Migdal et al, Equation (64)

Average Error: 0.5 → 0.4

Time: 10.9s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double th) {
        double r110026 = th;
        double r110027 = cos(r110026);
        double r110028 = 2.0;
        double r110029 = sqrt(r110028);
        double r110030 = r110027 / r110029;
        double r110031 = a1;
        double r110032 = r110031 * r110031;
        double r110033 = r110030 * r110032;
        double r110034 = a2;
        double r110035 = r110034 * r110034;
        double r110036 = r110030 * r110035;
        double r110037 = r110033 + r110036;
        return r110037;
}

↓

double f(double a1, double a2, double th) {
        double r110038 = th;
        double r110039 = cos(r110038);
        double r110040 = a2;
        double r110041 = 2.0;
        double r110042 = pow(r110040, r110041);
        double r110043 = a1;
        double r110044 = pow(r110043, r110041);
        double r110045 = r110042 + r110044;
        double r110046 = sqrt(r110045);
        double r110047 = 2.0;
        double r110048 = sqrt(r110047);
        double r110049 = cbrt(r110048);
        double r110050 = r110049 * r110049;
        double r110051 = r110046 / r110050;
        double r110052 = 1.0;
        double r110053 = cbrt(r110049);
        double r110054 = r110053 * r110053;
        double r110055 = r110052 / r110054;
        double r110056 = r110046 / r110053;
        double r110057 = r110055 * r110056;
        double r110058 = r110051 * r110057;
        double r110059 = r110039 * r110058;
        return r110059;
}

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 div-inv 0.5

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

5. Applied associate-*l* 0.5

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

6. Simplified 0.5

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

8. Applied add-cube-cbrt 0.5

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

9. Applied add-sqr-sqrt 0.5

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

10. Applied times-frac 0.5

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

12. Applied add-cube-cbrt 0.5

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

13. Applied *-un-lft-identity 0.5

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

14. Applied sqrt-prod 0.5

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

15. Applied times-frac 0.4

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

16. Simplified 0.4

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

17. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019322 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))