Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.2% → 91.3%

Time: 10.2s

Alternatives: 11

Speedup: 7.7×

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

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

FPCore

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

Initial Program: 58.2% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \mathbf{if}\;u1 \leq 0.00019999999494757503:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (* (PI) 2.0) u2))))
   (if (<= u1 0.00019999999494757503)
     (* (pow (* u1 u1) 0.25) t_0)
     (* t_0 (sqrt (- (log (- 1.0 u1))))))))

Derivation

1. Split input into 2 regimes

2. if u1 < 1.99999995e-4

   1. Initial program 37.0%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 92.6

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

   5. Applied rewrites 92.6%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-*.f32 92.6

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in u1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f32 92.6

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

   10. Applied rewrites 92.6%

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

   if 1.99999995e-4 < u1

   1. Initial program 90.0%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 0.00019999999494757503:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9959999918937683:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{elif}\;1 - u1 \leq 0.9999998211860657:\\ \;\;\;\;\sqrt{\frac{1 - 0.25 \cdot \left(u1 \cdot u1\right)}{1 - -0.5 \cdot u1} \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{1}{\mathsf{log1p}\left(-u1\right)}}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (* (PI) 2.0) u2))))
   (if (<= (- 1.0 u1) 0.9959999918937683)
     (* t_0 (sqrt (- (log (- 1.0 u1)))))
     (if (<= (- 1.0 u1) 0.9999998211860657)
       (*
        (sqrt
         (-
          (* (/ (- 1.0 (* 0.25 (* u1 u1))) (- 1.0 (* -0.5 u1))) u1)
          (log1p (* (- u1) u1))))
        t_0)
       (* (sqrt (/ -1.0 (/ 1.0 (log1p (- u1))))) t_0)))))

Derivation

1. Split input into 3 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.995999992

   1. Initial program 95.0%

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

   if 0.995999992 < (-.f32 #s(literal 1 binary32) u1) < 0.999999821

   1. Initial program 65.5%

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

   4. Applied rewrites 34.3%

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

   5. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 19.4

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-0.5, u1, 1\right)} \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Applied rewrites 13.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-0.5, u1, 1\right) \cdot u1} - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 58.5%

      \[\leadsto \sqrt{\frac{1 - \left(u1 \cdot u1\right) \cdot 0.25}{1 - -0.5 \cdot u1} \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   if 0.999999821 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 15.3%

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

   4. Applied rewrites 44.8%

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

      1. lift--.f32 N/A

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

      2. lift-log1p.f32 N/A

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

      3. lift-log1p.f32 N/A

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

      4. diff-log N/A

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

      5. lift-*.f32 N/A

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

      6. lift-neg.f32 N/A

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

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

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

      8. metadata-eval N/A

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

      9. flip-+ N/A

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

      10. lift--.f32 N/A

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

      11. associate-/l/ N/A

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

      12. log-div N/A

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

      13. metadata-eval N/A

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

      14. cancel-sign-sub-inv N/A

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

      15. lift-neg.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. lift-log1p.f32 N/A

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

   6. Applied rewrites 11.0%

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

   8. Applied rewrites 48.4%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9959999918937683:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{elif}\;1 - u1 \leq 0.9999998211860657:\\ \;\;\;\;\sqrt{\frac{1 - 0.25 \cdot \left(u1 \cdot u1\right)}{1 - -0.5 \cdot u1} \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{1}{\mathsf{log1p}\left(-u1\right)}}} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9959999918937683:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{elif}\;1 - u1 \leq 0.9999998211860657:\\ \;\;\;\;\sqrt{\left(\left(\frac{1}{u1} - 0.5\right) \cdot u1\right) \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{1}{\mathsf{log1p}\left(-u1\right)}}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (* (PI) 2.0) u2))))
   (if (<= (- 1.0 u1) 0.9959999918937683)
     (* t_0 (sqrt (- (log (- 1.0 u1)))))
     (if (<= (- 1.0 u1) 0.9999998211860657)
       (*
        (sqrt (- (* (* (- (/ 1.0 u1) 0.5) u1) u1) (log1p (* (- u1) u1))))
        t_0)
       (* (sqrt (/ -1.0 (/ 1.0 (log1p (- u1))))) t_0)))))

Derivation

1. Split input into 3 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.995999992

   1. Initial program 95.0%

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

   if 0.995999992 < (-.f32 #s(literal 1 binary32) u1) < 0.999999821

   1. Initial program 65.5%

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

   4. Applied rewrites 35.9%

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

   5. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 12.1

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-0.5, u1, 1\right)} \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Applied rewrites 16.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-0.5, u1, 1\right) \cdot u1} - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. Taylor expanded in u1 around inf

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

   10. Applied rewrites 53.9%

       \[\leadsto \sqrt{\left(\left(\frac{1}{u1} - 0.5\right) \cdot u1\right) \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   if 0.999999821 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 15.3%

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

   4. Applied rewrites 43.5%

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

      1. lift--.f32 N/A

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

      2. lift-log1p.f32 N/A

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

      3. lift-log1p.f32 N/A

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

      4. diff-log N/A

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

      5. lift-*.f32 N/A

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

      6. lift-neg.f32 N/A

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

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

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

      8. metadata-eval N/A

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

      9. flip-+ N/A

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

      10. lift--.f32 N/A

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

      11. associate-/l/ N/A

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

      12. log-div N/A

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

      13. metadata-eval N/A

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

      14. cancel-sign-sub-inv N/A

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

      15. lift-neg.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. lift-log1p.f32 N/A

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

   6. Applied rewrites 10.9%

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

   8. Applied rewrites 49.5%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9959999918937683:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{elif}\;1 - u1 \leq 0.9999998211860657:\\ \;\;\;\;\sqrt{\left(\left(\frac{1}{u1} - 0.5\right) \cdot u1\right) \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{1}{\mathsf{log1p}\left(-u1\right)}}} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9959999918937683:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{elif}\;1 - u1 \leq 0.9999998211860657:\\ \;\;\;\;\sqrt{\left(-0.5 \cdot u1 + 1\right) \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{1}{\mathsf{log1p}\left(-u1\right)}}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (* (PI) 2.0) u2))))
   (if (<= (- 1.0 u1) 0.9959999918937683)
     (* t_0 (sqrt (- (log (- 1.0 u1)))))
     (if (<= (- 1.0 u1) 0.9999998211860657)
       (* (sqrt (- (* (+ (* -0.5 u1) 1.0) u1) (log1p (* (- u1) u1)))) t_0)
       (* (sqrt (/ -1.0 (/ 1.0 (log1p (- u1))))) t_0)))))

Derivation

1. Split input into 3 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.995999992

   1. Initial program 95.0%

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

   if 0.995999992 < (-.f32 #s(literal 1 binary32) u1) < 0.999999821

   1. Initial program 65.5%

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

   4. Applied rewrites 35.1%

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

   5. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 18.0

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-0.5, u1, 1\right)} \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Applied rewrites 14.2%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-0.5, u1, 1\right) \cdot u1} - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 56.2%

      \[\leadsto \sqrt{\left(-0.5 \cdot u1 + 1\right) \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   if 0.999999821 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 15.3%

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

   4. Applied rewrites 46.5%

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

      1. lift--.f32 N/A

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

      2. lift-log1p.f32 N/A

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

      3. lift-log1p.f32 N/A

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

      4. diff-log N/A

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

      5. lift-*.f32 N/A

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

      6. lift-neg.f32 N/A

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

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

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

      8. metadata-eval N/A

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

      9. flip-+ N/A

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

      10. lift--.f32 N/A

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

      11. associate-/l/ N/A

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

      12. log-div N/A

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

      13. metadata-eval N/A

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

      14. cancel-sign-sub-inv N/A

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

      15. lift-neg.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. lift-log1p.f32 N/A

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

   6. Applied rewrites 10.7%

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

   8. Applied rewrites 50.8%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9959999918937683:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{elif}\;1 - u1 \leq 0.9999998211860657:\\ \;\;\;\;\sqrt{\left(-0.5 \cdot u1 + 1\right) \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1}{\frac{1}{\mathsf{log1p}\left(-u1\right)}}} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \mathbf{if}\;u1 \leq 5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot t\_0\\ \mathbf{elif}\;u1 \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{\left(-0.5 \cdot u1 + 1\right) \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (* (PI) 2.0) u2))))
   (if (<= u1 5.000000058430487e-8)
     (* (pow (* u1 u1) 0.25) t_0)
     (if (<= u1 0.004000000189989805)
       (* (sqrt (- (* (+ (* -0.5 u1) 1.0) u1) (log1p (* (- u1) u1)))) t_0)
       (* t_0 (sqrt (- (log (- 1.0 u1)))))))))

Derivation

1. Split input into 3 regimes

2. if u1 < 5.00000006e-8

   1. Initial program 9.2%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 98.7

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

   5. Applied rewrites 98.7%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-*.f32 98.8

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in u1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f32 98.8

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

   10. Applied rewrites 98.8%

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

   if 5.00000006e-8 < u1 < 0.00400000019

   1. Initial program 62.3%

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

   4. Applied rewrites 37.2%

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

   5. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 17.5

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-0.5, u1, 1\right)} \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Applied rewrites 16.2%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-0.5, u1, 1\right) \cdot u1} - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 51.1%

      \[\leadsto \sqrt{\left(-0.5 \cdot u1 + 1\right) \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   if 0.00400000019 < u1

   1. Initial program 95.0%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \mathbf{elif}\;u1 \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{\left(-0.5 \cdot u1 + 1\right) \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\\ \mathbf{if}\;1 - u1 \leq 0.9998000264167786:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* (PI) 2.0) u2)))
   (if (<= (- 1.0 u1) 0.9998000264167786)
     (* t_0 (sqrt (- (log (- 1.0 u1)))))
     (* (pow (* u1 u1) 0.25) (sin t_0)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999800026

   1. Initial program 90.0%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 76.4

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

   5. Applied rewrites 76.4%

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

   if 0.999800026 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 37.0%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 92.6

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

   5. Applied rewrites 92.6%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-*.f32 92.6

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in u1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f32 92.6

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

   10. Applied rewrites 92.6%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998000264167786:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\\ \mathbf{if}\;1 - u1 \leq 0.9998000264167786:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* (PI) 2.0) u2)))
   (if (<= (- 1.0 u1) 0.9998000264167786)
     (* t_0 (sqrt (- (log (- 1.0 u1)))))
     (* (sqrt u1) (sin t_0)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999800026

   1. Initial program 90.0%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 76.4

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

   5. Applied rewrites 76.4%

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

   if 0.999800026 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 37.0%

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

   4. Applied rewrites 18.5%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 92.6

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

   7. Applied rewrites 92.6%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998000264167786:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.3% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt u1) (sin (* (* (PI) 2.0) u2))))

Derivation

1. Initial program 59.3%

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

4. Applied rewrites 15.5%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 75.3

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

7. Applied rewrites 75.3%

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

8. Final simplification 75.3%

   \[\leadsto \sqrt{u1} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]
9. Add Preprocessing

Alternative 9: 66.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot {\left(u1 \cdot u1\right)}^{0.25} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (* (* u2 (PI)) 2.0) (pow (* u1 u1) 0.25)))

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 75.3

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

5. Applied rewrites 75.3%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. pow-prod-down N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-*.f32 75.4

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

7. Applied rewrites 75.4%

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

8. Taylor expanded in u1 around 0

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

   1. unpow2 N/A

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

   2. lower-*.f32 75.4

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

10. Applied rewrites 75.4%

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

11. Taylor expanded in u2 around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f32 N/A

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

    5. lower-PI.f32 65.2

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

13. Applied rewrites 65.2%

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

14. Final simplification 65.2%

    \[\leadsto \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot {\left(u1 \cdot u1\right)}^{0.25} \]
15. Add Preprocessing

Alternative 10: 66.1% accurate, 7.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 75.3

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

5. Applied rewrites 75.3%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 65.2

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

8. Applied rewrites 65.2%

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

Alternative 11: 4.6% accurate, 8.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 N/A

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

   6. lower-sqrt.f32 4.0

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

5. Applied rewrites 4.0%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 4.7

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

8. Applied rewrites 4.7%

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

Reproduce

herbie shell --seed 2024271 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :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)))) (sin (* (* 2.0 (PI)) u2))))