Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 11.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r88194 = th;
        double r88195 = cos(r88194);
        double r88196 = 2.0;
        double r88197 = sqrt(r88196);
        double r88198 = r88195 / r88197;
        double r88199 = a1;
        double r88200 = r88199 * r88199;
        double r88201 = r88198 * r88200;
        double r88202 = a2;
        double r88203 = r88202 * r88202;
        double r88204 = r88198 * r88203;
        double r88205 = r88201 + r88204;
        return r88205;
}

↓

double f(double a1, double a2, double th) {
        double r88206 = th;
        double r88207 = cos(r88206);
        double r88208 = 2.0;
        double r88209 = sqrt(r88208);
        double r88210 = sqrt(r88209);
        double r88211 = sqrt(r88210);
        double r88212 = r88207 / r88211;
        double r88213 = a1;
        double r88214 = 2.0;
        double r88215 = pow(r88213, r88214);
        double r88216 = r88211 * r88210;
        double r88217 = r88215 / r88216;
        double r88218 = r88212 * r88217;
        double r88219 = a2;
        double r88220 = r88219 / r88209;
        double r88221 = r88207 * r88220;
        double r88222 = r88221 * r88219;
        double r88223 = r88218 + r88222;
        return r88223;
}

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

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

5. Applied div-inv 0.5

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

6. Applied associate-*l* 0.5

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

7. Simplified 0.5

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

9. Applied add-sqr-sqrt 0.5

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

10. Applied sqrt-prod 0.5

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

11. Applied associate-/r* 0.5

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

13. Applied add-sqr-sqrt 0.5

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

14. Applied sqrt-prod 0.5

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

15. Applied sqrt-prod 0.5

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

16. Applied div-inv 0.5

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

17. Applied times-frac 0.5

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

18. Applied associate-*l* 0.5

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

19. Simplified 0.4

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

20. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))