Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.9% → 91.3%

Time: 9.1s

Alternatives: 8

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: 57.9% 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: 91.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \mathbf{if}\;1 - u1 \leq 0.9998499751091003:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \frac{\left(t\_0 + 1\right) - \left(1 - t\_0\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(-u1\right) \cdot \left(-u1\right)\right)}^{0.25} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* (PI) 2.0)))))
   (if (<= (- 1.0 u1) 0.9998499751091003)
     (* (sqrt (- (log (- 1.0 u1)))) (/ (- (+ t_0 1.0) (- 1.0 t_0)) 2.0))
     (* (pow (* (- u1) (- u1)) 0.25) (cos (* (* 2.0 (PI)) u2))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999849975

   1. Initial program 87.3%

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

      1. lift-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. cos-mult N/A

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

      7. sin-mult N/A

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

      8. sub-div N/A

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

      9. lower-/.f32 N/A

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

   4. Applied rewrites 87.4%

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

   if 0.999849975 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 36.7%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 92.7

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

   5. Applied rewrites 92.7%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. metadata-eval 92.7

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

   7. Applied rewrites 92.7%

      \[\leadsto \color{blue}{{\left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right)\right)}^{0.25}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998499751091003:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \frac{\left(\cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) + 1\right) - \left(1 - \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(-u1\right) \cdot \left(-u1\right)\right)}^{0.25} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.013000000268220901:\\ \;\;\;\;{\left(\left(-u1\right) \cdot \left(-u1\right)\right)}^{0.25} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

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) 0.013000000268220901)
     (* (pow (* (- u1) (- u1)) 0.25) 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))) < 0.0130000003

   1. Initial program 39.4%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 90.8

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

   5. Applied rewrites 90.8%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. metadata-eval 90.8

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

   7. Applied rewrites 90.8%

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

   if 0.0130000003 < (*.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 87.7%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 77.9%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.013000000268220901:\\ \;\;\;\;{\left(\left(-u1\right) \cdot \left(-u1\right)\right)}^{0.25} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.013000000268220901:\\ \;\;\;\;\sqrt{\left(u1 \cdot u1\right) \cdot \frac{-1}{-u1}} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

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) 0.013000000268220901)
     (* (sqrt (* (* u1 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))) < 0.0130000003

   1. Initial program 39.4%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 90.8

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

   5. Applied rewrites 90.8%

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

      1. lift-neg.f32 N/A

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

      2. neg-sub0 N/A

         \[\leadsto \sqrt{\color{blue}{0 - \left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. flip-- N/A

         \[\leadsto \sqrt{\color{blue}{\frac{0 \cdot 0 - \left(-u1\right) \cdot \left(-u1\right)}{0 + \left(-u1\right)}}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. div-inv N/A

         \[\leadsto \sqrt{\color{blue}{\left(0 \cdot 0 - \left(-u1\right) \cdot \left(-u1\right)\right) \cdot \frac{1}{0 + \left(-u1\right)}}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\left(0 \cdot 0 - \left(-u1\right) \cdot \left(-u1\right)\right) \cdot \frac{1}{0 + \left(-u1\right)}}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{\left(\color{blue}{0} - \left(-u1\right) \cdot \left(-u1\right)\right) \cdot \frac{1}{0 + \left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. sub0-neg N/A

         \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(-u1\right) \cdot \left(-u1\right)\right)\right)} \cdot \frac{1}{0 + \left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-neg.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-\left(-u1\right) \cdot \left(-u1\right)\right)} \cdot \frac{1}{0 + \left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. pow2 N/A

         \[\leadsto \sqrt{\left(-\color{blue}{{\left(-u1\right)}^{2}}\right) \cdot \frac{1}{0 + \left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lower-pow.f32 N/A

          \[\leadsto \sqrt{\left(-\color{blue}{{\left(-u1\right)}^{2}}\right) \cdot \frac{1}{0 + \left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. +-lft-identity N/A

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

      12. lower-/.f32 90.8

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

   7. Applied rewrites 90.8%

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

   8. Taylor expanded in u1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f32 90.8

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

   10. Applied rewrites 90.8%

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

   if 0.0130000003 < (*.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 87.7%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 77.9%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.013000000268220901:\\ \;\;\;\;\sqrt{\left(u1 \cdot u1\right) \cdot \frac{-1}{-u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.013000000268220901:\\ \;\;\;\;\sqrt{u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

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) 0.013000000268220901) (* (sqrt 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))) < 0.0130000003

   1. Initial program 39.4%

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

   4. Applied rewrites 52.5%

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

   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 90.8

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

   7. Applied rewrites 90.8%

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

   if 0.0130000003 < (*.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 87.7%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 77.9%

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

Alternative 5: 75.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \leq 0.01080000028014183:\\ \;\;\;\;\sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 (PI)) u2))) 0.01080000028014183)
     (sqrt u1)
     (* 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))) < 0.0108000003

   1. Initial program 38.6%

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

   4. Applied rewrites 52.4%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-log1p.f32 72.5

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

   7. Applied rewrites 72.5%

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

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 72.5%

       \[\leadsto \sqrt{u1} \]

   if 0.0108000003 < (*.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 87.0%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 77.0%

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

Alternative 6: 91.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998499751091003:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \frac{1}{\frac{1}{\cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(-u1\right) \cdot \left(-u1\right)\right)}^{0.25} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9998499751091003)
   (* (sqrt (- (log (- 1.0 u1)))) (/ 1.0 (/ 1.0 (cos (* u2 (* (PI) 2.0))))))
   (* (pow (* (- u1) (- u1)) 0.25) (cos (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999849975

   1. Initial program 87.3%

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

      1. lift-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. flip-- N/A

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

      7. cos-sin-sum N/A

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

      8. lower-/.f32 N/A

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

   4. Applied rewrites 87.2%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lower-/.f32 87.1

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

      5. /-rgt-identity N/A

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

      6. lift--.f32 N/A

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

      7. lift-pow.f32 N/A

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

      8. sqr-pow N/A

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

      9. lift-pow.f32 N/A

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

      10. sqr-pow N/A

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

      11. cos-sin-sum N/A

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

   6. Applied rewrites 87.3%

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

   if 0.999849975 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 36.7%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 92.7

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

   5. Applied rewrites 92.7%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. metadata-eval 92.7

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

   7. Applied rewrites 92.7%

      \[\leadsto \color{blue}{{\left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right)\right)}^{0.25}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998499751091003:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \frac{1}{\frac{1}{\cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(-u1\right) \cdot \left(-u1\right)\right)}^{0.25} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9998499751091003:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(-u1\right) \cdot \left(-u1\right)\right)}^{0.25} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (- 1.0 u1) 0.9998499751091003)
     (* (sqrt (- (log (- 1.0 u1)))) t_0)
     (* (pow (* (- u1) (- u1)) 0.25) t_0))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999849975

   1. Initial program 87.3%

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

   if 0.999849975 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 36.7%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 92.7

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

   5. Applied rewrites 92.7%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. metadata-eval 92.7

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

   7. Applied rewrites 92.7%

      \[\leadsto \color{blue}{{\left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right)\right)}^{0.25}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998499751091003:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(-u1\right) \cdot \left(-u1\right)\right)}^{0.25} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.7% 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 54.9%

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

4. Applied rewrites 45.5%

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

5. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-log1p.f32 66.1

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

7. Applied rewrites 66.1%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 66.1%

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

Reproduce

herbie shell --seed 2024323 
(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))))