Average Error: 0.5 → 0.4
Time: 25.5s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{1}{\frac{\sqrt{\sqrt{\sqrt{2}}}}{\cos th}} \cdot \frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{\sqrt{\sqrt{2}}} \cdot \sqrt{\sqrt{2}}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{1}{\frac{\sqrt{\sqrt{\sqrt{2}}}}{\cos th}} \cdot \frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{\sqrt{\sqrt{2}}} \cdot \sqrt{\sqrt{2}}}
double f(double a1, double a2, double th) {
        double r91335 = th;
        double r91336 = cos(r91335);
        double r91337 = 2.0;
        double r91338 = sqrt(r91337);
        double r91339 = r91336 / r91338;
        double r91340 = a1;
        double r91341 = r91340 * r91340;
        double r91342 = r91339 * r91341;
        double r91343 = a2;
        double r91344 = r91343 * r91343;
        double r91345 = r91339 * r91344;
        double r91346 = r91342 + r91345;
        return r91346;
}
↓
double f(double a1, double a2, double th) {
        double r91347 = 1.0;
        double r91348 = 2.0;
        double r91349 = sqrt(r91348);
        double r91350 = sqrt(r91349);
        double r91351 = sqrt(r91350);
        double r91352 = th;
        double r91353 = cos(r91352);
        double r91354 = r91351 / r91353;
        double r91355 = r91347 / r91354;
        double r91356 = a2;
        double r91357 = r91356 * r91356;
        double r91358 = a1;
        double r91359 = r91358 * r91358;
        double r91360 = r91357 + r91359;
        double r91361 = r91351 * r91350;
        double r91362 = r91360 / r91361;
        double r91363 = r91355 * r91362;
        return r91363;
}

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. Simplified0.5

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied sqrt-prod0.5

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

  6. Applied associate-/r*0.5

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

  8. Applied add-sqr-sqrt0.5

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

  9. Applied sqrt-prod0.5

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

  10. Applied sqrt-prod0.5

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

  11. Applied div-inv0.6

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

  12. Applied times-frac0.5

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

  13. Applied associate-*l*0.5

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

  14. Simplified0.4

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

  16. Applied clear-num0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2019304 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))