Average Error: 37.3 → 22.0
Time: 20.8s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(\cos x \cdot \sin \varepsilon + \sin x \cdot \cos \varepsilon\right) - \sin x\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(\cos x \cdot \sin \varepsilon + \sin x \cdot \cos \varepsilon\right) - \sin x
double f(double x, double eps) {
        double r132303 = x;
        double r132304 = eps;
        double r132305 = r132303 + r132304;
        double r132306 = sin(r132305);
        double r132307 = sin(r132303);
        double r132308 = r132306 - r132307;
        return r132308;
}
↓
double f(double x, double eps) {
        double r132309 = x;
        double r132310 = cos(r132309);
        double r132311 = eps;
        double r132312 = sin(r132311);
        double r132313 = r132310 * r132312;
        double r132314 = sin(r132309);
        double r132315 = cos(r132311);
        double r132316 = r132314 * r132315;
        double r132317 = r132313 + r132316;
        double r132318 = r132317 - r132314;
        return r132318;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target15.1
Herbie22.0
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.3

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

  3. Applied sin-sum22.0

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

  5. Applied *-un-lft-identity22.0

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

  6. Applied *-un-lft-identity22.0

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

  7. Applied distribute-lft-out--22.0

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

  8. Simplified0.4

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

  10. Applied add-cube-cbrt0.6

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

  11. Final simplification22.0

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

Reproduce

herbie shell --seed 2019297 
(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)))