Average Error: 37.1 → 0.4
Time: 25.7s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\frac{{\left(\sin x\right)}^{2}}{\cos x} + \cos x\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\frac{{\left(\sin x\right)}^{2}}{\cos x} + \cos x\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}
double f(double x, double eps) {
        double r118083 = x;
        double r118084 = eps;
        double r118085 = r118083 + r118084;
        double r118086 = tan(r118085);
        double r118087 = tan(r118083);
        double r118088 = r118086 - r118087;
        return r118088;
}
↓
double f(double x, double eps) {
        double r118089 = eps;
        double r118090 = sin(r118089);
        double r118091 = cos(r118089);
        double r118092 = r118090 / r118091;
        double r118093 = x;
        double r118094 = sin(r118093);
        double r118095 = 2.0;
        double r118096 = pow(r118094, r118095);
        double r118097 = cos(r118093);
        double r118098 = r118096 / r118097;
        double r118099 = r118098 + r118097;
        double r118100 = r118092 * r118099;
        double r118101 = 1.0;
        double r118102 = tan(r118093);
        double r118103 = tan(r118089);
        double r118104 = r118102 * r118103;
        double r118105 = r118101 - r118104;
        double r118106 = r118105 * r118097;
        double r118107 = r118100 / r118106;
        return r118107;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target15.0
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.1

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

  3. Applied tan-quot37.0

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

  4. Applied tan-sum22.1

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

  5. Applied frac-sub22.1

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

  6. Simplified22.1

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

  7. Taylor expanded around inf 0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

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

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