Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.9% → 91.3%

Time: 8.4s

Alternatives: 5

Speedup: 14.4×

Specification

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

Math

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

FPCore

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

Initial Program: 57.9% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 91.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998800158500671:\\ \;\;\;\;\cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{0.16666666666666666}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9998800158500671)
   (* (cos (* u2 (* (PI) 2.0))) (sqrt (- (log (- 1.0 u1)))))
   (*
    (sqrt u1)
    (cos
     (*
      (* (* (* (sqrt (PI)) u2) (pow (PI) 0.16666666666666666)) (cbrt (PI)))
      2.0)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999880016

   1. Initial program 88.8%

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

   if 0.999880016 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 36.7%

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

   4. Applied rewrites 60.9%

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

   5. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 93.3

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

   7. Applied rewrites 93.3%

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

   9. Applied rewrites 93.3%

      \[\leadsto \cos \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left({\mathsf{PI}\left(\right)}^{0.16666666666666666} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot u2\right)\right)\right) \cdot 2\right) \cdot \sqrt{u1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998800158500671:\\ \;\;\;\;\cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot u2\right) \cdot {\mathsf{PI}\left(\right)}^{0.16666666666666666}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998800158500671:\\ \;\;\;\;\cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{u1} \cdot u1}{u1} \cdot \cos \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9998800158500671)
   (* (cos (* u2 (* (PI) 2.0))) (sqrt (- (log (- 1.0 u1)))))
   (* (/ (* (sqrt u1) u1) u1) (cos (* (* u2 (PI)) 2.0)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999880016

   1. Initial program 88.8%

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

   if 0.999880016 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 36.7%

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 93.3

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

   7. Applied rewrites 93.3%

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

   9. Applied rewrites 93.0%

      \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \cdot \frac{0 + {u1}^{1.5}}{\color{blue}{0 + \left(u1 - 0\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 93.3%

       \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \cdot \frac{\sqrt{u1} \cdot u1}{\color{blue}{0} + \left(u1 - 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998800158500671:\\ \;\;\;\;\cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{u1} \cdot u1}{u1} \cdot \cos \left(\left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.4% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.3%

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

4. Applied rewrites 54.9%

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

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-cos.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-sqrt.f32 78.1

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

7. Applied rewrites 78.1%

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

9. Applied rewrites 77.9%

   \[\leadsto \cos \left(\left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot 2\right) \cdot \frac{0 + {u1}^{1.5}}{\color{blue}{0 + \left(u1 - 0\right)}} \]
10. Step-by-step derivation

11. Applied rewrites 78.1%

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

12. Final simplification 78.1%

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

Alternative 4: 76.4% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.3%

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

4. Applied rewrites 54.4%

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

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-cos.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-sqrt.f32 78.1

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

7. Applied rewrites 78.1%

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

8. Final simplification 78.1%

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

Alternative 5: 64.5% accurate, 14.4× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \sqrt{u1} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.3%

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

4. Applied rewrites 53.5%

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

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-cos.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-sqrt.f32 78.1

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

7. Applied rewrites 78.1%

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

8. Taylor expanded in u2 around 0

   \[\leadsto 1 \cdot \sqrt{\color{blue}{u1}} \]
9. Step-by-step derivation

10. Applied rewrites 68.4%

    \[\leadsto 1 \cdot \sqrt{\color{blue}{u1}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024304 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 (PI)) u2))))