Average Error: 36.9 → 0.4
Time: 6.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \frac{-\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \frac{-\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r145289 = x;
        double r145290 = eps;
        double r145291 = r145289 + r145290;
        double r145292 = sin(r145291);
        double r145293 = sin(r145289);
        double r145294 = r145292 - r145293;
        return r145294;
}
↓
double f(double x, double eps) {
        double r145295 = x;
        double r145296 = sin(r145295);
        double r145297 = eps;
        double r145298 = sin(r145297);
        double r145299 = r145298 * r145298;
        double r145300 = -r145299;
        double r145301 = cos(r145297);
        double r145302 = 1.0;
        double r145303 = r145301 + r145302;
        double r145304 = r145300 / r145303;
        double r145305 = r145296 * r145304;
        double r145306 = cos(r145295);
        double r145307 = r145306 * r145298;
        double r145308 = r145305 + r145307;
        return r145308;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
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 36.9

    \[\sin \left(x + \varepsilon\right) - \sin x\]
  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. Taylor expanded around inf 21.4

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

  5. Simplified0.4

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

  7. Applied add-cbrt-cube0.4

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

  8. Simplified0.4

    \[\leadsto \sin x \cdot \sqrt[3]{\color{blue}{{\left(\cos \varepsilon - 1\right)}^{3}}} + \cos x \cdot \sin \varepsilon\]
  9. Using strategy rm

  10. Applied flip--0.5

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

  11. Applied cube-div0.5

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

  12. Applied cbrt-div0.5

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

  13. Simplified0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

    \[\leadsto \sin x \cdot \frac{-\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]

Reproduce

herbie shell --seed 2019354 
(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)))