Average Error: 37.0 → 0.6
Time: 5.8s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \left(\left(\sqrt[3]{\cos \varepsilon - 1} \cdot \sqrt[3]{\cos \varepsilon - 1}\right) \cdot \sqrt[3]{\cos \varepsilon - 1}\right) + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \left(\left(\sqrt[3]{\cos \varepsilon - 1} \cdot \sqrt[3]{\cos \varepsilon - 1}\right) \cdot \sqrt[3]{\cos \varepsilon - 1}\right) + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r99386 = x;
        double r99387 = eps;
        double r99388 = r99386 + r99387;
        double r99389 = sin(r99388);
        double r99390 = sin(r99386);
        double r99391 = r99389 - r99390;
        return r99391;
}
↓
double f(double x, double eps) {
        double r99392 = x;
        double r99393 = sin(r99392);
        double r99394 = eps;
        double r99395 = cos(r99394);
        double r99396 = 1.0;
        double r99397 = r99395 - r99396;
        double r99398 = cbrt(r99397);
        double r99399 = r99398 * r99398;
        double r99400 = r99399 * r99398;
        double r99401 = r99393 * r99400;
        double r99402 = cos(r99392);
        double r99403 = sin(r99394);
        double r99404 = r99402 * r99403;
        double r99405 = r99401 + r99404;
        return r99405;
}

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
Target15.1
Herbie0.6
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.0

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

  3. Applied sin-sum21.8

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

  4. Applied associate--l+21.8

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

  5. Taylor expanded around inf 21.8

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

  6. Simplified0.4

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

  8. Applied add-cube-cbrt0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2020027 
(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)))