\[\tan \left(x + \varepsilon\right) - \tan x\]

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -5.547819297283335504428332134194072043683 \cdot 10^{-27}:\\ \;\;\;\;\frac{\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 3.432024878078755517709005585063122603554 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \varepsilon, x + \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\tan \varepsilon\right)}^{3} + {\left(\tan x\right)}^{3}}{\mathsf{fma}\left(\tan \varepsilon - \tan x, \tan \varepsilon, \tan x \cdot \tan x\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} - \tan x\\ \end{array}\]

\tan \left(x + \varepsilon\right) - \tan x

↓

\begin{array}{l}
\mathbf{if}\;\varepsilon \le -5.547819297283335504428332134194072043683 \cdot 10^{-27}:\\
\;\;\;\;\frac{\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\

\mathbf{elif}\;\varepsilon \le 3.432024878078755517709005585063122603554 \cdot 10^{-26}:\\
\;\;\;\;\mathsf{fma}\left(x \cdot \varepsilon, x + \varepsilon, \varepsilon\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{{\left(\tan \varepsilon\right)}^{3} + {\left(\tan x\right)}^{3}}{\mathsf{fma}\left(\tan \varepsilon - \tan x, \tan \varepsilon, \tan x \cdot \tan x\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} - \tan x\\

\end{array}

double f(double x, double eps) {
        double r5544906 = x;
        double r5544907 = eps;
        double r5544908 = r5544906 + r5544907;
        double r5544909 = tan(r5544908);
        double r5544910 = tan(r5544906);
        double r5544911 = r5544909 - r5544910;
        return r5544911;
}

↓

double f(double x, double eps) {
        double r5544912 = eps;
        double r5544913 = -5.5478192972833355e-27;
        bool r5544914 = r5544912 <= r5544913;
        double r5544915 = x;
        double r5544916 = cos(r5544915);
        double r5544917 = tan(r5544912);
        double r5544918 = tan(r5544915);
        double r5544919 = r5544917 + r5544918;
        double r5544920 = r5544916 * r5544919;
        double r5544921 = 1.0;
        double r5544922 = r5544918 * r5544917;
        double r5544923 = r5544921 - r5544922;
        double r5544924 = sin(r5544915);
        double r5544925 = r5544923 * r5544924;
        double r5544926 = r5544920 - r5544925;
        double r5544927 = r5544923 * r5544916;
        double r5544928 = r5544926 / r5544927;
        double r5544929 = 3.4320248780787555e-26;
        bool r5544930 = r5544912 <= r5544929;
        double r5544931 = r5544915 * r5544912;
        double r5544932 = r5544915 + r5544912;
        double r5544933 = fma(r5544931, r5544932, r5544912);
        double r5544934 = 3.0;
        double r5544935 = pow(r5544917, r5544934);
        double r5544936 = pow(r5544918, r5544934);
        double r5544937 = r5544935 + r5544936;
        double r5544938 = r5544917 - r5544918;
        double r5544939 = r5544918 * r5544918;
        double r5544940 = fma(r5544938, r5544917, r5544939);
        double r5544941 = r5544940 * r5544923;
        double r5544942 = r5544937 / r5544941;
        double r5544943 = r5544942 - r5544918;
        double r5544944 = r5544930 ? r5544933 : r5544943;
        double r5544945 = r5544914 ? r5544928 : r5544944;
        return r5544945;
}

Original	36.8
Target	15.1
Herbie	15.4

Derivation

Split input into 3 regimes
if eps < -5.5478192972833355e-27
1. Initial program 29.4
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Using strategy rm
3. Applied tan-quot29.2
  \[\leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
4. Applied tan-sum1.6
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \frac{\sin x}{\cos x}\]
5. Applied frac-sub1.7
  \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}}\]
if -5.5478192972833355e-27 < eps < 3.4320248780787555e-26
1. Initial program 45.1
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Taylor expanded around 0 31.4
  \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\varepsilon + {x}^{2} \cdot \varepsilon\right)}\]
3. Simplified31.2
  \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \varepsilon, \varepsilon + x, \varepsilon\right)}\]
if 3.4320248780787555e-26 < eps
1. Initial program 30.1
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Using strategy rm
3. Applied tan-sum1.9
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
4. Using strategy rm
5. Applied flip3-+2.1
  \[\leadsto \frac{\color{blue}{\frac{{\left(\tan x\right)}^{3} + {\left(\tan \varepsilon\right)}^{3}}{\tan x \cdot \tan x + \left(\tan \varepsilon \cdot \tan \varepsilon - \tan x \cdot \tan \varepsilon\right)}}}{1 - \tan x \cdot \tan \varepsilon} - \tan x\]
6. Applied associate-/l/2.1
  \[\leadsto \color{blue}{\frac{{\left(\tan x\right)}^{3} + {\left(\tan \varepsilon\right)}^{3}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan x + \left(\tan \varepsilon \cdot \tan \varepsilon - \tan x \cdot \tan \varepsilon\right)\right)}} - \tan x\]
7. Simplified2.1
  \[\leadsto \frac{{\left(\tan x\right)}^{3} + {\left(\tan \varepsilon\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\tan \varepsilon - \tan x, \tan \varepsilon, \tan x \cdot \tan x\right) \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}} - \tan x\]
Recombined 3 regimes into one program.
Final simplification15.4
\[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.547819297283335504428332134194072043683 \cdot 10^{-27}:\\ \;\;\;\;\frac{\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 3.432024878078755517709005585063122603554 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \varepsilon, x + \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\tan \varepsilon\right)}^{3} + {\left(\tan x\right)}^{3}}{\mathsf{fma}\left(\tan \varepsilon - \tan x, \tan \varepsilon, \tan x \cdot \tan x\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} - \tan x\\ \end{array}\]

Error

Target

Derivation

`if eps < -5.5478192972833355e-27`

`if -5.5478192972833355e-27 < eps < 3.4320248780787555e-26`

`if 3.4320248780787555e-26 < eps`

Reproduce