Average Error: 36.8 → 0.3
Time: 32.5s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) + \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) + \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)
double f(double x, double eps) {
        double r6102908 = x;
        double r6102909 = eps;
        double r6102910 = r6102908 + r6102909;
        double r6102911 = sin(r6102910);
        double r6102912 = sin(r6102908);
        double r6102913 = r6102911 - r6102912;
        return r6102913;
}
↓
double f(double x, double eps) {
        double r6102914 = 2.0;
        double r6102915 = 0.5;
        double r6102916 = eps;
        double r6102917 = r6102915 * r6102916;
        double r6102918 = sin(r6102917);
        double r6102919 = r6102914 * r6102918;
        double r6102920 = cos(r6102917);
        double r6102921 = x;
        double r6102922 = cos(r6102921);
        double r6102923 = r6102920 * r6102922;
        double r6102924 = r6102919 * r6102923;
        double r6102925 = -2.0;
        double r6102926 = r6102925 * r6102918;
        double r6102927 = sin(r6102921);
        double r6102928 = r6102927 * r6102918;
        double r6102929 = r6102926 * r6102928;
        double r6102930 = r6102924 + r6102929;
        return r6102930;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.8
Target14.9
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.8

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

  3. Applied diff-sin37.1

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

  4. Simplified14.9

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

  5. Taylor expanded around inf 14.9

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

  6. Simplified14.9

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

  8. Applied cos-sum0.3

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

  10. Applied sub-neg0.3

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

  11. Applied distribute-rgt-in0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 
(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)))