Average Error: 0.5 → 0.4
Time: 15.9s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\cos th \cdot \frac{a1 \cdot a1}{\sqrt{2}} + \frac{\cos th}{\sqrt{\sqrt{\sqrt{2}}}} \cdot \frac{a2 \cdot a2}{\sqrt{\sqrt{\sqrt{2}}} \cdot \sqrt{\sqrt{2}}}\]
\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 \frac{a1 \cdot a1}{\sqrt{2}} + \frac{\cos th}{\sqrt{\sqrt{\sqrt{2}}}} \cdot \frac{a2 \cdot a2}{\sqrt{\sqrt{\sqrt{2}}} \cdot \sqrt{\sqrt{2}}}
double f(double a1, double a2, double th) {
        double r119022 = th;
        double r119023 = cos(r119022);
        double r119024 = 2.0;
        double r119025 = sqrt(r119024);
        double r119026 = r119023 / r119025;
        double r119027 = a1;
        double r119028 = r119027 * r119027;
        double r119029 = r119026 * r119028;
        double r119030 = a2;
        double r119031 = r119030 * r119030;
        double r119032 = r119026 * r119031;
        double r119033 = r119029 + r119032;
        return r119033;
}
↓
double f(double a1, double a2, double th) {
        double r119034 = th;
        double r119035 = cos(r119034);
        double r119036 = a1;
        double r119037 = r119036 * r119036;
        double r119038 = 2.0;
        double r119039 = sqrt(r119038);
        double r119040 = r119037 / r119039;
        double r119041 = r119035 * r119040;
        double r119042 = sqrt(r119039);
        double r119043 = sqrt(r119042);
        double r119044 = r119035 / r119043;
        double r119045 = a2;
        double r119046 = r119045 * r119045;
        double r119047 = r119043 * r119042;
        double r119048 = r119046 / r119047;
        double r119049 = r119044 * r119048;
        double r119050 = r119041 + r119049;
        return r119050;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus th

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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-inv0.6

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

  4. Applied associate-*l*0.6

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

  5. Simplified0.5

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

  7. Applied add-sqr-sqrt0.5

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

  8. Applied sqrt-prod0.5

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

  9. Applied associate-/r*0.5

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

  11. Applied add-sqr-sqrt0.5

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

  12. Applied sqrt-prod0.5

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

  13. Applied sqrt-prod0.5

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

  14. Applied div-inv0.5

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

  15. Applied times-frac0.5

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

  16. Applied associate-*l*0.5

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

  17. Simplified0.4

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

  18. Final simplification0.4

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

Reproduce

herbie shell --seed 2020043 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))