Average Error: 0.5 → 0.4
Time: 13.6s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} \cdot \left(a1 \cdot a1\right) + \left(\frac{\frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{\sqrt{2}}}}}{\sqrt{\sqrt{\sqrt{2}}}} \cdot a2\right) \cdot a2\]
\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}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} \cdot \left(a1 \cdot a1\right) + \left(\frac{\frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{\sqrt{2}}}}}{\sqrt{\sqrt{\sqrt{2}}}} \cdot a2\right) \cdot a2
double f(double a1, double a2, double th) {
        double r48355 = th;
        double r48356 = cos(r48355);
        double r48357 = 2.0;
        double r48358 = sqrt(r48357);
        double r48359 = r48356 / r48358;
        double r48360 = a1;
        double r48361 = r48360 * r48360;
        double r48362 = r48359 * r48361;
        double r48363 = a2;
        double r48364 = r48363 * r48363;
        double r48365 = r48359 * r48364;
        double r48366 = r48362 + r48365;
        return r48366;
}
↓
double f(double a1, double a2, double th) {
        double r48367 = th;
        double r48368 = cos(r48367);
        double r48369 = 2.0;
        double r48370 = sqrt(r48369);
        double r48371 = sqrt(r48370);
        double r48372 = r48368 / r48371;
        double r48373 = r48372 / r48371;
        double r48374 = a1;
        double r48375 = r48374 * r48374;
        double r48376 = r48373 * r48375;
        double r48377 = sqrt(r48371);
        double r48378 = r48372 / r48377;
        double r48379 = r48378 / r48377;
        double r48380 = a2;
        double r48381 = r48379 * r48380;
        double r48382 = r48381 * r48380;
        double r48383 = r48376 + r48382;
        return r48383;
}

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 add-sqr-sqrt0.5

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

  4. Applied sqrt-prod0.5

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

  5. Applied associate-/r*0.5

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

  7. Applied associate-*r*0.5

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

  9. Applied add-sqr-sqrt0.5

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

  10. Applied sqrt-prod0.5

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

  11. Applied associate-/r*0.5

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

  13. Applied add-sqr-sqrt0.5

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

  14. Applied sqrt-prod0.5

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

  15. Applied sqrt-prod0.5

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

  16. Applied associate-/r*0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))