Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.7% → 91.4%

Time: 11.3s

Alternatives: 9

Speedup: 8.9×

Specification

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

Math

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

FPCore

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

Initial Program: 57.7% 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: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \mathbf{if}\;u1 \leq 0.00015999999595806003:\\ \;\;\;\;t\_0 \cdot {\left(u1 \cdot u1\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* u2 (* (PI) 2.0)))))
   (if (<= u1 0.00015999999595806003)
     (* t_0 (pow (* u1 u1) 0.25))
     (* (sqrt (- (log (- 1.0 u1)))) t_0))))

Derivation

1. Split input into 2 regimes

2. if u1 < 1.59999996e-4

   1. Initial program 43.2%

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

   4. Applied rewrites 32.5%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto {\color{blue}{\left({u1}^{2}\right)}}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. 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 90.6

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

   7. Applied rewrites 90.6%

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

   if 1.59999996e-4 < u1

   1. Initial program 89.9%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 0.00015999999595806003:\\ \;\;\;\;\sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot {\left(u1 \cdot u1\right)}^{0.25}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \mathsf{PI}\left(\right)\\ t_1 := \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \mathbf{if}\;u1 \leq 4.4999998749517545 \cdot 10^{-7}:\\ \;\;\;\;t\_1 \cdot {\left(u1 \cdot u1\right)}^{0.25}\\ \mathbf{elif}\;u1 \leq 0.0038399999029934406:\\ \;\;\;\;\sqrt{\frac{1 - 0.25 \cdot \left(u1 \cdot u1\right)}{1 - -0.5 \cdot u1} \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sin t\_0 \cdot \cos t\_0\right) \cdot 2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (PI))) (t_1 (sin (* u2 (* (PI) 2.0)))))
   (if (<= u1 4.4999998749517545e-7)
     (* t_1 (pow (* u1 u1) 0.25))
     (if (<= u1 0.0038399999029934406)
       (*
        (sqrt
         (-
          (* (/ (- 1.0 (* 0.25 (* u1 u1))) (- 1.0 (* -0.5 u1))) u1)
          (log1p (* (- u1) u1))))
        t_1)
       (* (* (* (sin t_0) (cos t_0)) 2.0) (sqrt (- (log (- 1.0 u1)))))))))

Derivation

1. Split input into 3 regimes

2. if u1 < 4.49999987e-7

   1. Initial program 21.5%

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

   4. Applied rewrites 37.7%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto {\color{blue}{\left({u1}^{2}\right)}}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. 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 98.5

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

   7. Applied rewrites 98.5%

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

   if 4.49999987e-7 < u1 < 0.0038399999

   1. Initial program 66.2%

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

   4. Applied rewrites 33.9%

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

   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 16.3

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

   7. Applied rewrites 16.6%

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

   9. Applied rewrites 58.2%

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

   if 0.0038399999 < u1

   1. Initial program 94.7%

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

      1. lift-sin.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. sin-2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-sin.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 94.7

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

   4. Applied rewrites 94.7%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 4.4999998749517545 \cdot 10^{-7}:\\ \;\;\;\;\sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot {\left(u1 \cdot u1\right)}^{0.25}\\ \mathbf{elif}\;u1 \leq 0.0038399999029934406:\\ \;\;\;\;\sqrt{\frac{1 - 0.25 \cdot \left(u1 \cdot u1\right)}{1 - -0.5 \cdot u1} \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sin \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \cos \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \mathbf{if}\;u1 \leq 4.4999998749517545 \cdot 10^{-7}:\\ \;\;\;\;t\_0 \cdot {\left(u1 \cdot u1\right)}^{0.25}\\ \mathbf{elif}\;u1 \leq 0.0038399999029934406:\\ \;\;\;\;\sqrt{\frac{1 - 0.25 \cdot \left(u1 \cdot u1\right)}{1 - -0.5 \cdot u1} \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* u2 (* (PI) 2.0)))))
   (if (<= u1 4.4999998749517545e-7)
     (* t_0 (pow (* u1 u1) 0.25))
     (if (<= u1 0.0038399999029934406)
       (*
        (sqrt
         (-
          (* (/ (- 1.0 (* 0.25 (* u1 u1))) (- 1.0 (* -0.5 u1))) u1)
          (log1p (* (- u1) u1))))
        t_0)
       (* (sqrt (- (log (- 1.0 u1)))) t_0)))))

Derivation

1. Split input into 3 regimes

2. if u1 < 4.49999987e-7

   1. Initial program 21.5%

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

   4. Applied rewrites 35.1%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto {\color{blue}{\left({u1}^{2}\right)}}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. 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 98.5

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

   7. Applied rewrites 98.5%

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

   if 4.49999987e-7 < u1 < 0.0038399999

   1. Initial program 66.2%

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

   4. Applied rewrites 35.3%

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

   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 16.6

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

   7. Applied rewrites 16.1%

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

   9. Applied rewrites 59.4%

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

   if 0.0038399999 < u1

   1. Initial program 94.7%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 4.4999998749517545 \cdot 10^{-7}:\\ \;\;\;\;\sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot {\left(u1 \cdot u1\right)}^{0.25}\\ \mathbf{elif}\;u1 \leq 0.0038399999029934406:\\ \;\;\;\;\sqrt{\frac{1 - 0.25 \cdot \left(u1 \cdot u1\right)}{1 - -0.5 \cdot u1} \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \mathbf{if}\;u1 \leq 4.4999998749517545 \cdot 10^{-7}:\\ \;\;\;\;t\_0 \cdot {\left(u1 \cdot u1\right)}^{0.25}\\ \mathbf{elif}\;u1 \leq 0.0038399999029934406:\\ \;\;\;\;\sqrt{\left(-0.5 \cdot u1 + 1\right) \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* u2 (* (PI) 2.0)))))
   (if (<= u1 4.4999998749517545e-7)
     (* t_0 (pow (* u1 u1) 0.25))
     (if (<= u1 0.0038399999029934406)
       (* (sqrt (- (* (+ (* -0.5 u1) 1.0) u1) (log1p (* (- u1) u1)))) t_0)
       (* (sqrt (- (log (- 1.0 u1)))) t_0)))))

Derivation

1. Split input into 3 regimes

2. if u1 < 4.49999987e-7

   1. Initial program 21.5%

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

   4. Applied rewrites 36.4%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto {\color{blue}{\left({u1}^{2}\right)}}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. 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 98.5

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

   7. Applied rewrites 98.5%

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

   if 4.49999987e-7 < u1 < 0.0038399999

   1. Initial program 66.2%

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

   4. Applied rewrites 35.2%

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

   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 17.9

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

   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 51.2%

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

   if 0.0038399999 < u1

   1. Initial program 94.7%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 4.4999998749517545 \cdot 10^{-7}:\\ \;\;\;\;\sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot {\left(u1 \cdot u1\right)}^{0.25}\\ \mathbf{elif}\;u1 \leq 0.0038399999029934406:\\ \;\;\;\;\sqrt{\left(-0.5 \cdot u1 + 1\right) \cdot u1 - \mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)} \cdot \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9993000030517578:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot {\left(u1 \cdot u1\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* (PI) 2.0))))
   (if (<= (- 1.0 u1) 0.9993000030517578)
     (* t_0 (sqrt (- (log (- 1.0 u1)))))
     (* (sin t_0) (pow (* u1 u1) 0.25)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.9993

   1. Initial program 92.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 80.1

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

   5. Applied rewrites 80.1%

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

   if 0.9993 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 45.7%

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

   4. Applied rewrites 31.1%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto {\color{blue}{\left({u1}^{2}\right)}}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. 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 89.0

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

   7. Applied rewrites 89.0%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9993000030517578:\\ \;\;\;\;\left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot {\left(u1 \cdot u1\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9993000030517578:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* (PI) 2.0))))
   (if (<= (- 1.0 u1) 0.9993000030517578)
     (* 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) < 0.9993

   1. Initial program 92.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 80.1

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

   5. Applied rewrites 80.1%

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

   if 0.9993 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 45.7%

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

   4. Applied rewrites 32.3%

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

   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 88.9

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

   7. Applied rewrites 88.9%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9993000030517578:\\ \;\;\;\;\left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.5% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 62.5%

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

4. Applied rewrites 26.2%

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

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 74.3

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

7. Applied rewrites 74.3%

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

8. Final simplification 74.3%

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

Alternative 8: 66.5% 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 62.5%

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

4. Applied rewrites 26.7%

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

5. Taylor expanded in u1 around 0

   \[\leadsto {\color{blue}{\left({u1}^{2}\right)}}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
6. 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 74.3

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

7. Applied rewrites 74.3%

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

8. Taylor expanded in u2 around 0

   \[\leadsto {\left(u1 \cdot u1\right)}^{\frac{1}{4}} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
9. 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 65.6

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

10. Applied rewrites 65.6%

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

11. Final simplification 65.6%

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

Alternative 9: 66.5% accurate, 8.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 62.5%

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

4. Applied rewrites 27.1%

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

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 74.3

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

7. Applied rewrites 74.3%

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

8. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 65.6

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

10. Applied rewrites 65.6%

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

11. Final simplification 65.6%

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

Reproduce

herbie shell --seed 2024268 
(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))))