Average Error: 0.5 → 0.5
Time: 6.6s
Precision: 64
\[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right)\]
↓
\[\frac{\cos th \cdot \sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}}}\]
\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 \sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}}}
double f(double a1, double a2, double th) {
        double r82563 = th;
        double r82564 = cos(r82563);
        double r82565 = 2.0;
        double r82566 = sqrt(r82565);
        double r82567 = r82564 / r82566;
        double r82568 = a1;
        double r82569 = r82568 * r82568;
        double r82570 = r82567 * r82569;
        double r82571 = a2;
        double r82572 = r82571 * r82571;
        double r82573 = r82567 * r82572;
        double r82574 = r82570 + r82573;
        return r82574;
}
↓
double f(double a1, double a2, double th) {
        double r82575 = th;
        double r82576 = cos(r82575);
        double r82577 = a1;
        double r82578 = r82577 * r82577;
        double r82579 = a2;
        double r82580 = r82579 * r82579;
        double r82581 = r82578 + r82580;
        double r82582 = sqrt(r82581);
        double r82583 = r82576 * r82582;
        double r82584 = 2.0;
        double r82585 = sqrt(r82584);
        double r82586 = cbrt(r82585);
        double r82587 = r82586 * r82586;
        double r82588 = r82583 / r82587;
        double r82589 = r82582 / r82586;
        double r82590 = r82588 * r82589;
        return r82590;
}

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 associate-*l/0.5

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

  6. Applied add-sqr-sqrt0.5

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

  7. Applied associate-*r*0.5

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

  9. Applied add-cube-cbrt0.5

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

  10. Applied times-frac0.5

    \[\leadsto \color{blue}{\frac{\cos th \cdot \sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}}}}\]

  11. Final simplification0.5

    \[\leadsto \frac{\cos th \cdot \sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}} \cdot \frac{\sqrt{a1 \cdot a1 + a2 \cdot a2}}{\sqrt[3]{\sqrt{2}}}\]

Reproduce

herbie shell --seed 2019356 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2)) (* a1 a1)) (* (/ (cos th) (sqrt 2)) (* a2 a2))))