Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.6% → 99.2%

Time: 11.1s

Alternatives: 19

Speedup: 8.6×

Specification

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

Math

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

FPCore

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

Initial Program: 57.6% 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: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 59.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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. *-lft-identity N/A

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

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

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

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

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

   6. *-lft-identity N/A

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

   7. lower-log1p.f32 N/A

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

   8. lower-neg.f32 99.2

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

4. Applied rewrites 99.2%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. associate-*l* N/A

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

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

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

   11. lower-fma.f32 N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lift-PI.f32 N/A

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

   16. lower-/.f32 99.2

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

6. Applied rewrites 99.2%

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

7. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt-out N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-fma.f32 99.3

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

9. Applied rewrites 99.3%

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

Alternative 2: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0 \leq 0.1550000011920929:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1, u1, u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* (sqrt (- (log (- 1.0 u1)))) t_0) 0.1550000011920929)
     (*
      (sqrt (fma (* (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1) u1 u1))
      t_0)
     (*
      (sqrt (- (log1p (- 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))) < 0.155000001

   1. Initial program 50.1%

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

   3. Taylor expanded in 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 98.7

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

   5. Applied rewrites 98.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 99.0%

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

   if 0.155000001 < (*.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 96.6%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-lft-identity N/A

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

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

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

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

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 96.5

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

   7. Applied rewrites 96.5%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 59.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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. *-lft-identity N/A

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

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

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

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

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

   6. *-lft-identity N/A

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

   7. lower-log1p.f32 N/A

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

   8. lower-neg.f32 99.2

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

4. Applied rewrites 99.2%

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

Alternative 4: 97.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.009499999694526196:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.009499999694526196)
   (* (sqrt (- (log1p (- u1)))) (fma (* -2.0 (* u2 u2)) (* (PI) (PI)) 1.0))
   (*
    (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
    (cos (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00949999969

   1. Initial program 60.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-lft-identity N/A

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

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

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

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

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 99.4

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

   7. Applied rewrites 99.4%

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

   if 0.00949999969 < u2

   1. Initial program 52.0%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 93.4

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

   5. Applied rewrites 93.4%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.009499999694526196:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.009499999694526196)
   (* (sqrt (- (log1p (- u1)))) (fma (* -2.0 (* u2 u2)) (* (PI) (PI)) 1.0))
   (*
    (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1))
    (sin (* (PI) (fma -2.0 u2 0.5))))))

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00949999969

   1. Initial program 60.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-lft-identity N/A

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

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

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

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

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 99.4

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

   7. Applied rewrites 99.4%

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

   if 0.00949999969 < u2

   1. Initial program 52.0%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-lft-identity N/A

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

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

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

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

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 97.5

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

   4. Applied rewrites 97.5%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. associate-*l* N/A

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

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

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

      11. lower-fma.f32 N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lower-/.f32 97.6

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

   6. Applied rewrites 97.6%

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

   7. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-fma.f32 98.2

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

   9. Applied rewrites 98.2%

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

   10. Taylor expanded in u1 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f32 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f32 92.7

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

   12. Applied rewrites 92.7%

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

Alternative 6: 97.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.009499999694526196:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1, u1, u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.009499999694526196)
   (* (sqrt (- (log1p (- u1)))) (fma (* -2.0 (* u2 u2)) (* (PI) (PI)) 1.0))
   (*
    (sqrt (fma (* (fma 0.3333333333333333 u1 0.5) u1) u1 u1))
    (cos (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00949999969

   1. Initial program 60.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-lft-identity N/A

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

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

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

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

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 99.4

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

   7. Applied rewrites 99.4%

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

   if 0.00949999969 < u2

   1. Initial program 52.0%

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

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

   5. Applied rewrites 92.3%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 92.4%

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

Alternative 7: 97.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.009499999694526196:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.009499999694526196)
   (* (sqrt (- (log1p (- u1)))) (fma (* -2.0 (* u2 u2)) (* (PI) (PI)) 1.0))
   (*
    (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1))
    (cos (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00949999969

   1. Initial program 60.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-lft-identity N/A

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

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

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

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

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 99.4

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

   7. Applied rewrites 99.4%

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

   if 0.00949999969 < u2

   1. Initial program 52.0%

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

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

   5. Applied rewrites 92.3%

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

Alternative 8: 96.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.013799999840557575:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.013799999840557575)
   (* (sqrt (- (log1p (- u1)))) (fma (* -2.0 (* u2 u2)) (* (PI) (PI)) 1.0))
   (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* (PI) (fma -2.0 u2 0.5))))))

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0137999998

   1. Initial program 60.7%

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

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

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

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

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

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

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 99.1

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

   7. Applied rewrites 99.1%

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

   if 0.0137999998 < u2

   1. Initial program 51.4%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-lft-identity N/A

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

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

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

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

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 97.2

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

   4. Applied rewrites 97.2%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. associate-*l* N/A

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

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

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

      11. lower-fma.f32 N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lower-/.f32 97.3

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

   6. Applied rewrites 97.3%

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

   7. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-fma.f32 98.0

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

   9. Applied rewrites 98.0%

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

   10. Taylor expanded in u1 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f32 91.1

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

   12. Applied rewrites 91.1%

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

Alternative 9: 94.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.00024199999461416155:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.00024199999461416155)
   (* (sqrt (- (log1p (- u1)))) 1.0)
   (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* (PI) (fma -2.0 u2 0.5))))))

Derivation

1. Split input into 2 regimes

2. if u2 < 2.41999995e-4

   1. Initial program 58.9%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-lft-identity N/A

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

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

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

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

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in u2 around 0

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

   7. Applied rewrites 98.9%

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

   if 2.41999995e-4 < u2

   1. Initial program 59.7%

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

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

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

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

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

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

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. associate-*l* N/A

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

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

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

      11. lower-fma.f32 N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lower-/.f32 98.4

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

   6. Applied rewrites 98.4%

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

   7. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-fma.f32 98.7

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

   9. Applied rewrites 98.7%

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

   10. Taylor expanded in u1 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f32 87.6

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

   12. Applied rewrites 87.6%

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

Alternative 10: 93.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot 1\\ \mathbf{elif}\;u2 \leq 0.010999999940395355:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, u2, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 9.999999747378752e-5)
   (* (sqrt (- (log1p (- u1)))) 1.0)
   (if (<= u2 0.010999999940395355)
     (*
      (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
      (fma (* -2.0 (* u2 u2)) (* (PI) (PI)) 1.0))
     (* (sqrt (- (- u1))) (sin (* (PI) (fma -2.0 u2 0.5)))))))

Derivation

1. Split input into 3 regimes

2. if u2 < 9.99999975e-5

   1. Initial program 58.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-lft-identity N/A

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

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

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

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

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in u2 around 0

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

   7. Applied rewrites 99.6%

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

   if 9.99999975e-5 < u2 < 0.0109999999

   1. Initial program 66.3%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-lft-identity N/A

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

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

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

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

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 98.4

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 93.5

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

   10. Applied rewrites 93.5%

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

   if 0.0109999999 < u2

   1. Initial program 51.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-lft-identity N/A

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

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

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

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

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 97.4

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

   4. Applied rewrites 97.4%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. associate-*l* N/A

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

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

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

      11. lower-fma.f32 N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lower-/.f32 97.5

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

   6. Applied rewrites 97.5%

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

   7. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-fma.f32 98.1

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

   9. Applied rewrites 98.1%

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

   10. Taylor expanded in u1 around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f32 80.5

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

   12. Applied rewrites 80.5%

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

Alternative 11: 94.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.00024199999461416155:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.00024199999461416155)
   (* (sqrt (- (log1p (- u1)))) 1.0)
   (* (sqrt (* (fma 0.5 u1 1.0) u1)) (cos (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if u2 < 2.41999995e-4

   1. Initial program 58.9%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-lft-identity N/A

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

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

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

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

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in u2 around 0

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

   7. Applied rewrites 98.9%

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

   if 2.41999995e-4 < u2

   1. Initial program 59.7%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 87.5

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

   5. Applied rewrites 87.5%

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

Alternative 12: 93.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot 1\\ \mathbf{elif}\;u2 \leq 0.010999999940395355:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 9.999999747378752e-5)
   (* (sqrt (- (log1p (- u1)))) 1.0)
   (if (<= u2 0.010999999940395355)
     (*
      (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
      (fma (* -2.0 (* u2 u2)) (* (PI) (PI)) 1.0))
     (* (sqrt u1) (cos (* (* 2.0 (PI)) u2))))))

Derivation

1. Split input into 3 regimes

2. if u2 < 9.99999975e-5

   1. Initial program 58.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-lft-identity N/A

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

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

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

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

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in u2 around 0

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

   7. Applied rewrites 99.6%

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

   if 9.99999975e-5 < u2 < 0.0109999999

   1. Initial program 66.3%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-lft-identity N/A

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

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

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

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

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 98.4

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 93.5

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

   10. Applied rewrites 93.5%

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

   if 0.0109999999 < u2

   1. Initial program 51.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 97.3

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in 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 80.1

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

   7. Applied rewrites 80.1%

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

Alternative 13: 86.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 9.999999747378752e-5)
   (* (sqrt (- (log1p (- u1)))) 1.0)
   (*
    (sqrt
     (-
      (*
       (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
       u1)))
    (fma (* -2.0 (* u2 u2)) (* (PI) (PI)) 1.0))))

Derivation

1. Split input into 2 regimes

2. if u2 < 9.99999975e-5

   1. Initial program 58.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-lft-identity N/A

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

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

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

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

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in u2 around 0

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

   7. Applied rewrites 99.6%

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

   if 9.99999975e-5 < u2

   1. Initial program 59.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-lft-identity N/A

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

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

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

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

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f32 N/A

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

      8. lower-neg.f32 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 77.3

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

   7. Applied rewrites 77.3%

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

   8. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 74.1

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

   10. Applied rewrites 74.1%

       \[\leadsto \sqrt{-\color{blue}{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1}} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 83.7% accurate, 3.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 59.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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. *-lft-identity N/A

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

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

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

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

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

   6. *-lft-identity N/A

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

   7. lower-log1p.f32 N/A

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

   8. lower-neg.f32 99.2

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 90.5

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

7. Applied rewrites 90.5%

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

8. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 85.7

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

10. Applied rewrites 85.7%

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

Alternative 15: 82.2% accurate, 4.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in 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 91.4

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

5. Applied rewrites 91.4%

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

6. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 83.5

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

8. Applied rewrites 83.5%

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

Alternative 16: 79.1% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 59.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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. *-lft-identity N/A

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

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

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

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

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

   6. *-lft-identity N/A

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

   7. lower-log1p.f32 N/A

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

   8. lower-neg.f32 99.2

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 90.5

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

7. Applied rewrites 90.5%

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

8. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 79.4

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

10. Applied rewrites 79.4%

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

Alternative 17: 75.1% accurate, 7.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(fmaf((fmaf(0.3333333333333333f, u1, 0.5f) * u1), u1, u1)) * 1.0f;
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(fma(Float32(fma(Float32(0.3333333333333333), u1, Float32(0.5)) * u1), u1, u1)) * Float32(1.0))
end

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in 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 91.4

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

5. Applied rewrites 91.4%

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

6. Taylor expanded in u2 around 0

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

8. Applied rewrites 77.4%

   \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \color{blue}{1} \]
9. Step-by-step derivation

10. Applied rewrites 77.4%

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

Alternative 18: 75.0% accurate, 7.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((fmaf(fmaf(0.3333333333333333f, u1, 0.5f), u1, 1.0f) * u1)) * 1.0f;
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(fma(fma(Float32(0.3333333333333333), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * Float32(1.0))
end

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in 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 91.4

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

5. Applied rewrites 91.4%

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

6. Taylor expanded in u2 around 0

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

8. Applied rewrites 77.4%

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

Alternative 19: 72.5% accurate, 8.6× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(fmaf((0.5f * u1), u1, u1)) * 1.0f;
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(fma(Float32(Float32(0.5) * u1), u1, u1)) * Float32(1.0))
end

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in 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 91.4

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

5. Applied rewrites 91.4%

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

6. Taylor expanded in u2 around 0

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

8. Applied rewrites 77.4%

   \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \color{blue}{1} \]
9. Step-by-step derivation

10. Applied rewrites 77.4%

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

11. Taylor expanded in u1 around 0

    \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2} \cdot u1, u1, u1\right)} \cdot 1 \]
12. Step-by-step derivation

13. Applied rewrites 74.3%

    \[\leadsto \sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)} \cdot 1 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024364 
(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))))