Average Error: 36.3 → 0.4
Time: 41.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\left(\cos x \cdot \cos \left(\frac{\varepsilon}{2}\right) - \log \left(e^{\sin x \cdot \sin \left(\frac{\varepsilon}{2}\right)}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\left(\cos x \cdot \cos \left(\frac{\varepsilon}{2}\right) - \log \left(e^{\sin x \cdot \sin \left(\frac{\varepsilon}{2}\right)}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2
double f(double x, double eps) {
        double r6225808 = x;
        double r6225809 = eps;
        double r6225810 = r6225808 + r6225809;
        double r6225811 = sin(r6225810);
        double r6225812 = sin(r6225808);
        double r6225813 = r6225811 - r6225812;
        return r6225813;
}
↓
double f(double x, double eps) {
        double r6225814 = x;
        double r6225815 = cos(r6225814);
        double r6225816 = eps;
        double r6225817 = 2.0;
        double r6225818 = r6225816 / r6225817;
        double r6225819 = cos(r6225818);
        double r6225820 = r6225815 * r6225819;
        double r6225821 = sin(r6225814);
        double r6225822 = sin(r6225818);
        double r6225823 = r6225821 * r6225822;
        double r6225824 = exp(r6225823);
        double r6225825 = log(r6225824);
        double r6225826 = r6225820 - r6225825;
        double r6225827 = r6225826 * r6225822;
        double r6225828 = r6225827 * r6225817;
        return r6225828;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.3
Target15.2
Herbie0.4
\[2.0 \cdot \left(\cos \left(x + \frac{\varepsilon}{2.0}\right) \cdot \sin \left(\frac{\varepsilon}{2.0}\right)\right)\]

Derivation

  1. Initial program 36.3

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

  3. Applied diff-sin36.7

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

  4. Simplified15.3

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

  5. Taylor expanded around inf 15.2

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

  6. Simplified15.2

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

  8. Applied cos-sum0.3

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

  10. Applied add-log-exp0.4

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

  11. Final simplification0.4

    \[\leadsto \left(\left(\cos x \cdot \cos \left(\frac{\varepsilon}{2}\right) - \log \left(e^{\sin x \cdot \sin \left(\frac{\varepsilon}{2}\right)}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

  (- (sin (+ x eps)) (sin x)))