Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.5% → 98.3%

Time: 9.7s

Alternatives: 10

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: 57.5% 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: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 63.1%

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

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

   7. log-prod N/A

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

   8. lower-+.f32 N/A

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

   9. log-rec N/A

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

   10. lower-neg.f32 N/A

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

   11. metadata-eval N/A

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

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

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

   13. lower-log1p.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-neg.f32 N/A

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

   16. lower-log1p.f32 98.4

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

4. Applied rewrites 98.4%

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

Alternative 2: 98.3% 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 63.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing
3. 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 3: 93.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
  (sin (* (* 2.0 (PI)) u2))))

Derivation

1. Initial program 63.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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 94.0

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

5. Applied rewrites 94.0%

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

Alternative 4: 91.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1))
  (sin (* (* 2.0 (PI)) u2))))

Derivation

1. Initial program 63.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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 92.3

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

5. Applied rewrites 92.3%

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

Alternative 5: 88.2% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 63.1%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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 88.8

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

5. Applied rewrites 88.8%

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

Alternative 6: 85.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.00139999995008111:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \sqrt{u1}\right) \cdot u2, \mathsf{PI}\left(\right), \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot \mathsf{fma}\left(0.5, \sqrt{\frac{1}{u1}}, 0.3333333333333333 \cdot \sqrt{u1}\right)\right) \cdot \left(u1 \cdot u1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.00139999995008111)
   (fma
    (* (* 2.0 (sqrt u1)) u2)
    (PI)
    (*
     (*
      (* (PI) u2)
      (fma 0.5 (sqrt (/ 1.0 u1)) (* 0.3333333333333333 (sqrt u1))))
     (* u1 u1)))
   (* (sqrt u1) (sin (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00139999995

   1. Initial program 64.2%

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

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. log-prod N/A

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

      8. lower-+.f32 N/A

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

      9. log-rec N/A

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

      10. lower-neg.f32 N/A

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

      11. metadata-eval N/A

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

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

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

      13. lower-log1p.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-neg.f32 N/A

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

      16. lower-log1p.f32 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. count-2-rev N/A

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

      3. distribute-lft-in N/A

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

      4. count-2-rev N/A

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

      5. lower-*.f32 N/A

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

      6. count-2-rev N/A

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

      7. distribute-lft-in N/A

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

      8. count-2-rev N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lower-PI.f32 N/A

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

      15. lower-sqrt.f32 N/A

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

      16. lower--.f32 N/A

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

      17. lower-log1p.f32 N/A

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

      18. mul-1-neg N/A

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in u1 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 74.3%

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

   11. Taylor expanded in u1 around 0

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

   13. Applied rewrites 90.5%

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

   if 0.00139999995 < u2

   1. Initial program 60.4%

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

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

      5. flip-- N/A

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

      6. associate-/r/ N/A

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

      7. log-prod N/A

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

      8. lower-+.f32 N/A

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

      9. log-rec N/A

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

      10. lower-neg.f32 N/A

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

      11. metadata-eval N/A

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

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

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

      13. lower-log1p.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-neg.f32 N/A

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

      16. lower-log1p.f32 98.1

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

   4. Applied rewrites 98.1%

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

   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 76.0

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

   7. Applied rewrites 76.0%

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

Alternative 7: 76.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(2 \cdot \sqrt{u1}\right) \cdot u2, \mathsf{PI}\left(\right), \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot \mathsf{fma}\left(0.5, \sqrt{\frac{1}{u1}}, 0.3333333333333333 \cdot \sqrt{u1}\right)\right) \cdot \left(u1 \cdot u1\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (fma
  (* (* 2.0 (sqrt u1)) u2)
  (PI)
  (*
   (* (* (PI) u2) (fma 0.5 (sqrt (/ 1.0 u1)) (* 0.3333333333333333 (sqrt u1))))
   (* u1 u1))))

Derivation

1. Initial program 63.1%

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

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

   7. log-prod N/A

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

   8. lower-+.f32 N/A

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

   9. log-rec N/A

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

   10. lower-neg.f32 N/A

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

   11. metadata-eval N/A

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

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

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

   13. lower-log1p.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-neg.f32 N/A

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

   16. lower-log1p.f32 98.4

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. count-2-rev N/A

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

   3. distribute-lft-in N/A

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

   4. count-2-rev N/A

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

   5. lower-*.f32 N/A

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

   6. count-2-rev N/A

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

   7. distribute-lft-in N/A

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

   8. count-2-rev N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 N/A

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

   15. lower-sqrt.f32 N/A

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

   16. lower--.f32 N/A

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

   17. lower-log1p.f32 N/A

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

   18. mul-1-neg N/A

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

7. Applied rewrites 82.0%

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

8. Taylor expanded in u1 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 65.7%

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

11. Taylor expanded in u1 around 0

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

13. Applied rewrites 77.4%

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

Alternative 8: 66.7% accurate, 8.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 63.1%

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

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

   7. log-prod N/A

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

   8. lower-+.f32 N/A

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

   9. log-rec N/A

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

   10. lower-neg.f32 N/A

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

   11. metadata-eval N/A

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

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

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

   13. lower-log1p.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-neg.f32 N/A

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

   16. lower-log1p.f32 98.4

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. count-2-rev N/A

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

   3. distribute-lft-in N/A

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

   4. count-2-rev N/A

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

   5. lower-*.f32 N/A

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

   6. count-2-rev N/A

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

   7. distribute-lft-in N/A

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

   8. count-2-rev N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 N/A

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

   15. lower-sqrt.f32 N/A

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

   16. lower--.f32 N/A

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

   17. lower-log1p.f32 N/A

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

   18. mul-1-neg N/A

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

7. Applied rewrites 82.0%

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

8. Taylor expanded in u1 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 65.7%

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

Alternative 9: 66.7% accurate, 8.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 63.1%

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

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

   7. log-prod N/A

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

   8. lower-+.f32 N/A

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

   9. log-rec N/A

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

   10. lower-neg.f32 N/A

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

   11. metadata-eval N/A

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

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

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

   13. lower-log1p.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-neg.f32 N/A

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

   16. lower-log1p.f32 98.4

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. count-2-rev N/A

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

   3. distribute-lft-in N/A

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

   4. count-2-rev N/A

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

   5. lower-*.f32 N/A

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

   6. count-2-rev N/A

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

   7. distribute-lft-in N/A

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

   8. count-2-rev N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 N/A

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

   15. lower-sqrt.f32 N/A

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

   16. lower--.f32 N/A

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

   17. lower-log1p.f32 N/A

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

   18. mul-1-neg N/A

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

7. Applied rewrites 82.0%

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

8. Taylor expanded in u1 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 65.7%

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

11. Taylor expanded in u1 around 0

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

13. Applied rewrites 65.7%

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

Alternative 10: 66.7% 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 63.1%

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

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

   7. log-prod N/A

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

   8. lower-+.f32 N/A

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

   9. log-rec N/A

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

   10. lower-neg.f32 N/A

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

   11. metadata-eval N/A

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

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

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

   13. lower-log1p.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-neg.f32 N/A

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

   16. lower-log1p.f32 98.4

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. count-2-rev N/A

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

   3. distribute-lft-in N/A

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

   4. count-2-rev N/A

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

   5. lower-*.f32 N/A

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

   6. count-2-rev N/A

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

   7. distribute-lft-in N/A

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

   8. count-2-rev N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 N/A

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

   15. lower-sqrt.f32 N/A

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

   16. lower--.f32 N/A

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

   17. lower-log1p.f32 N/A

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

   18. mul-1-neg N/A

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

7. Applied rewrites 82.0%

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

8. Taylor expanded in u1 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 65.7%

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

12. Applied rewrites 65.7%

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

Reproduce

herbie shell --seed 2025006 
(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))))