Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.6% → 99.1%

Time: 10.1s

Alternatives: 14

Speedup: 4.9×

Specification

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

Math

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

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: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(-\mathsf{log1p}\left(-u1\right)\right)}^{2}\\ {\left(t\_0 \cdot t\_0\right)}^{0.125} \cdot \sin \left(\mathsf{fma}\left(-2 \cdot \mathsf{PI}\left(\right), u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (pow (- (log1p (- u1))) 2.0)))
   (* (pow (* t_0 t_0) 0.125) (sin (fma (* -2.0 (PI)) u2 (/ (PI) 2.0))))))

Derivation

1. Initial program 57.2%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. lower-pow.f32 57.2

      \[\leadsto \color{blue}{{\left(-\log \left(1 - u1\right)\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. lift-log.f32 N/A

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

   5. lift--.f32 N/A

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

   6. *-lft-identity N/A

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

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

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

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

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

   9. *-lft-identity N/A

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

   10. lower-log1p.f32 N/A

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

   11. lower-neg.f32 98.8

       \[\leadsto {\left(-\mathsf{log1p}\left(\color{blue}{-u1}\right)\right)}^{0.5} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{{\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
5. Step-by-step derivation

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. cos-neg-rev N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. associate-*l* N/A

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

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

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

   10. metadata-eval N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. cos-neg-rev N/A

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

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

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

   17. lower-sin.f32 N/A

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

6. Applied rewrites 98.9%

   \[\leadsto {\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.5} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-2 \cdot \mathsf{PI}\left(\right), u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]
7. Step-by-step derivation

   1. lift-pow.f32 N/A

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

   2. metadata-eval N/A

      \[\leadsto {\left(-\mathsf{log1p}\left(-u1\right)\right)}^{\color{blue}{\left(2 \cdot \frac{1}{4}\right)}} \cdot \sin \left(\mathsf{fma}\left(-2 \cdot \mathsf{PI}\left(\right), u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   3. pow-sqr N/A

      \[\leadsto \color{blue}{\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{\frac{1}{4}} \cdot {\left(-\mathsf{log1p}\left(-u1\right)\right)}^{\frac{1}{4}}\right)} \cdot \sin \left(\mathsf{fma}\left(-2 \cdot \mathsf{PI}\left(\right), u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   4. pow-prod-down N/A

      \[\leadsto \color{blue}{{\left(\left(-\mathsf{log1p}\left(-u1\right)\right) \cdot \left(-\mathsf{log1p}\left(-u1\right)\right)\right)}^{\frac{1}{4}}} \cdot \sin \left(\mathsf{fma}\left(-2 \cdot \mathsf{PI}\left(\right), u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto {\left(\left(-\mathsf{log1p}\left(-u1\right)\right) \cdot \left(-\mathsf{log1p}\left(-u1\right)\right)\right)}^{\color{blue}{\left(\frac{1}{8} + \frac{1}{8}\right)}} \cdot \sin \left(\mathsf{fma}\left(-2 \cdot \mathsf{PI}\left(\right), u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto {\left(\left(-\mathsf{log1p}\left(-u1\right)\right) \cdot \left(-\mathsf{log1p}\left(-u1\right)\right)\right)}^{\left(\color{blue}{{\frac{1}{2}}^{3}} + \frac{1}{8}\right)} \cdot \sin \left(\mathsf{fma}\left(-2 \cdot \mathsf{PI}\left(\right), u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]

   7. metadata-eval N/A

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

   8. pow-prod-up N/A

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

   9. pow-prod-down N/A

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

   10. lower-pow.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. pow2 N/A

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

   13. lower-pow.f32 N/A

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

   14. pow2 N/A

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

   15. lower-pow.f32 N/A

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

8. Applied rewrites 99.0%

   \[\leadsto \color{blue}{{\left({\left(-\mathsf{log1p}\left(-u1\right)\right)}^{2} \cdot {\left(-\mathsf{log1p}\left(-u1\right)\right)}^{2}\right)}^{0.125}} \cdot \sin \left(\mathsf{fma}\left(-2 \cdot \mathsf{PI}\left(\right), u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \]
9. Add Preprocessing

Alternative 2: 97.8% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 50.0%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower--.f32 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f32 98.6

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

   5. Applied rewrites 98.6%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

      3. cos-neg-rev N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. associate-*l* N/A

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

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

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

      10. metadata-eval N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f32 N/A

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

      15. cos-neg-rev N/A

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

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

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

      17. lower-sin.f32 N/A

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

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-fma.f32 98.7

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

   10. Applied rewrites 98.7%

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

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

   1. Initial program 97.7%

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

   3. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 93.4

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

   5. Applied rewrites 93.4%

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

Alternative 3: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.20000000298023224:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.20000000298023224)
     (*
      (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
      t_1)
     (* t_0 (fma (* (* u2 u2) -2.0) (* (PI) (PI)) 1.0)))))

Derivation

1. Split input into 2 regimes

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

   1. Initial program 50.0%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 98.6

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

   5. Applied rewrites 98.6%

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

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

   1. Initial program 97.7%

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

   3. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 93.4

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

   5. Applied rewrites 93.4%

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

Alternative 4: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.10999999940395355:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.10999999940395355)
     (* (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1)) t_1)
     (* t_0 (fma (* (* u2 u2) -2.0) (* (PI) (PI)) 1.0)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.109999999

   1. Initial program 49.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 98.2

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

   5. Applied rewrites 98.2%

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

   if 0.109999999 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 97.2%

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

   3. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 93.3

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

   5. Applied rewrites 93.3%

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

Alternative 5: 95.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.06499999761581421:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.06499999761581421)
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) t_1)
     (* t_0 (fma (* (* u2 u2) -2.0) (* (PI) (PI)) 1.0)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0649999976

   1. Initial program 46.4%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 96.9

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

   5. Applied rewrites 96.9%

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

   if 0.0649999976 < (*.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 95.1%

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

   3. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 90.9

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

   5. Applied rewrites 90.9%

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

Alternative 6: 94.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.11999999731779099:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (* t_0 t_1) 0.11999999731779099)
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) t_1)
     (* t_0 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))) < 0.119999997

   1. Initial program 49.4%

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

   3. 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 95.6

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

   5. Applied rewrites 95.6%

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

   if 0.119999997 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 97.3%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 85.3%

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

Alternative 7: 86.9% accurate, 0.6× speedup

Math

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

FPCore

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

   1. Initial program 50.9%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower--.f32 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f32 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 86.1

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

   8. Applied rewrites 86.1%

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

   if 0.25 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 98.3%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 85.8%

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

Alternative 8: 99.2% accurate, 0.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.2%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. lower-pow.f32 57.2

      \[\leadsto \color{blue}{{\left(-\log \left(1 - u1\right)\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. lift-log.f32 N/A

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

   5. lift--.f32 N/A

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

   6. *-lft-identity N/A

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

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

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

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

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

   9. *-lft-identity N/A

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

   10. lower-log1p.f32 N/A

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

   11. lower-neg.f32 98.8

       \[\leadsto {\left(-\mathsf{log1p}\left(\color{blue}{-u1}\right)\right)}^{0.5} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{{\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
5. Step-by-step derivation

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. cos-neg-rev N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. associate-*l* N/A

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

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

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

   10. metadata-eval N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. cos-neg-rev N/A

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

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

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

   17. lower-sin.f32 N/A

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

6. Applied rewrites 98.9%

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

Alternative 9: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.2%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. lower-pow.f32 57.2

      \[\leadsto \color{blue}{{\left(-\log \left(1 - u1\right)\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. lift-log.f32 N/A

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

   5. lift--.f32 N/A

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

   6. *-lft-identity N/A

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

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

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

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

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

   9. *-lft-identity N/A

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

   10. lower-log1p.f32 N/A

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

   11. lower-neg.f32 98.8

       \[\leadsto {\left(-\mathsf{log1p}\left(\color{blue}{-u1}\right)\right)}^{0.5} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{{\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
5. Step-by-step derivation

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. cos-neg-rev N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. associate-*l* N/A

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

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

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

   10. metadata-eval N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. cos-neg-rev N/A

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

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

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

   17. lower-sin.f32 N/A

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

6. Applied rewrites 98.9%

   \[\leadsto {\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.5} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-2 \cdot \mathsf{PI}\left(\right), u2, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]
7. Step-by-step derivation

   1. lift-pow.f32 N/A

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

   2. unpow1/2 N/A

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

   3. lower-sqrt.f32 98.9

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

8. Applied rewrites 98.9%

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

Alternative 10: 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 57.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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 98.7

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

4. Applied rewrites 98.7%

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

Alternative 11: 90.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.008999999612569809:\\ \;\;\;\;\sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.008999999612569809)
   (*
    (sqrt
     (-
      (*
       (- (* (- (* (- (* -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 2 regimes

2. if u2 < 0.00899999961

   1. Initial program 56.6%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower--.f32 N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f32 93.3

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 93.2

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

   8. Applied rewrites 93.2%

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

   if 0.00899999961 < u2

   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-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. lower-pow.f32 59.2

         \[\leadsto \color{blue}{{\left(-\log \left(1 - u1\right)\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. lift-log.f32 N/A

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

      5. lift--.f32 N/A

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

      6. *-lft-identity N/A

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

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

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

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

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

      9. *-lft-identity N/A

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

      10. lower-log1p.f32 N/A

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

      11. lower-neg.f32 97.1

          \[\leadsto {\left(-\mathsf{log1p}\left(\color{blue}{-u1}\right)\right)}^{0.5} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. Applied rewrites 97.1%

      \[\leadsto \color{blue}{{\left(-\mathsf{log1p}\left(-u1\right)\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   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 76.0

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

   7. Applied rewrites 76.0%

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

Alternative 12: 84.1% accurate, 3.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt
   (-
    (* (- (* (- (* (- (* -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 57.2%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. metadata-eval N/A

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

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

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

   11. lower--.f32 N/A

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

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

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

   13. metadata-eval N/A

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

   14. lower-*.f32 93.5

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

5. Applied rewrites 93.5%

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

6. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 82.7

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

8. Applied rewrites 82.7%

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

Alternative 13: 76.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot 1 \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-(((((((-0.25f * u1) - 0.3333333333333333f) * u1) - 0.5f) * u1) - 1.0f) * u1)) * 1.0f;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(-((((((((-0.25e0) * u1) - 0.3333333333333333e0) * u1) - 0.5e0) * u1) - 1.0e0) * u1)) * 1.0e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.25) * u1) - Float32(0.3333333333333333)) * u1) - Float32(0.5)) * u1) - Float32(1.0)) * u1))) * Float32(1.0))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-(((((((single(-0.25) * u1) - single(0.3333333333333333)) * u1) - single(0.5)) * u1) - single(1.0)) * u1)) * single(1.0);
end

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. metadata-eval N/A

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

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

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

   11. lower--.f32 N/A

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

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

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

   13. metadata-eval N/A

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

   14. lower-*.f32 93.5

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

5. Applied rewrites 93.5%

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

6. Taylor expanded in u2 around 0

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

8. Applied rewrites 74.5%

   \[\leadsto \sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \color{blue}{1} \]
9. Add Preprocessing

Alternative 14: 4.9% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \left(-\sqrt{u1}\right) \cdot 1 \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = -sqrt(u1) * 1.0e0
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = -sqrt(u1) * single(1.0);
end

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in u2 around 0

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

5. Applied rewrites 47.9%

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

6. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 4.7

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

8. Applied rewrites 4.7%

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

9. Taylor expanded in u1 around 0

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

    1. *-commutative N/A

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

    2. unpow2 N/A

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

    3. rem-square-sqrt N/A

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

    4. mul-1-neg N/A

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

    5. lower-neg.f32 N/A

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

    6. lower-sqrt.f32 4.7

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

11. Applied rewrites 4.7%

    \[\leadsto \color{blue}{\left(-\sqrt{u1}\right)} \cdot 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024352 
(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))))