Average Error: 36.9 → 0.4
Time: 15.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \frac{\left(-\sin \varepsilon\right) \cdot \sin \varepsilon}{1 + \cos \varepsilon} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \frac{\left(-\sin \varepsilon\right) \cdot \sin \varepsilon}{1 + \cos \varepsilon} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r91241 = x;
        double r91242 = eps;
        double r91243 = r91241 + r91242;
        double r91244 = sin(r91243);
        double r91245 = sin(r91241);
        double r91246 = r91244 - r91245;
        return r91246;
}
↓
double f(double x, double eps) {
        double r91247 = x;
        double r91248 = sin(r91247);
        double r91249 = eps;
        double r91250 = sin(r91249);
        double r91251 = -r91250;
        double r91252 = r91251 * r91250;
        double r91253 = 1.0;
        double r91254 = cos(r91249);
        double r91255 = r91253 + r91254;
        double r91256 = r91252 / r91255;
        double r91257 = r91248 * r91256;
        double r91258 = cos(r91247);
        double r91259 = r91258 * r91250;
        double r91260 = r91257 + r91259;
        return r91260;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
Target15.1
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.9

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

  2. Simplified36.9

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

  4. Applied sin-sum21.6

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

  5. Applied associate--l+0.4

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

  6. Simplified0.4

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

  8. Applied flip--0.5

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

  9. Simplified0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(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)))