Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.7% → 98.4%

Time: 10.2s

Alternatives: 9

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: 57.7% 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: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{2}\right)}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (pow (pow (- (log1p (- u1))) 2.0) 0.25) (sin (* (* 2.0 (PI)) u2))))

Derivation

1. Initial program 57.5%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f32 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f32 N/A

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

   8. lift-log.f32 N/A

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

   9. lift--.f32 N/A

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

   10. *-lft-identity N/A

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

   11. fp-cancel-sub-sign-inv N/A

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

   12. distribute-lft-neg-in N/A

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

   13. *-lft-identity N/A

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

   14. lower-log1p.f32 N/A

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

   15. lower-neg.f32 N/A

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

   16. metadata-eval 98.5

       \[\leadsto {\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{2}\right)}^{\color{blue}{0.25}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 98.5%

   \[\leadsto \color{blue}{{\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{2}\right)}^{0.25}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
5. Add Preprocessing

Alternative 2: 95.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.07000000029802322:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (log (- 1.0 u1)) -0.07000000029802322)
   (* (* (* (PI) u2) 2.0) (sqrt (- (log1p (- u1)))))
   (*
    (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1))
    (sin (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0700000003

   1. Initial program 97.7%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 97.4

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

   4. Applied rewrites 97.4%

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

   5. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. log-rec N/A

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

      10. lower-neg.f32 N/A

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

      11. *-lft-identity N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. metadata-eval N/A

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

      14. mul-1-neg N/A

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

      15. lower-log1p.f32 N/A

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

      16. lower-neg.f32 85.9

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

   7. Applied rewrites 85.9%

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

   if -0.0700000003 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 51.1%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f32 97.5

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Applied rewrites 97.5%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (sin (* (* 2.0 (PI)) u2))))

Derivation

1. Initial program 57.5%

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

   1. lift-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. *-lft-identity N/A

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

   4. fp-cancel-sub-sign-inv N/A

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

   5. distribute-lft-neg-in N/A

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

   6. *-lft-identity N/A

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

   7. lower-log1p.f32 N/A

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

   8. lower-neg.f32 98.4

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

4. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
5. Add Preprocessing

Alternative 4: 96.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.06800000369548798:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.06800000369548798)
   (*
    (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
    (sin (* (* 2.0 (PI)) u2)))
   (* (* (* (PI) u2) 2.0) (sqrt (- (log1p (- u1)))))))

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0680000037

   1. Initial program 51.1%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right)} \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1} + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right)} \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1} + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u1 + \frac{1}{3}}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lower-fma.f32 98.1

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right)}, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Applied rewrites 98.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   if 0.0680000037 < u1

   1. Initial program 97.7%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 97.4

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

   4. Applied rewrites 97.4%

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

   5. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. log-rec N/A

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

      10. lower-neg.f32 N/A

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

      11. *-lft-identity N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. metadata-eval N/A

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

      14. mul-1-neg N/A

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

      15. lower-log1p.f32 N/A

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

      16. lower-neg.f32 85.9

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

   7. Applied rewrites 85.9%

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

Alternative 5: 94.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0006000000284984708:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0006000000284984708)
   (* (* (* (PI) u2) 2.0) (sqrt (- (log1p (- u1)))))
   (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if u2 < 6.00000028e-4

   1. Initial program 55.5%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 53.3

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

   4. Applied rewrites 53.3%

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

   5. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. log-rec N/A

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

      10. lower-neg.f32 N/A

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

      11. *-lft-identity N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. metadata-eval N/A

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

      14. mul-1-neg N/A

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

      15. lower-log1p.f32 N/A

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

      16. lower-neg.f32 97.4

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

   7. Applied rewrites 97.4%

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

   if 6.00000028e-4 < u2

   1. Initial program 60.8%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 88.6

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

   5. Applied rewrites 88.6%

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

Alternative 6: 90.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0017000000225380063:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0017000000225380063)
   (* (* (* (PI) u2) 2.0) (sqrt (- (log1p (- u1)))))
   (* (sqrt u1) (sin (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00170000002

   1. Initial program 55.8%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 53.5

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

   4. Applied rewrites 53.5%

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

   5. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. log-rec N/A

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

      10. lower-neg.f32 N/A

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

      11. *-lft-identity N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. metadata-eval N/A

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

      14. mul-1-neg N/A

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

      15. lower-log1p.f32 N/A

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

      16. lower-neg.f32 95.9

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

   7. Applied rewrites 95.9%

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

   if 0.00170000002 < u2

   1. Initial program 61.3%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 58.2

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

   4. Applied rewrites 58.2%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 74.8

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

   7. Applied rewrites 74.8%

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

Alternative 7: 76.8% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.5%

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

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lower-log.f32 N/A

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

   5. lower-/.f32 55.0

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

4. Applied rewrites 55.0%

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

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 76.9

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

7. Applied rewrites 76.9%

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

Alternative 8: 66.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {\left(u1 \cdot u1\right)}^{0.25} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (pow (* u1 u1) 0.25) (* (* (PI) u2) 2.0)))

Derivation

1. Initial program 57.5%

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

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lower-log.f32 N/A

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

   5. lower-/.f32 55.0

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

4. Applied rewrites 55.0%

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

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 76.9

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

7. Applied rewrites 76.9%

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

8. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 66.4

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

10. Applied rewrites 66.4%

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

12. Applied rewrites 66.4%

    \[\leadsto {\left(u1 \cdot u1\right)}^{\color{blue}{0.25}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \]
13. Add Preprocessing

Alternative 9: 66.3% accurate, 8.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.5%

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

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lower-log.f32 N/A

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

   5. lower-/.f32 55.0

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

4. Applied rewrites 55.0%

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

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 76.9

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

7. Applied rewrites 76.9%

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

8. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 66.4

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

10. Applied rewrites 66.4%

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

Reproduce

herbie shell --seed 2024352 
(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))))