Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.5% → 97.7%

Time: 10.0s

Alternatives: 10

Speedup: 11.6×

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: 57.5% 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: 97.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;1 - u1 \leq 0.9943000078201294:\\ \;\;\;\;\cos \left(\left({t\_0}^{2} \cdot \left(2 \cdot u2\right)\right) \cdot t\_0\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\frac{1 \cdot u1}{-1 - -0.5 \cdot u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cbrt (PI))))
   (if (<= (- 1.0 u1) 0.9943000078201294)
     (* (cos (* (* (pow t_0 2.0) (* 2.0 u2)) t_0)) (sqrt (- (log (- 1.0 u1)))))
     (*
      (cos (* (* (PI) 2.0) u2))
      (sqrt (- (/ (* 1.0 u1) (- -1.0 (* -0.5 u1)))))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.994300008

   1. Initial program 96.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*r* N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-cbrt.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lower-cbrt.f32 96.3

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

   4. Applied rewrites 96.3%

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

   if 0.994300008 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 44.5%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 69.2

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

   5. Applied rewrites 69.5%

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

   7. Applied rewrites 98.0%

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

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 98.5%

       \[\leadsto \sqrt{-\frac{1 \cdot u1}{-1 - -0.5 \cdot u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9943000078201294:\\ \;\;\;\;\cos \left(\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2} \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\frac{1 \cdot u1}{-1 - -0.5 \cdot u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999899864196777:\\ \;\;\;\;\sqrt{u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{-\frac{\left(1 - \left(u1 \cdot u1\right) \cdot 0.25\right) \cdot u1}{-1 - -0.5 \cdot u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* (PI) 2.0) u2))))
   (if (<= t_0 0.9999899864196777)
     (* (sqrt u1) t_0)
     (*
      1.0
      (sqrt (- (/ (* (- 1.0 (* (* u1 u1) 0.25)) u1) (- -1.0 (* -0.5 u1)))))))))

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999989986

   1. Initial program 58.1%

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

   4. Applied rewrites 10.8%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 77.5

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

   7. Applied rewrites 77.5%

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

   if 0.999989986 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 56.6%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 76.1

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

   5. Applied rewrites 75.8%

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

   7. Applied rewrites 87.2%

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

   8. Taylor expanded in u2 around 0

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

   10. Applied rewrites 86.1%

       \[\leadsto \sqrt{-\frac{\left(1 - \left(u1 \cdot u1\right) \cdot 0.25\right) \cdot u1}{-1 - -0.5 \cdot u1}} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \leq 0.9999899864196777:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{-\frac{\left(1 - \left(u1 \cdot u1\right) \cdot 0.25\right) \cdot u1}{-1 - -0.5 \cdot u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9943000078201294:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{-\frac{1 \cdot u1}{-1 - -0.5 \cdot u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* (PI) 2.0) u2))))
   (if (<= (- 1.0 u1) 0.9943000078201294)
     (* t_0 (sqrt (- (log (- 1.0 u1)))))
     (* t_0 (sqrt (- (/ (* 1.0 u1) (- -1.0 (* -0.5 u1)))))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.994300008

   1. Initial program 96.3%

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

   if 0.994300008 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 44.5%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 69.7

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

   5. Applied rewrites 69.0%

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

   7. Applied rewrites 98.0%

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

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 98.5%

       \[\leadsto \sqrt{-\frac{1 \cdot u1}{-1 - -0.5 \cdot u1}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9943000078201294:\\ \;\;\;\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\frac{1 \cdot u1}{-1 - -0.5 \cdot u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.0% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.0%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f32 60.9

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

5. Applied rewrites 60.6%

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

7. Applied rewrites 87.9%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 89.7%

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

11. Final simplification 89.7%

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

Alternative 5: 88.3% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (+ (* (* 0.5 u1) u1) u1)) (cos (* (* (PI) 2.0) u2))))

Derivation

1. Initial program 57.0%

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

3. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 76.5

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

5. Applied rewrites 76.5%

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

6. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lft-mult-inverse N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. distribute-rgt-in N/A

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

   7. lft-mult-inverse N/A

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

   8. lower-fma.f32 61.9

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

8. Applied rewrites 61.2%

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

10. Applied rewrites 87.9%

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

11. Final simplification 87.9%

    \[\leadsto \sqrt{\left(0.5 \cdot u1\right) \cdot u1 + u1} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]
12. Add Preprocessing

Alternative 6: 88.2% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.0%

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

3. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 76.5

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

5. Applied rewrites 76.5%

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

6. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lft-mult-inverse N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. distribute-rgt-in N/A

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

   7. lft-mult-inverse N/A

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

   8. lower-fma.f32 61.2

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

8. Applied rewrites 61.6%

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

10. Applied rewrites 87.8%

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

11. Final simplification 87.8%

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

Alternative 7: 72.4% accurate, 4.2× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 1.0f * sqrtf(-(((1.0f - ((u1 * u1) * 0.25f)) * u1) / (-1.0f - (-0.5f * 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(-(((1.0e0 - ((u1 * u1) * 0.25e0)) * u1) / ((-1.0e0) - ((-0.5e0) * u1))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(1.0) * sqrt(Float32(-Float32(Float32(Float32(Float32(1.0) - Float32(Float32(u1 * u1) * Float32(0.25))) * u1) / Float32(Float32(-1.0) - Float32(Float32(-0.5) * u1))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(1.0) * sqrt(-(((single(1.0) - ((u1 * u1) * single(0.25))) * u1) / (single(-1.0) - (single(-0.5) * u1))));
end

Derivation

1. Initial program 57.0%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f32 60.8

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

5. Applied rewrites 61.0%

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

7. Applied rewrites 87.9%

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

8. Taylor expanded in u2 around 0

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

10. Applied rewrites 73.4%

    \[\leadsto \sqrt{-\frac{\left(1 - \left(u1 \cdot u1\right) \cdot 0.25\right) \cdot u1}{-1 - -0.5 \cdot u1}} \cdot \color{blue}{1} \]

11. Final simplification 73.4%

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

Alternative 8: 72.4% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.0%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f32 60.4

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

5. Applied rewrites 60.4%

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

7. Applied rewrites 87.8%

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

8. Taylor expanded in u2 around 0

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

10. Applied rewrites 73.3%

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

Alternative 9: 64.5% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.0%

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

3. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 76.5

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

5. Applied rewrites 76.5%

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

6. Taylor expanded in u2 around 0

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

8. Applied rewrites 65.8%

   \[\leadsto \sqrt{-\left(-u1\right)} \cdot \color{blue}{1} \]
9. Add Preprocessing

Alternative 10: 5.0% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.0%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 N/A

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

   6. lower-sqrt.f32 3.5

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

5. Applied rewrites 3.5%

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

6. Taylor expanded in u2 around 0

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

8. Applied rewrites 4.9%

   \[\leadsto \left(-\sqrt{u1}\right) \cdot \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024267 
(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))))