Average Error: 36.8 → 0.3
Time: 19.2s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[2 \cdot \left(\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
2 \cdot \left(\left(\cos x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right) - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)
double f(double x, double eps) {
        double r5769416 = x;
        double r5769417 = eps;
        double r5769418 = r5769416 + r5769417;
        double r5769419 = sin(r5769418);
        double r5769420 = sin(r5769416);
        double r5769421 = r5769419 - r5769420;
        return r5769421;
}
↓
double f(double x, double eps) {
        double r5769422 = 2.0;
        double r5769423 = x;
        double r5769424 = cos(r5769423);
        double r5769425 = 0.5;
        double r5769426 = eps;
        double r5769427 = r5769425 * r5769426;
        double r5769428 = cos(r5769427);
        double r5769429 = r5769424 * r5769428;
        double r5769430 = sin(r5769427);
        double r5769431 = sin(r5769423);
        double r5769432 = r5769430 * r5769431;
        double r5769433 = r5769429 - r5769432;
        double r5769434 = r5769433 * r5769430;
        double r5769435 = r5769422 * r5769434;
        return r5769435;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.8
Target15.0
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.8

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

  3. Applied diff-sin37.2

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

  4. Simplified15.0

    \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}\]
  5. Using strategy rm

  6. Applied add-cbrt-cube15.2

    \[\leadsto 2 \cdot \left(\color{blue}{\sqrt[3]{\left(\cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)}} \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
  7. Using strategy rm

  8. Applied add-log-exp15.3

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

  9. Simplified15.2

    \[\leadsto 2 \cdot \left(\log \color{blue}{\left(e^{\cos \left(\frac{1}{2} \cdot \varepsilon + x\right)}\right)} \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
  10. Using strategy rm

  11. Applied cos-sum0.6

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

  12. Applied exp-diff0.6

    \[\leadsto 2 \cdot \left(\log \color{blue}{\left(\frac{e^{\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x}}{e^{\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x}}\right)} \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

  13. Applied log-div0.6

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

  14. Simplified0.4

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

  15. Simplified0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

  (- (sin (+ x eps)) (sin x)))