Average Error: 36.4 → 0.4
Time: 20.9s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[2 \cdot \left(\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \log \left(e^{\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
2 \cdot \left(\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \log \left(e^{\sin x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)
double f(double x, double eps) {
        double r4488227 = x;
        double r4488228 = eps;
        double r4488229 = r4488227 + r4488228;
        double r4488230 = sin(r4488229);
        double r4488231 = sin(r4488227);
        double r4488232 = r4488230 - r4488231;
        return r4488232;
}
↓
double f(double x, double eps) {
        double r4488233 = 2.0;
        double r4488234 = 0.5;
        double r4488235 = eps;
        double r4488236 = r4488234 * r4488235;
        double r4488237 = cos(r4488236);
        double r4488238 = x;
        double r4488239 = cos(r4488238);
        double r4488240 = r4488237 * r4488239;
        double r4488241 = sin(r4488238);
        double r4488242 = sin(r4488236);
        double r4488243 = r4488241 * r4488242;
        double r4488244 = exp(r4488243);
        double r4488245 = log(r4488244);
        double r4488246 = r4488240 - r4488245;
        double r4488247 = r4488235 / r4488233;
        double r4488248 = sin(r4488247);
        double r4488249 = r4488246 * r4488248;
        double r4488250 = r4488233 * r4488249;
        return r4488250;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.4
Target14.7
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.4

    \[\sin \left(x + \varepsilon\right) - \sin x\]
  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. Simplified14.7

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

  5. Taylor expanded around -inf 14.7

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

  7. Applied cos-sum0.3

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

  9. Applied add-log-exp0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019134 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

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