Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.3% → 98.7%

Time: 9.1s

Alternatives: 10

Speedup: 21.0×

Specification

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

Math

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

FPCore

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

Initial Program: 57.3% 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: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9580000042915344:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1} \cdot \sin \left(\left(\left(-u2\right) \cdot \mathsf{PI}\left(\right) - u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9580000042915344)
   (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 (PI)) u2)))
   (*
    (sqrt
     (-
      (*
       (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
       u1)))
    (sin (+ (- (* (- u2) (PI)) (* u2 (PI))) (/ (PI) 2.0))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.958000004

   1. Initial program 98.4%

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

   if 0.958000004 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 45.2%

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

      1. lift-cos.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. cos-2 N/A

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

      6. lower--.f32 N/A

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

      7. pow2 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-cos.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f32 N/A

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

      14. lower-sin.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 45.2

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

   4. Applied rewrites 45.2%

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

      1. lift--.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. unpow2 N/A

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

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

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

      5. lift-pow.f32 N/A

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

      6. unpow2 N/A

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

      7. lift-cos.f32 N/A

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

      8. cos-neg-rev N/A

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

      9. lift-cos.f32 N/A

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

      10. lift-sin.f32 N/A

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

      11. sin-neg-rev N/A

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

      12. lift-sin.f32 N/A

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

      13. cos-diff N/A

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

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

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

   6. Applied rewrites 45.2%

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

   7. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 98.9

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

   9. Applied rewrites 98.9%

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

Alternative 2: 94.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.08460000157356262:\\ \;\;\;\;\sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<=
      (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 (PI)) u2)))
      0.08460000157356262)
   (* (sqrt (- (* (- (* -0.5 u1) 1.0) u1))) (cos (* (PI) (+ u2 u2))))
   (sqrt (log (/ 1.0 (- 1.0 u1))))))

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))) < 0.0846000016

   1. Initial program 43.2%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. lower--.f32 N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f32 96.9

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

   5. Applied rewrites 96.9%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. count-2-rev N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

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

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

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

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

      10. lift-*.f32 N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lower-neg.f32 N/A

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

      14. lower-neg.f32 81.0

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

   7. Applied rewrites 81.0%

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

      1. lift-fma.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lift-neg.f32 N/A

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

      5. lift-neg.f32 N/A

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

      6. remove-double-neg N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f32 N/A

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

      9. lower-+.f32 96.9

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

   9. Applied rewrites 96.9%

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

   if 0.0846000016 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 96.1%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 95.4

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

   4. Applied rewrites 95.4%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. log-rec N/A

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

      3. lower-neg.f32 N/A

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

      4. lower-log.f32 N/A

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

      5. lower--.f32 77.6

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

   7. Applied rewrites 77.6%

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

   9. Applied rewrites 77.7%

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

Alternative 3: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999994039535522:\\ \;\;\;\;\sqrt{u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 (PI)) u2))))
   (if (<= t_0 0.9999994039535522)
     (* (sqrt u1) t_0)
     (sqrt
      (-
       (*
        (- (* (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0)
        u1))))))

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999999404

   1. Initial program 52.8%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 50.5

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in 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 77.7

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

   7. Applied rewrites 77.7%

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

   if 0.999999404 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 50.7%

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

      1. lift-neg.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. neg-log N/A

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

      4. lower-log.f32 N/A

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

      5. lower-/.f32 48.5

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

   4. Applied rewrites 48.5%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. log-rec N/A

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

      3. lower-neg.f32 N/A

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

      4. lower-log.f32 N/A

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

      5. lower--.f32 50.7

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

   7. Applied rewrites 50.7%

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

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 96.2%

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

Alternative 4: 94.1% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 51.4%

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

   1. lift-cos.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. cos-2 N/A

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

   6. lower--.f32 N/A

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

   7. pow2 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-cos.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. pow2 N/A

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

   13. lower-pow.f32 N/A

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

   14. lower-sin.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 51.4

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

4. Applied rewrites 51.4%

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

   1. lift--.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. unpow2 N/A

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

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

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

   5. lift-pow.f32 N/A

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

   6. unpow2 N/A

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

   7. lift-cos.f32 N/A

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

   8. cos-neg-rev N/A

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

   9. lift-cos.f32 N/A

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

   10. lift-sin.f32 N/A

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

   11. sin-neg-rev N/A

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

   12. lift-sin.f32 N/A

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

   13. cos-diff N/A

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

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

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

6. Applied rewrites 51.4%

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

7. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-*.f32 94.9

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

9. Applied rewrites 94.9%

   \[\leadsto \sqrt{-\color{blue}{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1}} \cdot \sin \left(\left(\left(-u2\right) \cdot \mathsf{PI}\left(\right) - u2 \cdot \mathsf{PI}\left(\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]
10. Add Preprocessing

Alternative 5: 94.1% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 51.4%

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

3. 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 94.9

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

5. Applied rewrites 94.9%

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

Alternative 6: 92.3% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 51.4%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. metadata-eval N/A

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

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

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

   8. *-commutative N/A

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

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

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

   10. lower--.f32 N/A

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

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

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

   12. *-commutative N/A

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

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

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

   14. metadata-eval N/A

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

   15. lower-*.f32 93.6

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

5. Applied rewrites 93.6%

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

Alternative 7: 76.5% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    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))
end function

Julia

function code(cosTheta_i, u1, u2)
	return 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)))
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));
end

Derivation

1. Initial program 51.4%

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

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lower-log.f32 N/A

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

   5. lower-/.f32 49.2

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

4. Applied rewrites 49.2%

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

5. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. log-rec N/A

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

   3. lower-neg.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. lower--.f32 43.9

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

7. Applied rewrites 43.9%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 79.4%

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

Alternative 8: 75.3% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 51.4%

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

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lower-log.f32 N/A

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

   5. lower-/.f32 49.2

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

4. Applied rewrites 49.2%

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

5. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. log-rec N/A

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

   3. lower-neg.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. lower--.f32 43.9

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

7. Applied rewrites 43.9%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 78.6%

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

Alternative 9: 72.8% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 51.4%

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

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lower-log.f32 N/A

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

   5. lower-/.f32 49.2

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

4. Applied rewrites 49.2%

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

5. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. log-rec N/A

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

   3. lower-neg.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. lower--.f32 43.9

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

7. Applied rewrites 43.9%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 76.4%

    \[\leadsto \sqrt{-\left(-0.5 \cdot u1 - 1\right) \cdot u1} \]
11. Add Preprocessing

Alternative 10: 64.9% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt u1))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1);
end

Derivation

1. Initial program 51.4%

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

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lower-log.f32 N/A

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

   5. lower-/.f32 49.2

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

4. Applied rewrites 49.2%

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

5. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. log-rec N/A

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

   3. lower-neg.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. lower--.f32 43.9

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

7. Applied rewrites 43.9%

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

9. Applied rewrites 41.4%

   \[\leadsto \sqrt{-\left(\log \left(1 - {u1}^{4}\right) - \log \left(\left(1 + u1 \cdot u1\right) \cdot \left(u1 + 1\right)\right)\right)} \]

10. Taylor expanded in u1 around 0

    \[\leadsto \sqrt{u1} \]
11. Step-by-step derivation

12. Applied rewrites 69.0%

    \[\leadsto \sqrt{u1} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(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))))