Average Error: 39.4 → 0.4
Time: 23.1s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x + \cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot -2\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x + \cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot -2
double f(double x, double eps) {
        double r2281069 = x;
        double r2281070 = eps;
        double r2281071 = r2281069 + r2281070;
        double r2281072 = cos(r2281071);
        double r2281073 = cos(r2281069);
        double r2281074 = r2281072 - r2281073;
        return r2281074;
}
↓
double f(double x, double eps) {
        double r2281075 = 0.5;
        double r2281076 = eps;
        double r2281077 = r2281075 * r2281076;
        double r2281078 = cos(r2281077);
        double r2281079 = x;
        double r2281080 = sin(r2281079);
        double r2281081 = r2281078 * r2281080;
        double r2281082 = cos(r2281079);
        double r2281083 = sin(r2281077);
        double r2281084 = r2281082 * r2281083;
        double r2281085 = r2281081 + r2281084;
        double r2281086 = 2.0;
        double r2281087 = r2281076 / r2281086;
        double r2281088 = sin(r2281087);
        double r2281089 = r2281085 * r2281088;
        double r2281090 = -2.0;
        double r2281091 = r2281089 * r2281090;
        return r2281091;
}

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 39.4

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

  3. Applied diff-cos33.6

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

  4. Simplified14.7

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

  5. Taylor expanded around inf 14.7

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

  6. Simplified14.7

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

  8. Applied fma-udef14.7

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

  9. Applied sin-sum0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))