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

↓

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

\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 r5545330 = x;
        double r5545331 = eps;
        double r5545332 = r5545330 + r5545331;
        double r5545333 = tan(r5545332);
        double r5545334 = tan(r5545330);
        double r5545335 = r5545333 - r5545334;
        return r5545335;
}

↓

double f(double x, double eps) {
        double r5545336 = eps;
        double r5545337 = -5.5478192972833355e-27;
        bool r5545338 = r5545336 <= r5545337;
        double r5545339 = x;
        double r5545340 = cos(r5545339);
        double r5545341 = tan(r5545336);
        double r5545342 = tan(r5545339);
        double r5545343 = r5545341 + r5545342;
        double r5545344 = r5545340 * r5545343;
        double r5545345 = 1.0;
        double r5545346 = r5545342 * r5545341;
        double r5545347 = r5545345 - r5545346;
        double r5545348 = sin(r5545339);
        double r5545349 = r5545347 * r5545348;
        double r5545350 = r5545344 - r5545349;
        double r5545351 = r5545347 * r5545340;
        double r5545352 = r5545350 / r5545351;
        double r5545353 = 3.4320248780787555e-26;
        bool r5545354 = r5545336 <= r5545353;
        double r5545355 = r5545339 * r5545336;
        double r5545356 = r5545339 + r5545336;
        double r5545357 = fma(r5545355, r5545356, r5545336);
        double r5545358 = 3.0;
        double r5545359 = pow(r5545341, r5545358);
        double r5545360 = pow(r5545342, r5545358);
        double r5545361 = r5545359 + r5545360;
        double r5545362 = r5545341 - r5545342;
        double r5545363 = r5545342 * r5545342;
        double r5545364 = fma(r5545362, r5545341, r5545363);
        double r5545365 = r5545364 * r5545347;
        double r5545366 = r5545361 / r5545365;
        double r5545367 = r5545366 - r5545342;
        double r5545368 = r5545354 ? r5545357 : r5545367;
        double r5545369 = r5545338 ? r5545352 : r5545368;
        return r5545369;
}

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