Average Error: 36.6 → 12.9
Time: 44.9s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sqrt[3]{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x\right) \cdot \left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x\right)\right)}}{\cos x}} + \mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{1}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}}, \frac{-\sin x}{\cos x}\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sqrt[3]{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x\right) \cdot \left(\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x\right)\right)}}{\cos x}} + \mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{1}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}}, \frac{-\sin x}{\cos x}\right)
double f(double x, double eps) {
        double r2457657 = x;
        double r2457658 = eps;
        double r2457659 = r2457657 + r2457658;
        double r2457660 = tan(r2457659);
        double r2457661 = tan(r2457657);
        double r2457662 = r2457660 - r2457661;
        return r2457662;
}
↓
double f(double x, double eps) {
        double r2457663 = eps;
        double r2457664 = sin(r2457663);
        double r2457665 = cos(r2457663);
        double r2457666 = r2457664 / r2457665;
        double r2457667 = 1.0;
        double r2457668 = x;
        double r2457669 = sin(r2457668);
        double r2457670 = r2457666 * r2457669;
        double r2457671 = r2457670 * r2457670;
        double r2457672 = r2457670 * r2457671;
        double r2457673 = cbrt(r2457672);
        double r2457674 = cos(r2457668);
        double r2457675 = r2457673 / r2457674;
        double r2457676 = r2457667 - r2457675;
        double r2457677 = r2457666 / r2457676;
        double r2457678 = r2457669 / r2457674;
        double r2457679 = r2457670 / r2457674;
        double r2457680 = r2457667 - r2457679;
        double r2457681 = r2457667 / r2457680;
        double r2457682 = -r2457669;
        double r2457683 = r2457682 / r2457674;
        double r2457684 = fma(r2457678, r2457681, r2457683);
        double r2457685 = r2457677 + r2457684;
        return r2457685;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.6
Target14.6
Herbie12.9
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.6

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

  3. Applied tan-sum22.0

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

  4. Taylor expanded around inf 22.1

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

  5. Simplified12.9

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

  7. Applied add-cbrt-cube12.9

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

  9. Applied div-inv12.9

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

  10. Applied fma-neg12.9

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

  11. Final simplification12.9

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

Reproduce

herbie shell --seed 2019156 +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)))