Beckmann Distribution sample, tan2theta, alphax != alphay, u1 <= 0.5

Average Error: 12.6 → 0.6

Time: 49.8s

Precision: binary32

\[0.0001 \leq alphax \land alphax \leq 1 \land 0.0001 \leq alphay \land alphay \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1 \land 0 \leq cos2phi \land cos2phi \leq 1 \land 0 \leq sin2phi\]

Math

\[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\]

↓

\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9445134401321411:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\left(alphay \cdot alphay\right) \cdot cos2phi + sin2phi \cdot \left(alphax \cdot alphax\right)} \cdot \left(\left(alphay \cdot alphay\right) \cdot \left(alphax \cdot alphax\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 - \left({u0}^{4} \cdot -0.25 - \left(u0 \cdot u0\right) \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array}\]

FPCore

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (/
  (- (log (- 1.0 u0)))
  (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))

↓

(FPCore (alphax alphay u0 cos2phi sin2phi)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9445134401321411)
   (*
    (/
     (- (log (- 1.0 u0)))
     (+ (* (* alphay alphay) cos2phi) (* sin2phi (* alphax alphax))))
    (* (* alphay alphay) (* alphax alphax)))
   (/
    (-
     u0
     (-
      (* (pow u0 4.0) -0.25)
      (* (* u0 u0) (+ 0.5 (* u0 0.3333333333333333)))))
    (+ (/ cos2phi (* alphax alphax)) (/ (/ sin2phi alphay) alphay)))))

C

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	return -logf(1.0f - u0) / ((cos2phi / (alphax * alphax)) + (sin2phi / (alphay * alphay)));
}

↓

float code(float alphax, float alphay, float u0, float cos2phi, float sin2phi) {
	float tmp;
	if ((1.0f - u0) <= 0.9445134401321411f) {
		tmp = (-logf(1.0f - u0) / (((alphay * alphay) * cos2phi) + (sin2phi * (alphax * alphax)))) * ((alphay * alphay) * (alphax * alphax));
	} else {
		tmp = (u0 - ((powf(u0, 4.0f) * -0.25f) - ((u0 * u0) * (0.5f + (u0 * 0.3333333333333333f))))) / ((cos2phi / (alphax * alphax)) + ((sin2phi / alphay) / alphay));
	}
	return tmp;
}

Error

Bits error versus alphax

Bits error versus alphay

Bits error versus u0

Bits error versus cos2phi

Bits error versus sin2phi

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u0) < 0.94451344

   1. Initial program 1.5

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\]
   2. Using strategy rm

   3. Applied frac-add_binary32 1.5

      \[\leadsto \frac{-\log \left(1 - u0\right)}{\color{blue}{\frac{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi}{\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)}}}\]

   4. Applied associate-/r/_binary32 1.0

      \[\leadsto \color{blue}{\frac{-\log \left(1 - u0\right)}{cos2phi \cdot \left(alphay \cdot alphay\right) + \left(alphax \cdot alphax\right) \cdot sin2phi} \cdot \left(\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)\right)}\]

   5. Simplified 1.0

      \[\leadsto \color{blue}{\frac{-\log \left(1 - u0\right)}{\left(alphay \cdot alphay\right) \cdot cos2phi + sin2phi \cdot \left(alphax \cdot alphax\right)}} \cdot \left(\left(alphax \cdot alphax\right) \cdot \left(alphay \cdot alphay\right)\right)\]

   if 0.94451344 < (-.f32 1 u0)

   1. Initial program 14.3

      \[\frac{-\log \left(1 - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\]

   2. Taylor expanded around 0 0.6

      \[\leadsto \frac{-\color{blue}{\left(-\left(u0 + \left(0.25 \cdot {u0}^{4} + \left(0.5 \cdot {u0}^{2} + 0.3333333333333333 \cdot {u0}^{3}\right)\right)\right)\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\]

   3. Simplified 0.6

      \[\leadsto \frac{-\color{blue}{\left(\left({u0}^{4} \cdot -0.25 - \left(u0 \cdot u0\right) \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right)\right) - u0\right)}}{\frac{cos2phi}{alphax \cdot alphax} + \frac{sin2phi}{alphay \cdot alphay}}\]
   4. Using strategy rm

   5. Applied associate-/r*_binary32 0.6

      \[\leadsto \frac{-\left(\left({u0}^{4} \cdot -0.25 - \left(u0 \cdot u0\right) \cdot \left(0.5 + 0.3333333333333333 \cdot u0\right)\right) - u0\right)}{\frac{cos2phi}{alphax \cdot alphax} + \color{blue}{\frac{\frac{sin2phi}{alphay}}{alphay}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9445134401321411:\\ \;\;\;\;\frac{-\log \left(1 - u0\right)}{\left(alphay \cdot alphay\right) \cdot cos2phi + sin2phi \cdot \left(alphax \cdot alphax\right)} \cdot \left(\left(alphay \cdot alphay\right) \cdot \left(alphax \cdot alphax\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{u0 - \left({u0}^{4} \cdot -0.25 - \left(u0 \cdot u0\right) \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)}{\frac{cos2phi}{alphax \cdot alphax} + \frac{\frac{sin2phi}{alphay}}{alphay}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021175 
(FPCore (alphax alphay u0 cos2phi sin2phi)
  :name "Beckmann Distribution sample, tan2theta, alphax != alphay, u1 <= 0.5"
  :precision binary32
  :pre (and (<= 0.0001 alphax 1.0) (<= 0.0001 alphay 1.0) (<= 2.328306437e-10 u0 1.0) (<= 0.0 cos2phi 1.0) (<= 0.0 sin2phi))
  (/ (- (log (- 1.0 u0))) (+ (/ cos2phi (* alphax alphax)) (/ sin2phi (* alphay alphay)))))