Average Error: 0.5 → 0.5
Time: 49.6s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\cos th \cdot \left(a2 \cdot a2\right)}{\sqrt{2}} + \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}}\]
\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)
↓
\frac{\cos th \cdot \left(a2 \cdot a2\right)}{\sqrt{2}} + \frac{\cos th \cdot \left(a1 \cdot a1\right)}{\sqrt{2}}
double f(double a1, double a2, double th) {
        double r1369329 = th;
        double r1369330 = cos(r1369329);
        double r1369331 = 2.0;
        double r1369332 = sqrt(r1369331);
        double r1369333 = r1369330 / r1369332;
        double r1369334 = a1;
        double r1369335 = r1369334 * r1369334;
        double r1369336 = r1369333 * r1369335;
        double r1369337 = a2;
        double r1369338 = r1369337 * r1369337;
        double r1369339 = r1369333 * r1369338;
        double r1369340 = r1369336 + r1369339;
        return r1369340;
}
↓
double f(double a1, double a2, double th) {
        double r1369341 = th;
        double r1369342 = cos(r1369341);
        double r1369343 = a2;
        double r1369344 = r1369343 * r1369343;
        double r1369345 = r1369342 * r1369344;
        double r1369346 = 2.0;
        double r1369347 = sqrt(r1369346);
        double r1369348 = r1369345 / r1369347;
        double r1369349 = a1;
        double r1369350 = r1369349 * r1369349;
        double r1369351 = r1369342 * r1369350;
        double r1369352 = r1369351 / r1369347;
        double r1369353 = r1369348 + r1369352;
        return r1369353;
}

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

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

  4. Applied associate-/r*0.5

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

  5. Taylor expanded around -inf 0.5

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

  6. Simplified0.5

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

  7. Taylor expanded around inf 0.5

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

  8. Simplified0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2019128 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))