Average Error: 37.1 → 0.4
Time: 10.3s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon} + \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}
double f(double x, double eps) {
        double r114670 = x;
        double r114671 = eps;
        double r114672 = r114670 + r114671;
        double r114673 = tan(r114672);
        double r114674 = tan(r114670);
        double r114675 = r114673 - r114674;
        return r114675;
}
↓
double f(double x, double eps) {
        double r114676 = eps;
        double r114677 = sin(r114676);
        double r114678 = x;
        double r114679 = cos(r114678);
        double r114680 = r114677 * r114679;
        double r114681 = cos(r114676);
        double r114682 = r114680 / r114681;
        double r114683 = sin(r114678);
        double r114684 = 2.0;
        double r114685 = pow(r114683, r114684);
        double r114686 = r114685 * r114677;
        double r114687 = r114679 * r114681;
        double r114688 = r114686 / r114687;
        double r114689 = r114682 + r114688;
        double r114690 = 1.0;
        double r114691 = tan(r114678);
        double r114692 = tan(r114676);
        double r114693 = r114691 * r114692;
        double r114694 = r114690 - r114693;
        double r114695 = r114694 * r114679;
        double r114696 = r114689 / r114695;
        return r114696;
}

Error

Bits error versus x

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target15.0
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.1

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

  3. Applied tan-quot37.1

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

  4. Applied tan-sum22.1

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

  5. Applied frac-sub22.1

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

  6. Taylor expanded around inf 0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019347 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

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

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