Average Error: 37.0 → 0.4
Time: 5.7s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\frac{\sqrt[3]{{\left(-1 \cdot \left(\sin x \cdot {\left(\sin \varepsilon\right)}^{2}\right)\right)}^{3}}}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\frac{\sqrt[3]{{\left(-1 \cdot \left(\sin x \cdot {\left(\sin \varepsilon\right)}^{2}\right)\right)}^{3}}}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r84288 = x;
        double r84289 = eps;
        double r84290 = r84288 + r84289;
        double r84291 = sin(r84290);
        double r84292 = sin(r84288);
        double r84293 = r84291 - r84292;
        return r84293;
}
↓
double f(double x, double eps) {
        double r84294 = -1.0;
        double r84295 = x;
        double r84296 = sin(r84295);
        double r84297 = eps;
        double r84298 = sin(r84297);
        double r84299 = 2.0;
        double r84300 = pow(r84298, r84299);
        double r84301 = r84296 * r84300;
        double r84302 = r84294 * r84301;
        double r84303 = 3.0;
        double r84304 = pow(r84302, r84303);
        double r84305 = cbrt(r84304);
        double r84306 = cos(r84297);
        double r84307 = 1.0;
        double r84308 = r84306 + r84307;
        double r84309 = r84305 / r84308;
        double r84310 = cos(r84295);
        double r84311 = r84310 * r84298;
        double r84312 = r84309 + r84311;
        return r84312;
}

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-sum21.4

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

  5. Applied *-un-lft-identity21.4

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

  6. Applied *-un-lft-identity21.4

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

  7. Applied distribute-lft-out--21.4

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

  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 flip--0.5

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

  11. Applied associate-*r/0.5

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

  12. Simplified0.4

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

  14. Applied add-cbrt-cube0.4

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

  15. Applied add-cbrt-cube0.4

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

  16. Applied cbrt-unprod0.4

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

  17. Simplified0.4

    \[\leadsto 1 \cdot \left(\frac{\sqrt[3]{\color{blue}{{\left(-1 \cdot \left(\sin x \cdot {\left(\sin \varepsilon\right)}^{2}\right)\right)}^{3}}}}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\right)\]

  18. Final simplification0.4

    \[\leadsto \frac{\sqrt[3]{{\left(-1 \cdot \left(\sin x \cdot {\left(\sin \varepsilon\right)}^{2}\right)\right)}^{3}}}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]

Reproduce

herbie shell --seed 2020060 
(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)))