Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 22.5s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r1856205 = th;
        double r1856206 = cos(r1856205);
        double r1856207 = 2.0;
        double r1856208 = sqrt(r1856207);
        double r1856209 = r1856206 / r1856208;
        double r1856210 = a1;
        double r1856211 = r1856210 * r1856210;
        double r1856212 = r1856209 * r1856211;
        double r1856213 = a2;
        double r1856214 = r1856213 * r1856213;
        double r1856215 = r1856209 * r1856214;
        double r1856216 = r1856212 + r1856215;
        return r1856216;
}

↓

double f(double a1, double a2, double th) {
        double r1856217 = a2;
        double r1856218 = r1856217 * r1856217;
        double r1856219 = 2.0;
        double r1856220 = sqrt(r1856219);
        double r1856221 = r1856218 / r1856220;
        double r1856222 = th;
        double r1856223 = cos(r1856222);
        double r1856224 = r1856221 * r1856223;
        double r1856225 = a1;
        double r1856226 = r1856225 / r1856220;
        double r1856227 = r1856225 * r1856226;
        double r1856228 = r1856227 * r1856223;
        double r1856229 = r1856224 + r1856228;
        return r1856229;
}

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 div-inv 0.5

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

4. Applied associate-*l* 0.5

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

5. Simplified 0.5

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

7. Applied div-inv 0.5

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

8. Applied associate-*l* 0.5

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

9. Simplified 0.5

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

11. Applied add-sqr-sqrt 32.1

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

12. Applied *-un-lft-identity 32.1

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

13. Applied sqrt-prod 32.1

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

14. Applied times-frac 32.1

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

15. Applied add-sqr-sqrt 32.2

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

16. Applied times-frac 32.2

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

17. Simplified 32.1

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

18. Simplified 0.5

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

19. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))