Average Error: 37.0 → 13.4
Time: 29.0s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin x}{\cos x}}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \left(\frac{\sin x}{\cos x} - \frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin x}{\cos x}}{\frac{\cos \varepsilon}{\sin \varepsilon}}}\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin x}{\cos x}}{\frac{\cos \varepsilon}{\sin \varepsilon}}} - \left(\frac{\sin x}{\cos x} - \frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin x}{\cos x}}{\frac{\cos \varepsilon}{\sin \varepsilon}}}\right)
double f(double x, double eps) {
        double r5488628 = x;
        double r5488629 = eps;
        double r5488630 = r5488628 + r5488629;
        double r5488631 = tan(r5488630);
        double r5488632 = tan(r5488628);
        double r5488633 = r5488631 - r5488632;
        return r5488633;
}
↓
double f(double x, double eps) {
        double r5488634 = eps;
        double r5488635 = sin(r5488634);
        double r5488636 = cos(r5488634);
        double r5488637 = r5488635 / r5488636;
        double r5488638 = 1.0;
        double r5488639 = x;
        double r5488640 = sin(r5488639);
        double r5488641 = cos(r5488639);
        double r5488642 = r5488640 / r5488641;
        double r5488643 = r5488636 / r5488635;
        double r5488644 = r5488642 / r5488643;
        double r5488645 = r5488638 - r5488644;
        double r5488646 = r5488637 / r5488645;
        double r5488647 = r5488642 / r5488645;
        double r5488648 = r5488642 - r5488647;
        double r5488649 = r5488646 - r5488648;
        return r5488649;
}

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
Target14.6
Herbie13.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.0

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

  3. Applied tan-sum22.5

    \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
  4. Using strategy rm

  5. Applied add-cube-cbrt22.6

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

  6. Applied associate-*l*22.6

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

  7. Taylor expanded around inf 22.6

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

  8. Simplified22.6

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

  9. Taylor expanded around inf 22.6

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

  10. Simplified13.4

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

  11. Final simplification13.4

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

Reproduce

herbie shell --seed 2019164 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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