Average Error: 0.5 → 0.5
Time: 7.6s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\frac{\cos th}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}} \cdot \left(a1 \cdot a1\right) + \left(\frac{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2\]
\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{\cos th}{\sqrt{2}} \cdot a2\right) \cdot a2
double f(double a1, double a2, double th) {
        double r101472 = th;
        double r101473 = cos(r101472);
        double r101474 = 2.0;
        double r101475 = sqrt(r101474);
        double r101476 = r101473 / r101475;
        double r101477 = a1;
        double r101478 = r101477 * r101477;
        double r101479 = r101476 * r101478;
        double r101480 = a2;
        double r101481 = r101480 * r101480;
        double r101482 = r101476 * r101481;
        double r101483 = r101479 + r101482;
        return r101483;
}
↓
double f(double a1, double a2, double th) {
        double r101484 = th;
        double r101485 = cos(r101484);
        double r101486 = 2.0;
        double r101487 = sqrt(r101486);
        double r101488 = sqrt(r101487);
        double r101489 = r101485 / r101488;
        double r101490 = r101489 / r101488;
        double r101491 = a1;
        double r101492 = r101491 * r101491;
        double r101493 = r101490 * r101492;
        double r101494 = r101485 / r101487;
        double r101495 = a2;
        double r101496 = r101494 * r101495;
        double r101497 = r101496 * r101495;
        double r101498 = r101493 + r101497;
        return r101498;
}

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 associate-*r*0.5

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

  5. Applied add-sqr-sqrt0.5

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

  6. Applied sqrt-prod0.5

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

  7. Applied associate-/r*0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2019354 +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))))