Average Error: 0.5 → 0.4
Time: 30.4s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\left(a1 \cdot a1\right) \cdot \cos th}{\sqrt{2}} + \frac{\left(\cos th \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right) \cdot \left(a2 \cdot a2\right)}{\left|\sqrt[3]{\sqrt{2}}\right|}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{\left(a1 \cdot a1\right) \cdot \cos th}{\sqrt{2}} + \frac{\left(\cos th \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right) \cdot \left(a2 \cdot a2\right)}{\left|\sqrt[3]{\sqrt{2}}\right|}
double f(double a1, double a2, double th) {
        double r2855034 = th;
        double r2855035 = cos(r2855034);
        double r2855036 = 2.0;
        double r2855037 = sqrt(r2855036);
        double r2855038 = r2855035 / r2855037;
        double r2855039 = a1;
        double r2855040 = r2855039 * r2855039;
        double r2855041 = r2855038 * r2855040;
        double r2855042 = a2;
        double r2855043 = r2855042 * r2855042;
        double r2855044 = r2855038 * r2855043;
        double r2855045 = r2855041 + r2855044;
        return r2855045;
}
↓
double f(double a1, double a2, double th) {
        double r2855046 = a1;
        double r2855047 = r2855046 * r2855046;
        double r2855048 = th;
        double r2855049 = cos(r2855048);
        double r2855050 = r2855047 * r2855049;
        double r2855051 = 2.0;
        double r2855052 = sqrt(r2855051);
        double r2855053 = r2855050 / r2855052;
        double r2855054 = 1.0;
        double r2855055 = sqrt(r2855052);
        double r2855056 = r2855054 / r2855055;
        double r2855057 = cbrt(r2855052);
        double r2855058 = sqrt(r2855057);
        double r2855059 = r2855056 / r2855058;
        double r2855060 = r2855049 * r2855059;
        double r2855061 = a2;
        double r2855062 = r2855061 * r2855061;
        double r2855063 = r2855060 * r2855062;
        double r2855064 = fabs(r2855057);
        double r2855065 = r2855063 / r2855064;
        double r2855066 = r2855053 + r2855065;
        return r2855066;
}

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

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

  5. Applied add-sqr-sqrt0.5

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

  6. Applied sqrt-prod0.5

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

  7. Applied associate-/r*0.5

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

  9. Applied add-cube-cbrt0.5

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

  10. Applied sqrt-prod0.5

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

  11. Applied div-inv0.5

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

  12. Applied times-frac0.5

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

  13. Simplified0.5

    \[\leadsto \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}} + \left(\color{blue}{\frac{\cos th}{\left|\sqrt[3]{\sqrt{2}}\right|}} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}}\right) \cdot \left(a2 \cdot a2\right)\]
  14. Using strategy rm

  15. Applied associate-*l/0.5

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

  16. Applied associate-*l/0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2019158 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))