Average Error: 36.9 → 0.6
Time: 17.7s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\cos x \cdot \sin \varepsilon + \left(\log \left(\sqrt{{\left(e^{\sin x}\right)}^{\left(\cos \varepsilon - 1\right)}}\right) + \log \left(\sqrt{{\left(e^{\sin x}\right)}^{\left(\cos \varepsilon - 1\right)}}\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\cos x \cdot \sin \varepsilon + \left(\log \left(\sqrt{{\left(e^{\sin x}\right)}^{\left(\cos \varepsilon - 1\right)}}\right) + \log \left(\sqrt{{\left(e^{\sin x}\right)}^{\left(\cos \varepsilon - 1\right)}}\right)\right)
double f(double x, double eps) {
        double r97643 = x;
        double r97644 = eps;
        double r97645 = r97643 + r97644;
        double r97646 = sin(r97645);
        double r97647 = sin(r97643);
        double r97648 = r97646 - r97647;
        return r97648;
}
↓
double f(double x, double eps) {
        double r97649 = x;
        double r97650 = cos(r97649);
        double r97651 = eps;
        double r97652 = sin(r97651);
        double r97653 = r97650 * r97652;
        double r97654 = sin(r97649);
        double r97655 = exp(r97654);
        double r97656 = cos(r97651);
        double r97657 = 1.0;
        double r97658 = r97656 - r97657;
        double r97659 = pow(r97655, r97658);
        double r97660 = sqrt(r97659);
        double r97661 = log(r97660);
        double r97662 = r97661 + r97661;
        double r97663 = r97653 + r97662;
        return r97663;
}

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

Derivation

  1. Initial program 36.9

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

  2. Simplified36.9

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

  4. Applied sin-sum21.6

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

  5. Applied associate--l+0.4

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

  7. Applied add-log-exp14.5

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

  8. Applied add-log-exp0.5

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

  9. Applied diff-log0.5

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

  10. Simplified0.5

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

  12. Applied add-sqr-sqrt0.6

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

  13. Applied log-prod0.6

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

  14. Simplified0.6

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

  15. Simplified0.6

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

  16. Final simplification0.6

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

Reproduce

herbie shell --seed 2019174 
(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)))