Average Error: 36.1 → 12.8
Time: 48.3s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin x \cdot \sin \varepsilon}{\cos x}}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) + \frac{1}{\frac{\cos \varepsilon \cdot \left(1 - \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)}{\sin \varepsilon}}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\frac{\sin x \cdot \sin \varepsilon}{\cos x}}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) + \frac{1}{\frac{\cos \varepsilon \cdot \left(1 - \frac{\frac{\sin \varepsilon}{\cos x} \cdot \sin x}{\cos \varepsilon}\right)}{\sin \varepsilon}}
double f(double x, double eps) {
        double r2976647 = x;
        double r2976648 = eps;
        double r2976649 = r2976647 + r2976648;
        double r2976650 = tan(r2976649);
        double r2976651 = tan(r2976647);
        double r2976652 = r2976650 - r2976651;
        return r2976652;
}
↓
double f(double x, double eps) {
        double r2976653 = x;
        double r2976654 = sin(r2976653);
        double r2976655 = cos(r2976653);
        double r2976656 = r2976654 / r2976655;
        double r2976657 = 1.0;
        double r2976658 = eps;
        double r2976659 = sin(r2976658);
        double r2976660 = r2976654 * r2976659;
        double r2976661 = r2976660 / r2976655;
        double r2976662 = cos(r2976658);
        double r2976663 = r2976661 / r2976662;
        double r2976664 = r2976657 - r2976663;
        double r2976665 = r2976656 / r2976664;
        double r2976666 = r2976665 - r2976656;
        double r2976667 = r2976659 / r2976655;
        double r2976668 = r2976667 * r2976654;
        double r2976669 = r2976668 / r2976662;
        double r2976670 = r2976657 - r2976669;
        double r2976671 = r2976662 * r2976670;
        double r2976672 = r2976671 / r2976659;
        double r2976673 = r2976657 / r2976672;
        double r2976674 = r2976666 + r2976673;
        return r2976674;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.1
Target14.3
Herbie12.8
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.1

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

  3. Applied tan-sum21.7

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

  4. Taylor expanded around inf 21.8

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

  5. Simplified12.7

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

  7. Applied clear-num12.8

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

  8. Taylor expanded around inf 12.8

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

  9. Final simplification12.8

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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