Average Error: 37.1 → 0.5
Time: 6.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \frac{\cos \varepsilon \cdot \cos \varepsilon - 1}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \frac{\cos \varepsilon \cdot \cos \varepsilon - 1}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r112981 = x;
        double r112982 = eps;
        double r112983 = r112981 + r112982;
        double r112984 = sin(r112983);
        double r112985 = sin(r112981);
        double r112986 = r112984 - r112985;
        return r112986;
}
↓
double f(double x, double eps) {
        double r112987 = x;
        double r112988 = sin(r112987);
        double r112989 = eps;
        double r112990 = cos(r112989);
        double r112991 = r112990 * r112990;
        double r112992 = 1.0;
        double r112993 = r112991 - r112992;
        double r112994 = r112990 + r112992;
        double r112995 = r112993 / r112994;
        double r112996 = r112988 * r112995;
        double r112997 = cos(r112987);
        double r112998 = sin(r112989);
        double r112999 = r112997 * r112998;
        double r113000 = r112996 + r112999;
        return r113000;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target15.0
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.1

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

  3. Applied sin-sum22.0

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

  4. Taylor expanded around inf 22.0

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

  5. Simplified0.4

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

  7. Applied flip--0.5

    \[\leadsto \sin x \cdot \color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1}{\cos \varepsilon + 1}} + \cos x \cdot \sin \varepsilon\]

  8. Simplified0.5

    \[\leadsto \sin x \cdot \frac{\color{blue}{\cos \varepsilon \cdot \cos \varepsilon - 1}}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]

  9. Final simplification0.5

    \[\leadsto \sin x \cdot \frac{\cos \varepsilon \cdot \cos \varepsilon - 1}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]

Reproduce

herbie shell --seed 2020062 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

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

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