Average Error: 37.6 → 0.5
Time: 26.9s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\sin \varepsilon \cdot \frac{\cos x}{\cos \varepsilon} + \frac{\frac{1 - \cos \left(x + x\right)}{2}}{\cos x \cdot \cos \varepsilon} \cdot \sin \varepsilon}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\sin \varepsilon \cdot \frac{\cos x}{\cos \varepsilon} + \frac{\frac{1 - \cos \left(x + x\right)}{2}}{\cos x \cdot \cos \varepsilon} \cdot \sin \varepsilon}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}
double f(double x, double eps) {
        double r1582096 = x;
        double r1582097 = eps;
        double r1582098 = r1582096 + r1582097;
        double r1582099 = tan(r1582098);
        double r1582100 = tan(r1582096);
        double r1582101 = r1582099 - r1582100;
        return r1582101;
}
↓
double f(double x, double eps) {
        double r1582102 = eps;
        double r1582103 = sin(r1582102);
        double r1582104 = x;
        double r1582105 = cos(r1582104);
        double r1582106 = cos(r1582102);
        double r1582107 = r1582105 / r1582106;
        double r1582108 = r1582103 * r1582107;
        double r1582109 = 1.0;
        double r1582110 = r1582104 + r1582104;
        double r1582111 = cos(r1582110);
        double r1582112 = r1582109 - r1582111;
        double r1582113 = 2.0;
        double r1582114 = r1582112 / r1582113;
        double r1582115 = r1582105 * r1582106;
        double r1582116 = r1582114 / r1582115;
        double r1582117 = r1582116 * r1582103;
        double r1582118 = r1582108 + r1582117;
        double r1582119 = tan(r1582102);
        double r1582120 = tan(r1582104);
        double r1582121 = r1582119 * r1582120;
        double r1582122 = r1582109 - r1582121;
        double r1582123 = r1582122 * r1582105;
        double r1582124 = r1582118 / r1582123;
        return r1582124;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.6
Target15.3
Herbie0.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.6

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

  3. Applied tan-quot37.6

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

  4. Applied tan-sum22.3

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

  5. Applied frac-sub22.3

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

  6. Taylor expanded around -inf 0.4

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

  7. Simplified0.4

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

  9. Applied sin-mult0.5

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

  10. Simplified0.5

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

  11. Final simplification0.5

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

Reproduce

herbie shell --seed 2019128 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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