applied rewrites 88.5%

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

11. Final simplification 88.5%

    \[\leadsto \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\left(\frac{-1}{u1} - 0.5\right) \cdot \left(u1 \cdot u1\right)} \]
12. Add Preprocessing

Alternative 5: 87.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\left(-1 - 0.5 \cdot u1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (* (- -1.0 (* 0.5 u1)) u1))) (sin (* (* 2.0 (PI)) u2))))

Derivation

1. Initial program 56.0%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f32 38.2

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

5. Applied rewrites 36.8%

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

7. Applied rewrites 88.4%

   \[\leadsto \sqrt{-\left(-1 - 0.5 \cdot u1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
8. Add Preprocessing

Alternative 6: 76.7% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.0%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip3-- N/A

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

   4. log-div N/A

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

   5. lower--.f32 N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. cube-neg N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-log1p.f32 N/A

       \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{log1p}\left({\left(\mathsf{neg}\left(u1\right)\right)}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   12. lower-pow.f32 N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{{\left(\mathsf{neg}\left(u1\right)\right)}^{3}}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   13. lower-neg.f32 N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({\color{blue}{\left(-u1\right)}}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   14. metadata-eval N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({\left(-u1\right)}^{3}\right) - \log \left(\color{blue}{1} + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   15. lower-log1p.f32 N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({\left(-u1\right)}^{3}\right) - \color{blue}{\mathsf{log1p}\left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   16. *-lft-identity N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({\left(-u1\right)}^{3}\right) - \mathsf{log1p}\left(u1 \cdot u1 + \color{blue}{u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   17. lower-fma.f32 25.1

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({\left(-u1\right)}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 25.1%

   \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left({\left(-u1\right)}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sin.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-sqrt.f32 77.7

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

7. Applied rewrites 77.7%

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

8. Final simplification 77.7%

   \[\leadsto \sqrt{u1} \cdot \sin \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \]
9. Add Preprocessing

Alternative 7: 66.8% accurate, 8.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.0%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip3-- N/A

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

   4. log-div N/A

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

   5. lower--.f32 N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. cube-neg N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-log1p.f32 N/A

       \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{log1p}\left({\left(\mathsf{neg}\left(u1\right)\right)}^{3}\right)} - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   12. lower-pow.f32 N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{{\left(\mathsf{neg}\left(u1\right)\right)}^{3}}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   13. lower-neg.f32 N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({\color{blue}{\left(-u1\right)}}^{3}\right) - \log \left(1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   14. metadata-eval N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({\left(-u1\right)}^{3}\right) - \log \left(\color{blue}{1} + \left(u1 \cdot u1 + 1 \cdot u1\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   15. lower-log1p.f32 N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({\left(-u1\right)}^{3}\right) - \color{blue}{\mathsf{log1p}\left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   16. *-lft-identity N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({\left(-u1\right)}^{3}\right) - \mathsf{log1p}\left(u1 \cdot u1 + \color{blue}{u1}\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   17. lower-fma.f32 25.2

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({\left(-u1\right)}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 25.1%

   \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left({\left(-u1\right)}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sin.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-sqrt.f32 77.7

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

7. Applied rewrites 77.7%

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

8. Taylor expanded in u2 around 0

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

10. Applied rewrites 66.0%

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

11. Final simplification 66.0%

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

Reproduce

herbie shell --seed 2024312 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :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)))) (sin (* (* 2.0 (PI)) u2))))