Average Error: 36.9 → 12.8
Time: 42.3s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\sin x}{\left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right) \cdot \cos x} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(\frac{\sin x}{\left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}\right) \cdot \cos x} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos x}}
double f(double x, double eps) {
        double r4789728 = x;
        double r4789729 = eps;
        double r4789730 = r4789728 + r4789729;
        double r4789731 = tan(r4789730);
        double r4789732 = tan(r4789728);
        double r4789733 = r4789731 - r4789732;
        return r4789733;
}
↓
double f(double x, double eps) {
        double r4789734 = x;
        double r4789735 = sin(r4789734);
        double r4789736 = 1.0;
        double r4789737 = eps;
        double r4789738 = sin(r4789737);
        double r4789739 = r4789735 * r4789738;
        double r4789740 = cos(r4789737);
        double r4789741 = cos(r4789734);
        double r4789742 = r4789740 * r4789741;
        double r4789743 = r4789739 / r4789742;
        double r4789744 = r4789736 - r4789743;
        double r4789745 = r4789744 * r4789741;
        double r4789746 = r4789735 / r4789745;
        double r4789747 = r4789735 / r4789741;
        double r4789748 = r4789746 - r4789747;
        double r4789749 = r4789738 / r4789740;
        double r4789750 = r4789749 / r4789744;
        double r4789751 = r4789748 + r4789750;
        return r4789751;
}

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
Herbie12.8
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.9

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

  3. Applied tan-sum21.8

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

  4. Taylor expanded around -inf 21.9

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

  5. Simplified12.8

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

  7. Applied add-cbrt-cube12.8

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

  8. Applied add-cbrt-cube12.8

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

  9. Applied cbrt-unprod12.8

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

  10. Simplified12.8

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

  11. Taylor expanded around -inf 12.8

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

  12. Final simplification12.8

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

Reproduce

herbie shell --seed 2019144 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

  (- (tan (+ x eps)) (tan x)))