Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.9% → 91.4%

Time: 10.9s

Alternatives: 8

Speedup: 10.5×

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: 91.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \mathbf{if}\;1 - u1 \leq 0.9998149871826172:\\ \;\;\;\;\frac{\left(t\_2 + 1\right) - \frac{t\_1}{\frac{-1}{t\_2 - 1}} \cdot \frac{t\_1}{{t\_1}^{2}}}{2} \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 (PI)) u2)) (t_1 (sin t_0)) (t_2 (cos t_0)))
   (if (<= (- 1.0 u1) 0.9998149871826172)
     (*
      (/
       (- (+ t_2 1.0) (* (/ t_1 (/ -1.0 (- t_2 1.0))) (/ t_1 (pow t_1 2.0))))
       2.0)
      (sqrt (- (log (- 1.0 u1)))))
     (* (sqrt u1) t_2))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.99981499

   1. Initial program 87.9%

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

      1. lift-cos.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. cos-2 N/A

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

      6. cos-mult N/A

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

      7. sin-mult N/A

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

      8. sub-div N/A

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

      9. lower-/.f32 N/A

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

   4. Applied rewrites 88.0%

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

   6. Applied rewrites 88.1%

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

   if 0.99981499 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 38.2%

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

   4. Applied rewrites 52.0%

      \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   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 92.3

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

   7. Applied rewrites 92.3%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998149871826172:\\ \;\;\;\;\frac{\left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 1\right) - \frac{\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)}{\frac{-1}{\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) - 1}} \cdot \frac{\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)}{{\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)}^{2}}}{2} \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9998149871826172:\\ \;\;\;\;\left(\left(t\_0 - 1\right) \cdot 0.5 - \left(-1 - t\_0\right) \cdot 0.5\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (- 1.0 u1) 0.9998149871826172)
     (*
      (- (* (- t_0 1.0) 0.5) (* (- -1.0 t_0) 0.5))
      (sqrt (- (log (- 1.0 u1)))))
     (* (sqrt u1) t_0))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.99981499

   1. Initial program 87.9%

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

      1. lift-cos.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-*l* N/A

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

      5. cos-2 N/A

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

      6. cos-mult N/A

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

      7. sin-mult N/A

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

      8. sub-div N/A

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

      9. lower-/.f32 N/A

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

   4. Applied rewrites 88.0%

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

      1. lift-/.f32 N/A

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

      2. lift--.f32 N/A

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

      3. div-sub N/A

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

      4. lower--.f32 N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f32 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f32 88.0

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

   6. Applied rewrites 88.0%

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

   if 0.99981499 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 38.2%

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

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   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 92.3

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

   7. Applied rewrites 92.3%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998149871826172:\\ \;\;\;\;\left(\left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) - 1\right) \cdot 0.5 - \left(-1 - \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) \cdot 0.5\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ t_1 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.013500000350177288:\\ \;\;\;\;\sqrt{u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 (PI)) u2))) (t_1 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 t_1) 0.013500000350177288) (* (sqrt u1) t_0) (* 1.0 t_1))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0135000004

   1. Initial program 40.2%

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

   4. Applied rewrites 49.4%

      \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 90.6

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

   7. Applied rewrites 90.6%

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

   if 0.0135000004 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 87.5%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 75.8%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.013500000350177288:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot t\_0 \leq 0.013500000350177288:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, -0.5 \cdot u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* (cos (* (* 2.0 (PI)) u2)) t_0) 0.013500000350177288)
     (sqrt (fma u1 (* -0.5 u1) u1))
     (* 1.0 t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0135000004

   1. Initial program 40.2%

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

   4. Applied rewrites 49.4%

      \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-log1p.f32 74.2

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 73.7%

       \[\leadsto \sqrt{\mathsf{fma}\left(-0.5, u1, 1\right) \cdot u1} \]
   11. Step-by-step derivation

   12. Applied rewrites 73.6%

       \[\leadsto \sqrt{\mathsf{fma}\left(u1, -0.5 \cdot u1, u1\right)} \]

   if 0.0135000004 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 87.5%

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

   3. Taylor expanded in u2 around 0

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

   5. Applied rewrites 75.8%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.013500000350177288:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, -0.5 \cdot u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9998149871826172:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 (PI)) u2))))
   (if (<= (- 1.0 u1) 0.9998149871826172)
     (* t_0 (sqrt (- (log (- 1.0 u1)))))
     (* (sqrt u1) t_0))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.99981499

   1. Initial program 87.9%

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

   if 0.99981499 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 38.2%

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

   4. Applied rewrites 52.8%

      \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   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 92.3

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

   7. Applied rewrites 92.3%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998149871826172:\\ \;\;\;\;\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.7% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(fma(u1, Float32(Float32(-0.5) * u1), 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 42.4%

   \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-log1p.f32 66.6

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

7. Applied rewrites 66.6%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 66.6%

    \[\leadsto \sqrt{\mathsf{fma}\left(-0.5, u1, 1\right) \cdot u1} \]
11. Step-by-step derivation

12. Applied rewrites 66.6%

    \[\leadsto \sqrt{\mathsf{fma}\left(u1, -0.5 \cdot u1, u1\right)} \]
13. Add Preprocessing

Alternative 7: 64.7% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(fma(u1, Float32(-0.5), Float32(1.0)) * 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 42.7%

   \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-log1p.f32 66.6

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

7. Applied rewrites 66.6%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 66.2%

    \[\leadsto \sqrt{\mathsf{fma}\left(-0.5, u1, 1\right) \cdot u1} \]
11. Step-by-step derivation

12. Applied rewrites 66.2%

    \[\leadsto \sqrt{\mathsf{fma}\left(u1, -0.5, 1\right) \cdot u1} \]
13. Add Preprocessing

Alternative 8: -0.0% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(((single(-0.5) * u1) * 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 42.5%

   \[\leadsto \color{blue}{{\left(e^{\log \left(\mathsf{log1p}\left(u1\right)\right)}\right)}^{0.5}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-log1p.f32 66.6

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

7. Applied rewrites 66.6%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 66.6%

    \[\leadsto \sqrt{\mathsf{fma}\left(-0.5, u1, 1\right) \cdot u1} \]

11. Taylor expanded in u1 around inf

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

13. Applied rewrites -0.0%

    \[\leadsto \sqrt{\left(-0.5 \cdot u1\right) \cdot u1} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024294 
(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))))