cos(2*x)/(cos^2(x)*sin^2(x))

Average Error: 27.9 → 2.7

Time: 1.1m

Precision: 64

• could not determine a ground truth for program body

  1. x = 2.88733846674046e-150
  2. cos = -1.4028819597761525e+262
  3. sin = -7.995242360819531e+16

Math

\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]

↓

\[\frac{\frac{\cos \left(2 \cdot x\right)}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}}{\left|{cos}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot \left({cos}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)\right|}\]

C

double f(double x, double cos, double sin) {
        double r54038 = 2.0;
        double r54039 = x;
        double r54040 = r54038 * r54039;
        double r54041 = cos(r54040);
        double r54042 = cos;
        double r54043 = pow(r54042, r54038);
        double r54044 = sin;
        double r54045 = pow(r54044, r54038);
        double r54046 = r54039 * r54045;
        double r54047 = r54046 * r54039;
        double r54048 = r54043 * r54047;
        double r54049 = r54041 / r54048;
        return r54049;
}

↓

double f(double x, double cos, double sin) {
        double r54050 = 2.0;
        double r54051 = x;
        double r54052 = r54050 * r54051;
        double r54053 = cos(r54052);
        double r54054 = cos;
        double r54055 = 2.0;
        double r54056 = r54050 / r54055;
        double r54057 = pow(r54054, r54056);
        double r54058 = sin;
        double r54059 = pow(r54058, r54056);
        double r54060 = r54051 * r54059;
        double r54061 = r54057 * r54060;
        double r54062 = fabs(r54061);
        double r54063 = r54053 / r54062;
        double r54064 = r54056 / r54055;
        double r54065 = pow(r54054, r54064);
        double r54066 = r54065 * r54060;
        double r54067 = r54065 * r54066;
        double r54068 = fabs(r54067);
        double r54069 = r54063 / r54068;
        return r54069;
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

1. Initial program 27.9

   \[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
2. Using strategy rm

3. Applied sqr-pow 27.9

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot \color{blue}{\left({sin}^{\left(\frac{2}{2}\right)} \cdot {sin}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot x\right)}\]

4. Applied associate-*r* 21.6

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\color{blue}{\left(\left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot {sin}^{\left(\frac{2}{2}\right)}\right)} \cdot x\right)}\]
5. Using strategy rm

6. Applied add-sqr-sqrt 21.7

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\sqrt{{cos}^{2} \cdot \left(\left(\left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot x\right)} \cdot \sqrt{{cos}^{2} \cdot \left(\left(\left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot x\right)}}}\]

7. Simplified 21.7

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|} \cdot \sqrt{{cos}^{2} \cdot \left(\left(\left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot x\right)}}\]

8. Simplified 2.8

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right| \cdot \color{blue}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}}\]
9. Using strategy rm

10. Applied associate-/r* 2.6

    \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}}\]
11. Using strategy rm

12. Applied sqr-pow 2.7

    \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}}{\left|\color{blue}{\left({cos}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {cos}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}\]

13. Applied associate-*l* 2.7

    \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}}{\left|\color{blue}{{cos}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot \left({cos}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)}\right|}\]

14. Final simplification 2.7

    \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}}{\left|{cos}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot \left({cos}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)\right|}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  :precision binary64
  (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))))