Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 5.2s

Precision: 64

Math

\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]

↓

\[\frac{a1 \cdot a1 + a2 \cdot a2}{\frac{1}{\cos th} \cdot \sqrt{2}}\]

C

double code(double a1, double a2, double th) {
	return (((cos(th) / sqrt(2.0)) * (a1 * a1)) + ((cos(th) / sqrt(2.0)) * (a2 * a2)));
}

↓

double code(double a1, double a2, double th) {
	return (((a1 * a1) + (a2 * a2)) / ((1.0 / cos(th)) * sqrt(2.0)));
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Derivation

1. Initial program 0.5

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]

2. Simplified 0.5

   \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}\]
3. Using strategy rm

4. Applied clear-num 0.5

   \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{2}}{\cos th}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\]

5. Applied associate-*l/ 0.5

   \[\leadsto \color{blue}{\frac{1 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\frac{\sqrt{2}}{\cos th}}}\]

6. Simplified 0.5

   \[\leadsto \frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\frac{\sqrt{2}}{\cos th}}\]
7. Using strategy rm

8. Applied clear-num 0.5

   \[\leadsto \frac{a1 \cdot a1 + a2 \cdot a2}{\color{blue}{\frac{1}{\frac{\cos th}{\sqrt{2}}}}}\]
9. Using strategy rm

10. Applied associate-/r/ 0.5

    \[\leadsto \frac{a1 \cdot a1 + a2 \cdot a2}{\color{blue}{\frac{1}{\cos th} \cdot \sqrt{2}}}\]

11. Final simplification 0.5

    \[\leadsto \frac{a1 \cdot a1 + a2 \cdot a2}{\frac{1}{\cos th} \cdot \sqrt{2}}\]

Reproduce

herbie shell --seed 2020078 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))