Beckmann Sample, near normal, slope_x

Percentage Accurate: 58.0% → 99.1%

Time: 11.3s

Alternatives: 26

Speedup: 21.0×

Specification

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

Math

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

FPCore

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

Initial Program: 58.0% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (log1p u1) (log1p (* (- u1) u1))))
  (sin (* (PI) (fma -2.0 u2 0.5)))))

Derivation

1. Initial program 56.3%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. log-div N/A

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

   5. lower--.f32 N/A

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

   6. metadata-eval N/A

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

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

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

   8. lower-log1p.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-log1p.f32 98.6

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

4. Applied rewrites 98.6%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

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

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

   9. metadata-eval N/A

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

   10. lower-fma.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lower-/.f32 98.7

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

6. Applied rewrites 98.7%

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

7. Taylor expanded in u2 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-log1p.f32 N/A

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

   6. mul-1-neg N/A

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

   7. lower-log1p.f32 N/A

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

   8. unpow2 N/A

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

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

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

   10. lower-*.f32 N/A

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

   11. lower-neg.f32 N/A

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

   12. lower-sin.f32 N/A

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

9. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)} \]
10. Add Preprocessing

Alternative 2: 89.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.0024999999441206455:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right) \cdot \sqrt{u1}\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\frac{u1 - -1}{\mathsf{fma}\left(-u1, u1, 1\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 (PI)) u2)))))
   (if (<= t_0 0.0024999999441206455)
     (* (sin (* (PI) (fma -2.0 u2 0.5))) (sqrt u1))
     (if (<= t_0 0.20000000298023224)
       (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
       (sqrt (log (/ (- u1 -1.0) (fma (- u1) u1 1.0))))))))

Derivation

1. Split input into 3 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.00249999994

   1. Initial program 30.5%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log1p.f32 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-/.f32 98.3

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

   6. Applied rewrites 98.3%

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

   7. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. lower-fma.f32 N/A

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

      9. lower-sqrt.f32 95.0

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

   9. Applied rewrites 95.0%

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

   if 0.00249999994 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.200000003

   1. Initial program 77.5%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log1p.f32 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-log1p.f32 N/A

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

      4. mul-1-neg N/A

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

      5. lower-log1p.f32 N/A

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

      6. unpow2 N/A

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

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

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

      8. lower-*.f32 N/A

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

      9. lower-neg.f32 85.5

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 85.3%

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \]

   if 0.200000003 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 97.6%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log1p.f32 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-log1p.f32 N/A

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

      4. mul-1-neg N/A

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

      5. lower-log1p.f32 N/A

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

      6. unpow2 N/A

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

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

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

      8. lower-*.f32 N/A

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

      9. lower-neg.f32 85.7

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

   7. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 84.8%

      \[\leadsto \sqrt{\log \left(\frac{u1 - -1}{\mathsf{fma}\left(-u1, u1, 1\right)}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.0024999999441206455:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right) \cdot \sqrt{u1}\\ \mathbf{elif}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\frac{1 + u1}{1 - u1 \cdot u1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 (PI)) u2)))))
   (if (<= t_0 0.0024999999441206455)
     (* (sin (* (PI) (fma -2.0 u2 0.5))) (sqrt u1))
     (if (<= t_0 0.20000000298023224)
       (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
       (sqrt (log (/ (+ 1.0 u1) (- 1.0 (* u1 u1)))))))))

Derivation

1. Split input into 3 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.00249999994

   1. Initial program 30.5%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log1p.f32 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-/.f32 98.3

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

   6. Applied rewrites 98.3%

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

   7. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. lower-fma.f32 N/A

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

      9. lower-sqrt.f32 95.0

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

   9. Applied rewrites 95.0%

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

   if 0.00249999994 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.200000003

   1. Initial program 77.5%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log1p.f32 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-log1p.f32 N/A

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

      4. mul-1-neg N/A

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

      5. lower-log1p.f32 N/A

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

      6. unpow2 N/A

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

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

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

      8. lower-*.f32 N/A

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

      9. lower-neg.f32 85.5

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 85.3%

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \]

   if 0.200000003 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 97.6%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 97.2

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

   4. Applied rewrites 97.2%

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

      1. lift-/.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sub-sign N/A

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

      6. lift-neg.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f32 N/A

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

      10. lower-/.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lower-fma.f32 N/A

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

      14. +-commutative N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. lower--.f32 96.9

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

   6. Applied rewrites 96.9%

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

   7. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-log.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-+.f32 N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 84.8

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

   9. Applied rewrites 84.8%

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

Alternative 4: 89.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := t\_0 \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_1 \leq 0.0024999999441206455:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right) \cdot \sqrt{u1}\\ \mathbf{elif}\;t\_1 \leq 0.22499999403953552:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1)))))
        (t_1 (* t_0 (cos (* (* 2.0 (PI)) u2)))))
   (if (<= t_1 0.0024999999441206455)
     (* (sin (* (PI) (fma -2.0 u2 0.5))) (sqrt u1))
     (if (<= t_1 0.22499999403953552)
       (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
       t_0))))

Derivation

1. Split input into 3 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.00249999994

   1. Initial program 30.5%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log1p.f32 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-/.f32 98.3

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

   6. Applied rewrites 98.3%

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

   7. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. lower-fma.f32 N/A

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

      9. lower-sqrt.f32 95.0

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

   9. Applied rewrites 95.0%

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

   if 0.00249999994 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.224999994

   1. Initial program 77.9%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log1p.f32 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-log1p.f32 N/A

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

      4. mul-1-neg N/A

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

      5. lower-log1p.f32 N/A

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

      6. unpow2 N/A

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

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

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

      8. lower-*.f32 N/A

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

      9. lower-neg.f32 85.2

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

   7. Applied rewrites 85.2%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 85.0%

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \]

   if 0.224999994 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 97.7%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. log-rec N/A

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

      3. lower-neg.f32 N/A

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

      4. lower-log.f32 N/A

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

      5. lower--.f32 85.4

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

   7. Applied rewrites 85.4%

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

Alternative 5: 89.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{u1} \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq 0.22499999403953552:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1)))))
        (t_1 (cos (* (* 2.0 (PI)) u2)))
        (t_2 (* t_0 t_1)))
   (if (<= t_2 0.0024999999441206455)
     (* (sqrt u1) t_1)
     (if (<= t_2 0.22499999403953552)
       (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
       t_0))))

Derivation

1. Split input into 3 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.00249999994

   1. Initial program 30.5%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log1p.f32 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 94.6

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

   7. Applied rewrites 94.6%

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

   if 0.00249999994 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.224999994

   1. Initial program 77.9%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log1p.f32 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-log1p.f32 N/A

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

      4. mul-1-neg N/A

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

      5. lower-log1p.f32 N/A

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

      6. unpow2 N/A

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

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

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

      8. lower-*.f32 N/A

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

      9. lower-neg.f32 85.2

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

   7. Applied rewrites 85.2%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 85.0%

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \]

   if 0.224999994 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 97.7%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. log-rec N/A

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

      3. lower-neg.f32 N/A

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

      4. lower-log.f32 N/A

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

      5. lower--.f32 85.4

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

   7. Applied rewrites 85.4%

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

Alternative 6: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.25999999046325684:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<=
      (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 (PI)) u2)))
      0.25999999046325684)
   (*
    (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
    (sin (* (PI) (fma -2.0 u2 0.5))))
   (*
    (sqrt (log (/ 1.0 (- 1.0 u1))))
    (fma (* -2.0 (* u2 u2)) (* (PI) (PI)) 1.0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.25999999

   1. Initial program 50.4%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log1p.f32 98.6

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

   4. Applied rewrites 98.6%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-/.f32 98.7

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

   6. Applied rewrites 98.7%

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

   7. Taylor expanded in u2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-log1p.f32 N/A

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

      6. mul-1-neg N/A

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

      7. lower-log1p.f32 N/A

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

      8. unpow2 N/A

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

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

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

      10. lower-*.f32 N/A

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

      11. lower-neg.f32 N/A

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

      12. lower-sin.f32 N/A

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

   9. Applied rewrites 99.0%

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

   10. Taylor expanded in u1 around 0

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

   12. Applied rewrites 98.1%

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

   if 0.25999999 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 98.1%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 95.8

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

   7. Applied rewrites 95.8%

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

Alternative 7: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.25999999046325684:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 (PI)) u2))) 0.25999999046325684)
     (*
      (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
      (sin (* (PI) (fma -2.0 u2 0.5))))
     (* t_0 (fma (* (* u2 u2) -2.0) (* (PI) (PI)) 1.0)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.25999999

   1. Initial program 50.4%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log1p.f32 98.6

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

   4. Applied rewrites 98.6%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-/.f32 98.7

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

   6. Applied rewrites 98.7%

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

   7. Taylor expanded in u2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-log1p.f32 N/A

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

      6. mul-1-neg N/A

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

      7. lower-log1p.f32 N/A

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

      8. unpow2 N/A

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

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

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

      10. lower-*.f32 N/A

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

      11. lower-neg.f32 N/A

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

      12. lower-sin.f32 N/A

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

   9. Applied rewrites 99.0%

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

   10. Taylor expanded in u1 around 0

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

   12. Applied rewrites 98.1%

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

   if 0.25999999 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 98.1%

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

   3. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 95.7

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

   5. Applied rewrites 95.7%

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

Alternative 8: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.25999999046325684:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.25999999046325684)
     (*
      (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
      t_1)
     (* t_0 (fma (* (* u2 u2) -2.0) (* (PI) (PI)) 1.0)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.25999999

   1. Initial program 50.4%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 97.8

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

   5. Applied rewrites 97.8%

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

   if 0.25999999 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 98.1%

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

   3. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 95.7

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

   5. Applied rewrites 95.7%

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

Alternative 9: 97.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.12999999523162842:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 (PI)) u2))) 0.12999999523162842)
     (*
      (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1))
      (sin (* (PI) (fma -2.0 u2 0.5))))
     (* t_0 (fma (* (* u2 u2) -2.0) (* (PI) (PI)) 1.0)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.129999995

   1. Initial program 47.0%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log1p.f32 98.5

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

   4. Applied rewrites 98.5%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-/.f32 98.6

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in u2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-log1p.f32 N/A

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

      6. mul-1-neg N/A

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

      7. lower-log1p.f32 N/A

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

      8. unpow2 N/A

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

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

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

      10. lower-*.f32 N/A

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

      11. lower-neg.f32 N/A

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

      12. lower-sin.f32 N/A

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

   9. Applied rewrites 99.0%

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

   10. Taylor expanded in u1 around 0

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

   12. Applied rewrites 98.2%

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

   if 0.129999995 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 96.8%

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

   3. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 92.5

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

   5. Applied rewrites 92.5%

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

Alternative 10: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.12999999523162842:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.12999999523162842)
     (* (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1)) t_1)
     (* t_0 (fma (* (* u2 u2) -2.0) (* (PI) (PI)) 1.0)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.129999995

   1. Initial program 47.0%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f32 97.9

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

   5. Applied rewrites 97.9%

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

   if 0.129999995 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 96.8%

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

   3. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 92.5

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

   5. Applied rewrites 92.5%

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

Alternative 11: 95.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.054999999701976776:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 (PI)) u2))) 0.054999999701976776)
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* (PI) (fma -2.0 u2 0.5))))
     (* t_0 (fma (* (* u2 u2) -2.0) (* (PI) (PI)) 1.0)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0549999997

   1. Initial program 43.0%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log1p.f32 98.5

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

   4. Applied rewrites 98.5%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-/.f32 98.6

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in u2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-log1p.f32 N/A

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

      6. mul-1-neg N/A

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

      7. lower-log1p.f32 N/A

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

      8. unpow2 N/A

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

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

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

      10. lower-*.f32 N/A

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

      11. lower-neg.f32 N/A

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

      12. lower-sin.f32 N/A

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

   9. Applied rewrites 99.0%

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

   10. Taylor expanded in u1 around 0

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

   12. Applied rewrites 97.5%

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

   if 0.0549999997 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 94.7%

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

   3. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 90.0

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

   5. Applied rewrites 90.0%

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

Alternative 12: 94.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.08900000154972076:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 (PI)) u2))) 0.08900000154972076)
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* (PI) (fma -2.0 u2 0.5))))
     t_0)))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0890000015

   1. Initial program 46.0%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log1p.f32 98.5

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

   4. Applied rewrites 98.5%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-/.f32 98.6

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in u2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-log1p.f32 N/A

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

      6. mul-1-neg N/A

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

      7. lower-log1p.f32 N/A

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

      8. unpow2 N/A

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

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

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

      10. lower-*.f32 N/A

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

      11. lower-neg.f32 N/A

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

      12. lower-sin.f32 N/A

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

   9. Applied rewrites 99.0%

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

   10. Taylor expanded in u1 around 0

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

   12. Applied rewrites 96.5%

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

   if 0.0890000015 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 96.1%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 95.5

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. log-rec N/A

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

      3. lower-neg.f32 N/A

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

      4. lower-log.f32 N/A

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

      5. lower--.f32 83.0

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

   7. Applied rewrites 83.0%

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

Alternative 13: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.08900000154972076:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.08900000154972076)
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) t_1)
     t_0)))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0890000015

   1. Initial program 46.0%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 96.1

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

   5. Applied rewrites 96.1%

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

   if 0.0890000015 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 96.1%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 95.5

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. log-rec N/A

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

      3. lower-neg.f32 N/A

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

      4. lower-log.f32 N/A

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

      5. lower--.f32 83.0

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

   7. Applied rewrites 83.0%

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

Alternative 14: 80.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.22499999403953552:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 (PI)) u2))) 0.22499999403953552)
     (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
     t_0)))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.224999994

   1. Initial program 49.4%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f32 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. lower-log1p.f32 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-log1p.f32 N/A

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

      4. mul-1-neg N/A

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

      5. lower-log1p.f32 N/A

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

      6. unpow2 N/A

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

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

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

      8. lower-*.f32 N/A

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

      9. lower-neg.f32 76.8

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

   7. Applied rewrites 76.8%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 76.8%

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \]

   if 0.224999994 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 97.7%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 97.5

         \[\leadsto \sqrt{\log \color{blue}{\left(\frac{1}{1 - u1}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 97.5%

      \[\leadsto \sqrt{\color{blue}{\log \left(\frac{1}{1 - u1}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\log \left(\frac{1}{1 - u1}\right)}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(\frac{1}{1 - u1}\right)}} \]

      2. log-rec N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}} \]

      3. lower-neg.f32 N/A

         \[\leadsto \sqrt{\color{blue}{-\log \left(1 - u1\right)}} \]

      4. lower-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \]

      5. lower--.f32 85.4

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \]

   7. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\sqrt{-\log \left(1 - u1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \cos \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p u1) (log1p (* (- u1) u1)))) (cos (* -2.0 (* (PI) u2)))))

Derivation

1. Initial program 56.3%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. lift--.f32 N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. flip-- N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. log-div N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. lower--.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. metadata-eval N/A

      \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. fp-cancel-sub-sign-inv N/A

      \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. lower-log1p.f32 N/A

      \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. lower-neg.f32 N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   11. lower-log1p.f32 98.6

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 98.6%

   \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u2 around inf

   \[\leadsto \color{blue}{\cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   3. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   5. lower-log1p.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)} - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \log \left(1 + \color{blue}{\left(\mathsf{neg}\left({u1}^{2}\right)\right)}\right)} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   7. lower-log1p.f32 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{u1 \cdot u1}\right)\right)} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   9. distribute-lft-neg-in N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   11. lower-neg.f32 N/A

       \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right)} \cdot \cos \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   12. cos-neg-rev N/A

       \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   13. lower-cos.f32 N/A

       \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   14. distribute-lft-neg-in N/A

       \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   15. metadata-eval N/A

       \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \cos \left(\color{blue}{-2} \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

   16. lower-*.f32 N/A

       \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \cos \color{blue}{\left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

7. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \cos \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)} \]
8. Add Preprocessing

Alternative 16: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{log1p}\left(u1\right) - \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot u1, u1, -0.3333333333333333\right), u1 \cdot u1, -0.5\right) \cdot u1\right) \cdot u1 - 1\right) \cdot \left(u1 \cdot u1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt
   (-
    (log1p u1)
    (*
     (-
      (*
       (* (fma (fma (* -0.25 u1) u1 -0.3333333333333333) (* u1 u1) -0.5) u1)
       u1)
      1.0)
     (* u1 u1))))
  (sin (* (PI) (fma -2.0 u2 0.5)))))

Derivation

1. Initial program 56.3%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. lift--.f32 N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. flip-- N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. log-div N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. lower--.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. metadata-eval N/A

      \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. fp-cancel-sub-sign-inv N/A

      \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. lower-log1p.f32 N/A

      \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. lower-neg.f32 N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   11. lower-log1p.f32 98.6

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 98.6%

   \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
5. Step-by-step derivation

   1. lift-cos.f32 N/A

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

   2. cos-neg-rev N/A

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)} \]

   3. sin-+PI/2-rev N/A

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   4. lower-sin.f32 N/A

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   5. lift-*.f32 N/A

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   6. lift-*.f32 N/A

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)} \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   7. associate-*l* N/A

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   8. distribute-lft-neg-in N/A

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   9. metadata-eval N/A

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\color{blue}{-2} \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   10. lower-fma.f32 N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-2, \mathsf{PI}\left(\right) \cdot u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   11. *-commutative N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   12. lower-*.f32 N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   13. lift-PI.f32 N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \]

   14. lower-/.f32 98.7

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

6. Applied rewrites 98.7%

   \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

7. Taylor expanded in u2 around inf

   \[\leadsto \color{blue}{\sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

   3. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

   5. lower-log1p.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)} - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \log \left(1 + \color{blue}{\left(\mathsf{neg}\left({u1}^{2}\right)\right)}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

   7. lower-log1p.f32 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{u1 \cdot u1}\right)\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

   9. distribute-lft-neg-in N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

   11. lower-neg.f32 N/A

       \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

   12. lower-sin.f32 N/A

       \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \color{blue}{\sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

9. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)} \]

10. Taylor expanded in u1 around 0

    \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - {u1}^{2} \cdot \left({u1}^{2} \cdot \left({u1}^{2} \cdot \left(\frac{-1}{4} \cdot {u1}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, \frac{1}{2}\right)\right) \]
11. Step-by-step derivation

12. Applied rewrites 96.5%

    \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot u1, u1, -0.3333333333333333\right), u1 \cdot u1, -0.5\right) \cdot u1\right) \cdot u1 - 1\right) \cdot \left(u1 \cdot u1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right) \]
13. Add Preprocessing

Alternative 17: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)\\ \mathbf{if}\;u1 \leq 0.1599999964237213:\\ \;\;\;\;\sqrt{\mathsf{log1p}\left(u1\right) - \left(\left(\mathsf{fma}\left(-0.3333333333333333, u1 \cdot u1, -0.5\right) \cdot u1\right) \cdot u1 - 1\right) \cdot \left(u1 \cdot u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\frac{u1 - -1}{\mathsf{fma}\left(-u1, u1, 1\right)}\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (PI) (fma -2.0 u2 0.5)))))
   (if (<= u1 0.1599999964237213)
     (*
      (sqrt
       (-
        (log1p u1)
        (*
         (- (* (* (fma -0.3333333333333333 (* u1 u1) -0.5) u1) u1) 1.0)
         (* u1 u1))))
      t_0)
     (* (sqrt (log (/ (- u1 -1.0) (fma (- u1) u1 1.0)))) t_0))))

Derivation

1. Split input into 2 regimes

2. if u1 < 0.159999996

   1. Initial program 51.0%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. flip-- N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. log-div N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-log1p.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lower-neg.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. lower-log1p.f32 98.6

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 98.6%

      \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. lift-cos.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)} \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\color{blue}{-2} \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-2, \mathsf{PI}\left(\right) \cdot u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      13. lift-PI.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \]

      14. lower-/.f32 98.8

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

   6. Applied rewrites 98.8%

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   7. Taylor expanded in u2 around inf

      \[\leadsto \color{blue}{\sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

      3. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      5. lower-log1p.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)} - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \log \left(1 + \color{blue}{\left(\mathsf{neg}\left({u1}^{2}\right)\right)}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      7. lower-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{u1 \cdot u1}\right)\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      11. lower-neg.f32 N/A

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      12. lower-sin.f32 N/A

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \color{blue}{\sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

   9. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)} \]

   10. Taylor expanded in u1 around 0

       \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - {u1}^{2} \cdot \left({u1}^{2} \cdot \left(\frac{-1}{3} \cdot {u1}^{2} - \frac{1}{2}\right) - 1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, \frac{1}{2}\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 99.0%

       \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \left(\left(\mathsf{fma}\left(-0.3333333333333333, u1 \cdot u1, -0.5\right) \cdot u1\right) \cdot u1 - 1\right) \cdot \left(u1 \cdot u1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right) \]

   if 0.159999996 < u1

   1. Initial program 98.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. flip-- N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. log-div N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-log1p.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lower-neg.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. lower-log1p.f32 98.5

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 98.5%

      \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. lift-cos.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)} \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\color{blue}{-2} \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-2, \mathsf{PI}\left(\right) \cdot u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      13. lift-PI.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \]

      14. lower-/.f32 98.4

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

   6. Applied rewrites 98.4%

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   7. Taylor expanded in u2 around inf

      \[\leadsto \color{blue}{\sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

      3. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      5. lower-log1p.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)} - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \log \left(1 + \color{blue}{\left(\mathsf{neg}\left({u1}^{2}\right)\right)}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      7. lower-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{u1 \cdot u1}\right)\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      11. lower-neg.f32 N/A

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      12. lower-sin.f32 N/A

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \color{blue}{\sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

   9. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.5%

       \[\leadsto \sqrt{\log \left(\frac{u1 - -1}{\mathsf{fma}\left(-u1, u1, 1\right)}\right)} \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)\\ \mathbf{if}\;u1 \leq 0.05999999865889549:\\ \;\;\;\;\sqrt{\mathsf{log1p}\left(u1\right) - \left(\left(\left(-0.5 \cdot u1\right) \cdot u1 - 1\right) \cdot u1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\frac{u1 - -1}{\mathsf{fma}\left(-u1, u1, 1\right)}\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (PI) (fma -2.0 u2 0.5)))))
   (if (<= u1 0.05999999865889549)
     (* (sqrt (- (log1p u1) (* (* (- (* (* -0.5 u1) u1) 1.0) u1) u1))) t_0)
     (* (sqrt (log (/ (- u1 -1.0) (fma (- u1) u1 1.0)))) t_0))))

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0599999987

   1. Initial program 49.6%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. flip-- N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. log-div N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-log1p.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lower-neg.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. lower-log1p.f32 98.6

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 98.6%

      \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. lift-cos.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)} \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\color{blue}{-2} \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-2, \mathsf{PI}\left(\right) \cdot u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      13. lift-PI.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \]

      14. lower-/.f32 98.7

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

   6. Applied rewrites 98.7%

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   7. Taylor expanded in u2 around inf

      \[\leadsto \color{blue}{\sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

      3. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      5. lower-log1p.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)} - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \log \left(1 + \color{blue}{\left(\mathsf{neg}\left({u1}^{2}\right)\right)}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      7. lower-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{u1 \cdot u1}\right)\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      11. lower-neg.f32 N/A

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      12. lower-sin.f32 N/A

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \color{blue}{\sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

   9. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)} \]

   10. Taylor expanded in u1 around 0

       \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - {u1}^{2} \cdot \left(\frac{-1}{2} \cdot {u1}^{2} - 1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, \frac{1}{2}\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 99.0%

       \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \left(\left(\left(-0.5 \cdot u1\right) \cdot u1 - 1\right) \cdot u1\right) \cdot u1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right) \]

   if 0.0599999987 < u1

   1. Initial program 97.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. flip-- N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. log-div N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-log1p.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lower-neg.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. lower-log1p.f32 98.7

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 98.7%

      \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. lift-cos.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)} \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\color{blue}{-2} \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-2, \mathsf{PI}\left(\right) \cdot u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      13. lift-PI.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \]

      14. lower-/.f32 98.6

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

   6. Applied rewrites 98.6%

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   7. Taylor expanded in u2 around inf

      \[\leadsto \color{blue}{\sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

      3. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      5. lower-log1p.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)} - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \log \left(1 + \color{blue}{\left(\mathsf{neg}\left({u1}^{2}\right)\right)}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      7. lower-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{u1 \cdot u1}\right)\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      11. lower-neg.f32 N/A

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      12. lower-sin.f32 N/A

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \color{blue}{\sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

   9. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.0%

       \[\leadsto \sqrt{\log \left(\frac{u1 - -1}{\mathsf{fma}\left(-u1, u1, 1\right)}\right)} \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.03999999910593033:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.03999999910593033)
   (*
    (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
    (sin (* (PI) (fma -2.0 u2 0.5))))
   (* (sqrt (- (log (- 1.0 u1)))) (sin (fma -2.0 (* u2 (PI)) (/ (PI) 2.0))))))

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0399999991

   1. Initial program 47.9%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. flip-- N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. log-div N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-log1p.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lower-neg.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. lower-log1p.f32 98.6

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 98.6%

      \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. lift-cos.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)} \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\color{blue}{-2} \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-2, \mathsf{PI}\left(\right) \cdot u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      13. lift-PI.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \]

      14. lower-/.f32 98.7

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

   6. Applied rewrites 98.7%

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   7. Taylor expanded in u2 around inf

      \[\leadsto \color{blue}{\sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

      3. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      5. lower-log1p.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)} - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \log \left(1 + \color{blue}{\left(\mathsf{neg}\left({u1}^{2}\right)\right)}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      7. lower-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{u1 \cdot u1}\right)\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      11. lower-neg.f32 N/A

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      12. lower-sin.f32 N/A

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \color{blue}{\sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

   9. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)} \]

   10. Taylor expanded in u1 around 0

       \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)} \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{fma}\left(-2, u2, \frac{1}{2}\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 99.0%

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right) \]

   if 0.0399999991 < u1

   1. Initial program 97.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)} \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. lower-fma.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{neg}\left(2\right), \mathsf{PI}\left(\right) \cdot u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{fma}\left(\color{blue}{-2}, \mathsf{PI}\left(\right) \cdot u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{fma}\left(-2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{fma}\left(-2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      13. lift-PI.f32 N/A

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \]

      14. lower-/.f32 97.2

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

   4. Applied rewrites 97.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.03999999910593033:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.03999999910593033)
   (*
    (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
    (sin (* (PI) (fma -2.0 u2 0.5))))
   (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0399999991

   1. Initial program 47.9%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. flip-- N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. log-div N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-log1p.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lower-neg.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. lower-log1p.f32 98.6

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 98.6%

      \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. lift-cos.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)} \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\color{blue}{-2} \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-2, \mathsf{PI}\left(\right) \cdot u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      13. lift-PI.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \]

      14. lower-/.f32 98.7

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

   6. Applied rewrites 98.7%

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   7. Taylor expanded in u2 around inf

      \[\leadsto \color{blue}{\sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

      3. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      5. lower-log1p.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)} - \log \left(1 + -1 \cdot {u1}^{2}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \log \left(1 + \color{blue}{\left(\mathsf{neg}\left({u1}^{2}\right)\right)}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      7. lower-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{u1 \cdot u1}\right)\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      11. lower-neg.f32 N/A

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right)} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

      12. lower-sin.f32 N/A

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \color{blue}{\sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]

   9. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)} \]

   10. Taylor expanded in u1 around 0

       \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)} \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{fma}\left(-2, u2, \frac{1}{2}\right)\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 99.0%

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right) \]

   if 0.0399999991 < u1

   1. Initial program 97.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 87.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.00023999999393709004:\\ \;\;\;\;\sqrt{\mathsf{log1p}\left(u1\right) - \left(\left(\left(-0.5 \cdot u1\right) \cdot u1 - 1\right) \cdot u1\right) \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.00023999999393709004)
   (sqrt (- (log1p u1) (* (* (- (* (* -0.5 u1) u1) 1.0) u1) u1)))
   (* (sin (* (PI) (fma -2.0 u2 0.5))) (sqrt u1))))

Derivation

1. Split input into 2 regimes

2. if u2 < 2.39999994e-4

   1. Initial program 58.4%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. flip-- N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. log-div N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-log1p.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lower-neg.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. lower-log1p.f32 99.3

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 99.3%

      \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]

      2. lower--.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]

      3. lower-log1p.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)} - \log \left(1 + -1 \cdot {u1}^{2}\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \log \left(1 + \color{blue}{\left(\mathsf{neg}\left({u1}^{2}\right)\right)}\right)} \]

      5. lower-log1p.f32 N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{u1 \cdot u1}\right)\right)} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \]

      9. lower-neg.f32 98.8

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right)} \]

   7. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}} \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - {u1}^{2} \cdot \left(\frac{-1}{2} \cdot {u1}^{2} - 1\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.3%

       \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \left(\left(\left(-0.5 \cdot u1\right) \cdot u1 - 1\right) \cdot u1\right) \cdot u1} \]

   if 2.39999994e-4 < u2

   1. Initial program 53.0%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-log.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. flip-- N/A

         \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. log-div N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-log1p.f32 N/A

         \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lower-neg.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. lower-log1p.f32 97.5

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 97.5%

      \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. lift-cos.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)} \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\color{blue}{-2} \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-2, \mathsf{PI}\left(\right) \cdot u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

      13. lift-PI.f32 N/A

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) \]

      14. lower-/.f32 97.8

          \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)\right) \]

   6. Applied rewrites 97.8%

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-2, u2 \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   7. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{u1}} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{u1}} \]

      3. lower-sin.f32 N/A

         \[\leadsto \color{blue}{\sin \left(-2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{u1} \]

      4. associate-*r* N/A

         \[\leadsto \sin \left(\color{blue}{\left(-2 \cdot u2\right) \cdot \mathsf{PI}\left(\right)} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{u1} \]

      5. distribute-rgt-out N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(-2 \cdot u2 + \frac{1}{2}\right)\right)} \cdot \sqrt{u1} \]

      6. lower-*.f32 N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(-2 \cdot u2 + \frac{1}{2}\right)\right)} \cdot \sqrt{u1} \]

      7. lower-PI.f32 N/A

         \[\leadsto \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(-2 \cdot u2 + \frac{1}{2}\right)\right) \cdot \sqrt{u1} \]

      8. lower-fma.f32 N/A

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{fma}\left(-2, u2, \frac{1}{2}\right)}\right) \cdot \sqrt{u1} \]

      9. lower-sqrt.f32 77.4

         \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right) \cdot \color{blue}{\sqrt{u1}} \]

   9. Applied rewrites 77.4%

      \[\leadsto \color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right) \cdot \sqrt{u1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 76.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{u1}}\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.25 - \frac{0.0625}{u1}\right) \cdot \sqrt{u1}, 0.5, t\_0 \cdot 0.16666666666666666\right), u1, t\_0 \cdot 0.25\right), u1 \cdot u1, \sqrt{u1}\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 u1))))
   (fma
    (fma
     (fma (* (- 0.25 (/ 0.0625 u1)) (sqrt u1)) 0.5 (* t_0 0.16666666666666666))
     u1
     (* t_0 0.25))
    (* u1 u1)
    (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((1.0f / u1));
	return fmaf(fmaf(fmaf(((0.25f - (0.0625f / u1)) * sqrtf(u1)), 0.5f, (t_0 * 0.16666666666666666f)), u1, (t_0 * 0.25f)), (u1 * u1), sqrtf(u1));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(Float32(1.0) / u1))
	return fma(fma(fma(Float32(Float32(Float32(0.25) - Float32(Float32(0.0625) / u1)) * sqrt(u1)), Float32(0.5), Float32(t_0 * Float32(0.16666666666666666))), u1, Float32(t_0 * Float32(0.25))), Float32(u1 * u1), sqrt(u1))
end

Derivation

1. Initial program 56.3%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. lift--.f32 N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. flip-- N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. log-div N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. lower--.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. metadata-eval N/A

      \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. fp-cancel-sub-sign-inv N/A

      \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. lower-log1p.f32 N/A

      \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. lower-neg.f32 N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   11. lower-log1p.f32 98.6

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 98.6%

   \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]
6. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]

   2. lower--.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]

   3. lower-log1p.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)} - \log \left(1 + -1 \cdot {u1}^{2}\right)} \]

   4. mul-1-neg N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \log \left(1 + \color{blue}{\left(\mathsf{neg}\left({u1}^{2}\right)\right)}\right)} \]

   5. lower-log1p.f32 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \]

   6. unpow2 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{u1 \cdot u1}\right)\right)} \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \]

   8. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \]

   9. lower-neg.f32 78.2

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right)} \]

7. Applied rewrites 78.2%

   \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}} \]

8. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1} + \color{blue}{{u1}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{\frac{1}{u1}} + u1 \cdot \left(\frac{1}{6} \cdot \sqrt{\frac{1}{u1}} + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)} \]
9. Step-by-step derivation

10. Applied rewrites 74.1%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.25 - \frac{0.0625}{u1}\right) \cdot \sqrt{u1}, 0.5, \sqrt{\frac{1}{u1}} \cdot 0.16666666666666666\right), u1, \sqrt{\frac{1}{u1}} \cdot 0.25\right), \color{blue}{u1 \cdot u1}, \sqrt{u1}\right) \]
11. Add Preprocessing

Alternative 23: 77.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1))
end

Derivation

1. Initial program 56.3%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. lift--.f32 N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. flip-- N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. log-div N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. lower--.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. metadata-eval N/A

      \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. fp-cancel-sub-sign-inv N/A

      \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. lower-log1p.f32 N/A

      \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. lower-neg.f32 N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   11. lower-log1p.f32 98.6

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 98.6%

   \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]
6. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]

   2. lower--.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]

   3. lower-log1p.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)} - \log \left(1 + -1 \cdot {u1}^{2}\right)} \]

   4. mul-1-neg N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \log \left(1 + \color{blue}{\left(\mathsf{neg}\left({u1}^{2}\right)\right)}\right)} \]

   5. lower-log1p.f32 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \]

   6. unpow2 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{u1 \cdot u1}\right)\right)} \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \]

   8. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \]

   9. lower-neg.f32 78.2

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right)} \]

7. Applied rewrites 78.2%

   \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}} \]

8. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)} \]
9. Step-by-step derivation

10. Applied rewrites 73.9%

    \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \]
11. Add Preprocessing

Alternative 24: 75.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((fmaf(fmaf(0.3333333333333333f, u1, 0.5f), u1, 1.0f) * u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(fma(fma(Float32(0.3333333333333333), u1, Float32(0.5)), u1, Float32(1.0)) * u1))
end

Derivation

1. Initial program 56.3%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. lift--.f32 N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. flip-- N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. log-div N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. lower--.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. metadata-eval N/A

      \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. fp-cancel-sub-sign-inv N/A

      \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. lower-log1p.f32 N/A

      \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. lower-neg.f32 N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   11. lower-log1p.f32 98.6

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 98.6%

   \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]
6. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]

   2. lower--.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]

   3. lower-log1p.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)} - \log \left(1 + -1 \cdot {u1}^{2}\right)} \]

   4. mul-1-neg N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \log \left(1 + \color{blue}{\left(\mathsf{neg}\left({u1}^{2}\right)\right)}\right)} \]

   5. lower-log1p.f32 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \]

   6. unpow2 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{u1 \cdot u1}\right)\right)} \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \]

   8. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \]

   9. lower-neg.f32 78.2

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right)} \]

7. Applied rewrites 78.2%

   \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}} \]

8. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right)} \]
9. Step-by-step derivation

10. Applied rewrites 72.9%

    \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \]
11. Add Preprocessing

Alternative 25: 73.2% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (* (fma 0.5 u1 1.0) u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((fmaf(0.5f, u1, 1.0f) * u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1))
end

Derivation

1. Initial program 56.3%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. lift--.f32 N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. flip-- N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. log-div N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. lower--.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. metadata-eval N/A

      \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. fp-cancel-sub-sign-inv N/A

      \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. lower-log1p.f32 N/A

      \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. lower-neg.f32 N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   11. lower-log1p.f32 98.6

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 98.6%

   \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]
6. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]

   2. lower--.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]

   3. lower-log1p.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)} - \log \left(1 + -1 \cdot {u1}^{2}\right)} \]

   4. mul-1-neg N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \log \left(1 + \color{blue}{\left(\mathsf{neg}\left({u1}^{2}\right)\right)}\right)} \]

   5. lower-log1p.f32 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \]

   6. unpow2 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{u1 \cdot u1}\right)\right)} \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \]

   8. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \]

   9. lower-neg.f32 78.2

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right)} \]

7. Applied rewrites 78.2%

   \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}} \]

8. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \]
9. Step-by-step derivation

10. Applied rewrites 70.6%

    \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \]
11. Add Preprocessing

Alternative 26: 65.0% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt u1))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1);
end

Derivation

1. Initial program 56.3%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\log \left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. lift--.f32 N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. flip-- N/A

      \[\leadsto \sqrt{-\log \color{blue}{\left(\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. log-div N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. lower--.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(1 \cdot 1 - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. metadata-eval N/A

      \[\leadsto \sqrt{-\left(\log \left(\color{blue}{1} - u1 \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. fp-cancel-sub-sign-inv N/A

      \[\leadsto \sqrt{-\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. lower-log1p.f32 N/A

      \[\leadsto \sqrt{-\left(\color{blue}{\mathsf{log1p}\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)} - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. lower-neg.f32 N/A

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right) - \log \left(1 + u1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   11. lower-log1p.f32 98.6

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \color{blue}{\mathsf{log1p}\left(u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 98.6%

   \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right) - \mathsf{log1p}\left(u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]
6. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]

   2. lower--.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\log \left(1 + u1\right) - \log \left(1 + -1 \cdot {u1}^{2}\right)}} \]

   3. lower-log1p.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)} - \log \left(1 + -1 \cdot {u1}^{2}\right)} \]

   4. mul-1-neg N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \log \left(1 + \color{blue}{\left(\mathsf{neg}\left({u1}^{2}\right)\right)}\right)} \]

   5. lower-log1p.f32 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left({u1}^{2}\right)\right)}} \]

   6. unpow2 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\mathsf{neg}\left(\color{blue}{u1 \cdot u1}\right)\right)} \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \]

   8. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)} \]

   9. lower-neg.f32 78.2

      \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right)} \]

7. Applied rewrites 78.2%

   \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)}} \]

8. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1} \]
9. Step-by-step derivation

10. Applied rewrites 63.1%

    \[\leadsto \sqrt{u1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024360 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 (PI)) u2))))