Average Error: 37.0 → 0.4
Time: 6.4s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\sin x \cdot \cos \varepsilon + \left(-\sin x\right)\right) + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r113298 = x;
        double r113299 = eps;
        double r113300 = r113298 + r113299;
        double r113301 = sin(r113300);
        double r113302 = sin(r113298);
        double r113303 = r113301 - r113302;
        return r113303;
}
↓
double f(double x, double eps) {
        double r113304 = x;
        double r113305 = sin(r113304);
        double r113306 = eps;
        double r113307 = cos(r113306);
        double r113308 = r113305 * r113307;
        double r113309 = -r113305;
        double r113310 = r113308 + r113309;
        double r113311 = cos(r113304);
        double r113312 = sin(r113306);
        double r113313 = r113311 * r113312;
        double r113314 = r113310 + r113313;
        return r113314;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
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 37.0

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

  3. Applied sin-sum21.3

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

  4. Applied associate--l+21.3

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

  5. Taylor expanded around inf 21.3

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

  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 fma-udef0.4

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

  10. Applied sub-neg0.4

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

  11. Applied distribute-lft-in0.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2020033 +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)))