Average Error: 37.1 → 0.5
Time: 6.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \varepsilon - 1\right)\right), \cos x \cdot \sin \varepsilon\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \varepsilon - 1\right)\right), \cos x \cdot \sin \varepsilon\right)
double f(double x, double eps) {
        double r132129 = x;
        double r132130 = eps;
        double r132131 = r132129 + r132130;
        double r132132 = sin(r132131);
        double r132133 = sin(r132129);
        double r132134 = r132132 - r132133;
        return r132134;
}
↓
double f(double x, double eps) {
        double r132135 = x;
        double r132136 = sin(r132135);
        double r132137 = eps;
        double r132138 = cos(r132137);
        double r132139 = 1.0;
        double r132140 = r132138 - r132139;
        double r132141 = expm1(r132140);
        double r132142 = log1p(r132141);
        double r132143 = cos(r132135);
        double r132144 = sin(r132137);
        double r132145 = r132143 * r132144;
        double r132146 = fma(r132136, r132142, r132145);
        return r132146;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target14.6
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-sum22.3

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

  5. Applied *-un-lft-identity22.3

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

  6. Applied *-un-lft-identity22.3

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

  7. Applied distribute-lft-out--22.3

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

  8. Simplified0.4

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

  10. Applied log1p-expm1-u0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2019346 +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)))