Average Error: 37.0 → 12.7
Time: 32.8s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}
double f(double x, double eps) {
        double r2992068 = x;
        double r2992069 = eps;
        double r2992070 = r2992068 + r2992069;
        double r2992071 = tan(r2992070);
        double r2992072 = tan(r2992068);
        double r2992073 = r2992071 - r2992072;
        return r2992073;
}
↓
double f(double x, double eps) {
        double r2992074 = x;
        double r2992075 = sin(r2992074);
        double r2992076 = cos(r2992074);
        double r2992077 = r2992075 / r2992076;
        double r2992078 = 1.0;
        double r2992079 = eps;
        double r2992080 = sin(r2992079);
        double r2992081 = cos(r2992079);
        double r2992082 = r2992080 / r2992081;
        double r2992083 = r2992077 * r2992082;
        double r2992084 = r2992078 - r2992083;
        double r2992085 = r2992077 / r2992084;
        double r2992086 = r2992085 - r2992077;
        double r2992087 = r2992082 / r2992084;
        double r2992088 = r2992086 + r2992087;
        return r2992088;
}

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
Target15.4
Herbie12.7
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.0

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

  3. Applied tan-sum21.5

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

  5. Applied tan-quot21.5

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

  6. Applied associate-*r/21.5

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

  7. Taylor expanded around inf 21.6

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

  8. Simplified12.7

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

  9. Final simplification12.7

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

Reproduce

herbie shell --seed 2019152 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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