Migdal et al, Equation (64)

Average Error: 0.5 → 0.5

Time: 24.3s

Precision: 64

Math

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

↓

\[\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} \cdot \left(\frac{\mathsf{fma}\left(a2, a2, {a1}^{2}\right)}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}} \cdot \frac{\cos th}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}}\right)\]

C

double f(double a1, double a2, double th) {
        double r77212 = th;
        double r77213 = cos(r77212);
        double r77214 = 2.0;
        double r77215 = sqrt(r77214);
        double r77216 = r77213 / r77215;
        double r77217 = a1;
        double r77218 = r77217 * r77217;
        double r77219 = r77216 * r77218;
        double r77220 = a2;
        double r77221 = r77220 * r77220;
        double r77222 = r77216 * r77221;
        double r77223 = r77219 + r77222;
        return r77223;
}

↓

double f(double a1, double a2, double th) {
        double r77224 = 1.0;
        double r77225 = 2.0;
        double r77226 = r77224 / r77225;
        double r77227 = cbrt(r77226);
        double r77228 = sqrt(r77225);
        double r77229 = cbrt(r77228);
        double r77230 = r77229 * r77229;
        double r77231 = cbrt(r77230);
        double r77232 = r77227 / r77231;
        double r77233 = a2;
        double r77234 = a1;
        double r77235 = 2.0;
        double r77236 = pow(r77234, r77235);
        double r77237 = fma(r77233, r77233, r77236);
        double r77238 = cbrt(r77229);
        double r77239 = sqrt(r77238);
        double r77240 = r77237 / r77239;
        double r77241 = th;
        double r77242 = cos(r77241);
        double r77243 = r77242 / r77239;
        double r77244 = r77240 * r77243;
        double r77245 = r77232 * r77244;
        return r77245;
}

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

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

4. Applied add-cube-cbrt 0.5

   \[\leadsto \frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}}\]

5. Applied associate-/r* 0.5

   \[\leadsto \color{blue}{\frac{\frac{\cos th \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}}}\]

6. Taylor expanded around inf 0.6

   \[\leadsto \frac{\color{blue}{\left({a1}^{2} \cdot \cos th\right) \cdot {\left(\frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}^{\frac{1}{3}} + {\left(\frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}^{\frac{1}{3}} \cdot \left(\cos th \cdot {a2}^{2}\right)}}{\sqrt[3]{\sqrt{2}}}\]

7. Simplified 0.4

   \[\leadsto \frac{\color{blue}{\sqrt[3]{\frac{1}{2}} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \cos th\right)}}{\sqrt[3]{\sqrt{2}}}\]
8. Using strategy rm

9. Applied add-cube-cbrt 0.4

   \[\leadsto \frac{\sqrt[3]{\frac{1}{2}} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \cos th\right)}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}}}\]

10. Applied cbrt-prod 0.9

    \[\leadsto \frac{\sqrt[3]{\frac{1}{2}} \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \cos th\right)}{\color{blue}{\sqrt[3]{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{2}}}}}\]

11. Applied times-frac 0.4

    \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \cos th}{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}}\]
12. Using strategy rm

13. Applied add-sqr-sqrt 0.4

    \[\leadsto \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \cos th}{\color{blue}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}}}\]

14. Applied times-frac 0.5

    \[\leadsto \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}} \cdot \frac{\cos th}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}}\right)}\]

15. Simplified 0.5

    \[\leadsto \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} \cdot \left(\color{blue}{\frac{\mathsf{fma}\left(a2, a2, {a1}^{2}\right)}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}}} \cdot \frac{\cos th}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}}\right)\]

16. Final simplification 0.5

    \[\leadsto \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} \cdot \left(\frac{\mathsf{fma}\left(a2, a2, {a1}^{2}\right)}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}} \cdot \frac{\cos th}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt{2}}}}}\right)\]

Reproduce

herbie shell --seed 2019326 +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))))