Average Error: 0.5 → 0.4
Time: 44.9s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{a1 \cdot a1}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{\sqrt{2}}}} \cdot \frac{\cos th}{\sqrt{\sqrt{\sqrt{2}}}} + \frac{\cos th \cdot \left(a2 \cdot a2\right)}{\sqrt{2}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{a1 \cdot a1}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{\sqrt{2}}}} \cdot \frac{\cos th}{\sqrt{\sqrt{\sqrt{2}}}} + \frac{\cos th \cdot \left(a2 \cdot a2\right)}{\sqrt{2}}
double f(double a1, double a2, double th) {
        double r2968031 = th;
        double r2968032 = cos(r2968031);
        double r2968033 = 2.0;
        double r2968034 = sqrt(r2968033);
        double r2968035 = r2968032 / r2968034;
        double r2968036 = a1;
        double r2968037 = r2968036 * r2968036;
        double r2968038 = r2968035 * r2968037;
        double r2968039 = a2;
        double r2968040 = r2968039 * r2968039;
        double r2968041 = r2968035 * r2968040;
        double r2968042 = r2968038 + r2968041;
        return r2968042;
}
↓
double f(double a1, double a2, double th) {
        double r2968043 = a1;
        double r2968044 = r2968043 * r2968043;
        double r2968045 = 2.0;
        double r2968046 = sqrt(r2968045);
        double r2968047 = sqrt(r2968046);
        double r2968048 = sqrt(r2968047);
        double r2968049 = r2968047 * r2968048;
        double r2968050 = r2968044 / r2968049;
        double r2968051 = th;
        double r2968052 = cos(r2968051);
        double r2968053 = r2968052 / r2968048;
        double r2968054 = r2968050 * r2968053;
        double r2968055 = a2;
        double r2968056 = r2968055 * r2968055;
        double r2968057 = r2968052 * r2968056;
        double r2968058 = r2968057 / r2968046;
        double r2968059 = r2968054 + r2968058;
        return r2968059;
}

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 associate-*l/0.5

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

  5. Applied add-sqr-sqrt0.5

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

  6. Applied associate-/r*0.5

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

  8. Applied add-sqr-sqrt0.5

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

  9. Applied div-inv0.5

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

  10. Applied times-frac0.5

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

  11. Applied associate-*l*0.5

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019138 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))