Average Error: 37.1 → 0.4
Time: 20.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\cos x, \sin \varepsilon, \mathsf{fma}\left(\cos \varepsilon, \sin x, -\sin x\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\cos x, \sin \varepsilon, \mathsf{fma}\left(\cos \varepsilon, \sin x, -\sin x\right)\right)
double f(double x, double eps) {
        double r5411243 = x;
        double r5411244 = eps;
        double r5411245 = r5411243 + r5411244;
        double r5411246 = sin(r5411245);
        double r5411247 = sin(r5411243);
        double r5411248 = r5411246 - r5411247;
        return r5411248;
}
↓
double f(double x, double eps) {
        double r5411249 = x;
        double r5411250 = cos(r5411249);
        double r5411251 = eps;
        double r5411252 = sin(r5411251);
        double r5411253 = cos(r5411251);
        double r5411254 = sin(r5411249);
        double r5411255 = -r5411254;
        double r5411256 = fma(r5411253, r5411254, r5411255);
        double r5411257 = fma(r5411250, r5411252, r5411256);
        return r5411257;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target15.0
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

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. Applied associate--l+22.0

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

  5. Taylor expanded around inf 22.0

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

  6. Simplified0.4

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

  8. Applied fma-neg0.4

    \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, \color{blue}{\mathsf{fma}\left(\cos \varepsilon, \sin x, -\sin x\right)}\right)\]

  9. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, \mathsf{fma}\left(\cos \varepsilon, \sin x, -\sin x\right)\right)\]

Reproduce

herbie shell --seed 2019171 +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)))