Average Error: 0.5 → 0.4
Time: 42.0s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\left(\frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\right) \cdot \frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} + \frac{a1}{\frac{\sqrt[3]{\sqrt{2}}}{\cos th}} \cdot \frac{a1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\left(\frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt[3]{\sqrt{2}}}} \cdot \left(a2 \cdot a2\right)\right) \cdot \frac{\cos th}{\sqrt{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}} + \frac{a1}{\frac{\sqrt[3]{\sqrt{2}}}{\cos th}} \cdot \frac{a1}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}
double f(double a1, double a2, double th) {
        double r2974432 = th;
        double r2974433 = cos(r2974432);
        double r2974434 = 2.0;
        double r2974435 = sqrt(r2974434);
        double r2974436 = r2974433 / r2974435;
        double r2974437 = a1;
        double r2974438 = r2974437 * r2974437;
        double r2974439 = r2974436 * r2974438;
        double r2974440 = a2;
        double r2974441 = r2974440 * r2974440;
        double r2974442 = r2974436 * r2974441;
        double r2974443 = r2974439 + r2974442;
        return r2974443;
}
↓
double f(double a1, double a2, double th) {
        double r2974444 = 1.0;
        double r2974445 = 2.0;
        double r2974446 = sqrt(r2974445);
        double r2974447 = sqrt(r2974446);
        double r2974448 = r2974444 / r2974447;
        double r2974449 = cbrt(r2974446);
        double r2974450 = sqrt(r2974449);
        double r2974451 = r2974448 / r2974450;
        double r2974452 = a2;
        double r2974453 = r2974452 * r2974452;
        double r2974454 = r2974451 * r2974453;
        double r2974455 = th;
        double r2974456 = cos(r2974455);
        double r2974457 = r2974449 * r2974449;
        double r2974458 = sqrt(r2974457);
        double r2974459 = r2974456 / r2974458;
        double r2974460 = r2974454 * r2974459;
        double r2974461 = a1;
        double r2974462 = r2974449 / r2974456;
        double r2974463 = r2974461 / r2974462;
        double r2974464 = r2974461 / r2974457;
        double r2974465 = r2974463 * r2974464;
        double r2974466 = r2974460 + r2974465;
        return r2974466;
}

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 clear-num0.5

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

  4. Applied associate-*l/0.5

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

  5. Simplified0.5

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

  7. Applied add-sqr-sqrt0.5

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

  8. Applied sqrt-prod0.5

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

  9. Applied associate-/r*0.5

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

  11. Applied add-cube-cbrt0.5

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

  12. Applied sqrt-prod0.5

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

  13. Applied div-inv0.5

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

  14. Applied times-frac0.5

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

  15. Applied associate-*l*0.4

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

  17. Applied *-un-lft-identity0.4

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

  18. Applied add-cube-cbrt0.4

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

  19. Applied times-frac0.5

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

  20. Applied times-frac0.4

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

  21. Simplified0.4

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

  22. Final simplification0.4

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

Reproduce

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