Average Error: 37.3 → 0.4
Time: 6.2s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \sqrt[3]{{\left(\cos \varepsilon - 1\right)}^{3}}, \cos x \cdot \sin \varepsilon\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \sqrt[3]{{\left(\cos \varepsilon - 1\right)}^{3}}, \cos x \cdot \sin \varepsilon\right)
double f(double x, double eps) {
        double r120187 = x;
        double r120188 = eps;
        double r120189 = r120187 + r120188;
        double r120190 = sin(r120189);
        double r120191 = sin(r120187);
        double r120192 = r120190 - r120191;
        return r120192;
}
↓
double f(double x, double eps) {
        double r120193 = x;
        double r120194 = sin(r120193);
        double r120195 = eps;
        double r120196 = cos(r120195);
        double r120197 = 1.0;
        double r120198 = r120196 - r120197;
        double r120199 = 3.0;
        double r120200 = pow(r120198, r120199);
        double r120201 = cbrt(r120200);
        double r120202 = cos(r120193);
        double r120203 = sin(r120195);
        double r120204 = r120202 * r120203;
        double r120205 = fma(r120194, r120201, r120204);
        return r120205;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.3
Target15.3
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.3

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

  3. Applied sin-sum21.9

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

  4. Applied associate--l+21.9

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

  5. Taylor expanded around inf 21.9

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

  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 add-cbrt-cube0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \color{blue}{\sqrt[3]{\left(\left(\cos \varepsilon - 1\right) \cdot \left(\cos \varepsilon - 1\right)\right) \cdot \left(\cos \varepsilon - 1\right)}}, \cos x \cdot \sin \varepsilon\right)\]

  9. Simplified0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \sqrt[3]{\color{blue}{{\left(\cos \varepsilon - 1\right)}^{3}}}, \cos x \cdot \sin \varepsilon\right)\]

  10. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \sqrt[3]{{\left(\cos \varepsilon - 1\right)}^{3}}, \cos x \cdot \sin \varepsilon\right)\]

Reproduce

herbie shell --seed 2020049 +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)))