Average Error: 37.4 → 15.5
Time: 15.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.950013469467184 \cdot 10^{-38} \lor \neg \left(\varepsilon \le 2.8556285663292881 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.950013469467184 \cdot 10^{-38} \lor \neg \left(\varepsilon \le 2.8556285663292881 \cdot 10^{-18}\right):\\
\;\;\;\;\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}\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\

\end{array}
double f(double x, double eps) {
        double r125086 = x;
        double r125087 = eps;
        double r125088 = r125086 + r125087;
        double r125089 = tan(r125088);
        double r125090 = tan(r125086);
        double r125091 = r125089 - r125090;
        return r125091;
}
↓
double f(double x, double eps) {
        double r125092 = eps;
        double r125093 = -1.9500134694671836e-38;
        bool r125094 = r125092 <= r125093;
        double r125095 = 2.855628566329288e-18;
        bool r125096 = r125092 <= r125095;
        double r125097 = !r125096;
        bool r125098 = r125094 || r125097;
        double r125099 = x;
        double r125100 = tan(r125099);
        double r125101 = tan(r125092);
        double r125102 = r125100 + r125101;
        double r125103 = cos(r125099);
        double r125104 = r125102 * r125103;
        double r125105 = 1.0;
        double r125106 = r125100 * r125101;
        double r125107 = r125105 - r125106;
        double r125108 = sin(r125099);
        double r125109 = r125107 * r125108;
        double r125110 = r125104 - r125109;
        double r125111 = r125107 * r125103;
        double r125112 = r125110 / r125111;
        double r125113 = r125099 * r125092;
        double r125114 = r125092 + r125099;
        double r125115 = r125113 * r125114;
        double r125116 = r125115 + r125092;
        double r125117 = r125098 ? r125112 : r125116;
        return r125117;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.4
Target15.2
Herbie15.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.9500134694671836e-38 or 2.855628566329288e-18 < eps

    1. Initial program 30.2

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

    3. Applied tan-quot30.0

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

    4. Applied tan-sum2.2

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

    5. Applied frac-sub2.3

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

    if -1.9500134694671836e-38 < eps < 2.855628566329288e-18

    1. Initial program 45.9

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

    3. Applied tan-sum45.9

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

    4. Taylor expanded around 0 31.4

      \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\varepsilon + {x}^{2} \cdot \varepsilon\right)}\]

    5. Simplified31.2

      \[\leadsto \color{blue}{\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification15.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.950013469467184 \cdot 10^{-38} \lor \neg \left(\varepsilon \le 2.8556285663292881 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))