Average Error: 0.5 → 0.5
Time: 26.1s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\frac{\cos th \cdot \sqrt{a2 \cdot a2 + a1 \cdot a1}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}} \cdot \sqrt{a2 \cdot a2 + a1 \cdot a1}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{\frac{\cos th \cdot \sqrt{a2 \cdot a2 + a1 \cdot a1}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}} \cdot \sqrt{a2 \cdot a2 + a1 \cdot a1}
double f(double a1, double a2, double th) {
        double r2481495 = th;
        double r2481496 = cos(r2481495);
        double r2481497 = 2.0;
        double r2481498 = sqrt(r2481497);
        double r2481499 = r2481496 / r2481498;
        double r2481500 = a1;
        double r2481501 = r2481500 * r2481500;
        double r2481502 = r2481499 * r2481501;
        double r2481503 = a2;
        double r2481504 = r2481503 * r2481503;
        double r2481505 = r2481499 * r2481504;
        double r2481506 = r2481502 + r2481505;
        return r2481506;
}
↓
double f(double a1, double a2, double th) {
        double r2481507 = th;
        double r2481508 = cos(r2481507);
        double r2481509 = a2;
        double r2481510 = r2481509 * r2481509;
        double r2481511 = a1;
        double r2481512 = r2481511 * r2481511;
        double r2481513 = r2481510 + r2481512;
        double r2481514 = sqrt(r2481513);
        double r2481515 = r2481508 * r2481514;
        double r2481516 = 2.0;
        double r2481517 = sqrt(r2481516);
        double r2481518 = cbrt(r2481517);
        double r2481519 = r2481518 * r2481518;
        double r2481520 = r2481515 / r2481519;
        double r2481521 = r2481520 / r2481518;
        double r2481522 = r2481521 * r2481514;
        return r2481522;
}

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. Simplified0.8

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

  4. Applied add-sqr-sqrt0.8

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

  5. Applied *-un-lft-identity0.8

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

  6. Applied sqrt-prod0.8

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

  7. Applied times-frac0.8

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

  8. Applied *-un-lft-identity0.8

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

  9. Applied times-frac0.5

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

  10. Simplified0.5

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

  11. Simplified0.5

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

  13. Applied add-cube-cbrt0.5

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

  14. Applied associate-/r*0.5

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

  15. Final simplification0.5

    \[\leadsto \frac{\frac{\cos th \cdot \sqrt{a2 \cdot a2 + a1 \cdot a1}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt[3]{\sqrt{2}}} \cdot \sqrt{a2 \cdot a2 + a1 \cdot a1}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))