Average Error: 0.5 → 0.5
Time: 23.7s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\left(\frac{\cos th}{\sqrt{\sqrt{\sqrt{2}}}} \cdot \frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{\sqrt{2}}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\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)
double f(double a1, double a2, double th) {
        double r63387 = th;
        double r63388 = cos(r63387);
        double r63389 = 2.0;
        double r63390 = sqrt(r63389);
        double r63391 = r63388 / r63390;
        double r63392 = a1;
        double r63393 = r63392 * r63392;
        double r63394 = r63391 * r63393;
        double r63395 = a2;
        double r63396 = r63395 * r63395;
        double r63397 = r63391 * r63396;
        double r63398 = r63394 + r63397;
        return r63398;
}
↓
double f(double a1, double a2, double th) {
        double r63399 = th;
        double r63400 = cos(r63399);
        double r63401 = 2.0;
        double r63402 = sqrt(r63401);
        double r63403 = sqrt(r63402);
        double r63404 = sqrt(r63403);
        double r63405 = r63400 / r63404;
        double r63406 = 1.0;
        double r63407 = r63406 / r63403;
        double r63408 = r63407 / r63404;
        double r63409 = r63405 * r63408;
        double r63410 = a1;
        double r63411 = r63410 * r63410;
        double r63412 = a2;
        double r63413 = r63412 * r63412;
        double r63414 = r63411 + r63413;
        double r63415 = r63409 * r63414;
        return r63415;
}

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. Final simplification0.5

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

Reproduce

herbie shell --seed 2019326 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))