Average Error: 36.9 → 12.6
Time: 27.7s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(-\tan x, \frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} + \left(\frac{-\sin x}{\cos x} + \frac{\frac{\sin x}{\cos x}}{\mathsf{fma}\left(\frac{-\sin x}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(-\tan x, \frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} + \left(\frac{-\sin x}{\cos x} + \frac{\frac{\sin x}{\cos x}}{\mathsf{fma}\left(\frac{-\sin x}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}\right)
double f(double x, double eps) {
        double r4944659 = x;
        double r4944660 = eps;
        double r4944661 = r4944659 + r4944660;
        double r4944662 = tan(r4944661);
        double r4944663 = tan(r4944659);
        double r4944664 = r4944662 - r4944663;
        return r4944664;
}
↓
double f(double x, double eps) {
        double r4944665 = eps;
        double r4944666 = sin(r4944665);
        double r4944667 = cos(r4944665);
        double r4944668 = r4944666 / r4944667;
        double r4944669 = x;
        double r4944670 = tan(r4944669);
        double r4944671 = -r4944670;
        double r4944672 = 1.0;
        double r4944673 = fma(r4944671, r4944668, r4944672);
        double r4944674 = r4944668 / r4944673;
        double r4944675 = sin(r4944669);
        double r4944676 = -r4944675;
        double r4944677 = cos(r4944669);
        double r4944678 = r4944676 / r4944677;
        double r4944679 = r4944675 / r4944677;
        double r4944680 = fma(r4944678, r4944668, r4944672);
        double r4944681 = r4944679 / r4944680;
        double r4944682 = r4944678 + r4944681;
        double r4944683 = r4944674 + r4944682;
        return r4944683;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.9
Target15.1
Herbie12.6
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.9

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

  3. Applied tan-sum21.6

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

  4. Taylor expanded around inf 21.7

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

  5. Simplified12.6

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

  7. Applied quot-tan12.6

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

  8. Final simplification12.6

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(-\tan x, \frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} + \left(\frac{-\sin x}{\cos x} + \frac{\frac{\sin x}{\cos x}}{\mathsf{fma}\left(\frac{-\sin x}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}\right)\]

Reproduce

herbie shell --seed 2019192 +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)))