Average Error: 37.1 → 0.4
Time: 6.4s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \mathsf{expm1}\left(\mathsf{log1p}\left(\cos x \cdot \sin \varepsilon\right)\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \mathsf{expm1}\left(\mathsf{log1p}\left(\cos x \cdot \sin \varepsilon\right)\right)\right)
double f(double x, double eps) {
        double r127305 = x;
        double r127306 = eps;
        double r127307 = r127305 + r127306;
        double r127308 = sin(r127307);
        double r127309 = sin(r127305);
        double r127310 = r127308 - r127309;
        return r127310;
}
↓
double f(double x, double eps) {
        double r127311 = x;
        double r127312 = sin(r127311);
        double r127313 = eps;
        double r127314 = cos(r127313);
        double r127315 = 1.0;
        double r127316 = r127314 - r127315;
        double r127317 = cos(r127311);
        double r127318 = sin(r127313);
        double r127319 = r127317 * r127318;
        double r127320 = log1p(r127319);
        double r127321 = expm1(r127320);
        double r127322 = fma(r127312, r127316, r127321);
        return r127322;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target15.5
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-sum21.5

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

  4. Applied associate--l+21.5

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

  5. Taylor expanded around inf 21.5

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

  6. Simplified0.4

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

  8. Applied expm1-log1p-u0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos x \cdot \sin \varepsilon\right)\right)}\right)\]

  9. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \mathsf{expm1}\left(\mathsf{log1p}\left(\cos x \cdot \sin \varepsilon\right)\right)\right)\]

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(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)))