Average Error: 27.5 → 2.8
Time: 36.8s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos x \cdot \cos x}{\left(cos \cdot \left(sin \cdot x\right)\right) \cdot \left(cos \cdot \left(sin \cdot x\right)\right)} - \frac{\sin x \cdot \sin x}{\left(cos \cdot \left(sin \cdot x\right)\right) \cdot \left(cos \cdot \left(sin \cdot x\right)\right)}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\cos x \cdot \cos x}{\left(cos \cdot \left(sin \cdot x\right)\right) \cdot \left(cos \cdot \left(sin \cdot x\right)\right)} - \frac{\sin x \cdot \sin x}{\left(cos \cdot \left(sin \cdot x\right)\right) \cdot \left(cos \cdot \left(sin \cdot x\right)\right)}
double f(double x, double cos, double sin) {
        double r1145004 = 2.0;
        double r1145005 = x;
        double r1145006 = r1145004 * r1145005;
        double r1145007 = cos(r1145006);
        double r1145008 = cos;
        double r1145009 = pow(r1145008, r1145004);
        double r1145010 = sin;
        double r1145011 = pow(r1145010, r1145004);
        double r1145012 = r1145005 * r1145011;
        double r1145013 = r1145012 * r1145005;
        double r1145014 = r1145009 * r1145013;
        double r1145015 = r1145007 / r1145014;
        return r1145015;
}
↓
double f(double x, double cos, double sin) {
        double r1145016 = x;
        double r1145017 = cos(r1145016);
        double r1145018 = r1145017 * r1145017;
        double r1145019 = cos;
        double r1145020 = sin;
        double r1145021 = r1145020 * r1145016;
        double r1145022 = r1145019 * r1145021;
        double r1145023 = r1145022 * r1145022;
        double r1145024 = r1145018 / r1145023;
        double r1145025 = sin(r1145016);
        double r1145026 = r1145025 * r1145025;
        double r1145027 = r1145026 / r1145023;
        double r1145028 = r1145024 - r1145027;
        return r1145028;
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 27.5

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

  2. Simplified2.8

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

  3. Taylor expanded around -inf 30.9

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

  4. Simplified2.7

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

  6. Applied cos-22.8

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

  7. Applied div-sub2.8

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

  8. Final simplification2.8

    \[\leadsto \frac{\cos x \cdot \cos x}{\left(cos \cdot \left(sin \cdot x\right)\right) \cdot \left(cos \cdot \left(sin \cdot x\right)\right)} - \frac{\sin x \cdot \sin x}{\left(cos \cdot \left(sin \cdot x\right)\right) \cdot \left(cos \cdot \left(sin \cdot x\right)\right)}\]

Reproduce

herbie shell --seed 2019133 
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))))