Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 7.1s

Precision: binary64

Math

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

↓

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

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (+
  (* (/ (cos th) (sqrt 2.0)) (* a1 a1))
  (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))

↓

(FPCore (a1 a2 th)
 :precision binary64
 (+
  (* a1 (* (/ (cos th) (sqrt 2.0)) a1))
  (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))

C

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

↓

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

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. Using strategy rm

3. Applied associate-*r* 0.5

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

4. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020198 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))