Average Error: 39.1 → 0.5
Time: 29.2s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot \left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sin x\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot \left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sin x\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)
double f(double x, double eps) {
        double r20305 = x;
        double r20306 = eps;
        double r20307 = r20305 + r20306;
        double r20308 = cos(r20307);
        double r20309 = cos(r20305);
        double r20310 = r20308 - r20309;
        return r20310;
}
↓
double f(double x, double eps) {
        double r20311 = 0.5;
        double r20312 = eps;
        double r20313 = r20311 * r20312;
        double r20314 = sin(r20313);
        double r20315 = x;
        double r20316 = cos(r20315);
        double r20317 = r20314 * r20316;
        double r20318 = cos(r20313);
        double r20319 = cbrt(r20318);
        double r20320 = r20319 * r20319;
        double r20321 = sin(r20315);
        double r20322 = r20319 * r20321;
        double r20323 = r20320 * r20322;
        double r20324 = r20317 + r20323;
        double r20325 = -2.0;
        double r20326 = r20314 * r20325;
        double r20327 = r20324 * r20326;
        return r20327;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 39.1

    \[\cos \left(x + \varepsilon\right) - \cos x\]
  2. Using strategy rm

  3. Applied diff-cos33.5

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

  4. Simplified14.9

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

  5. Taylor expanded around inf 14.9

    \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}\]

  6. Simplified14.9

    \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \varepsilon + x\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)}\]
  7. Using strategy rm

  8. Applied sin-sum0.4

    \[\leadsto \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)} \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]
  9. Using strategy rm

  10. Applied add-cube-cbrt0.5

    \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \color{blue}{\left(\left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)}\right)} \cdot \sin x\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]

  11. Applied associate-*l*0.5

    \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \color{blue}{\left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot \left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sin x\right)}\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]

  12. Final simplification0.5

    \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot \left(\sqrt[3]{\cos \left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \sin x\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))