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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.063411921076795502138845593884088192778 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \le 7.60970569110987615975455365802284218756 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(x, {\varepsilon}^{2}, \mathsf{fma}\left({x}^{2}, \varepsilon, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \cdot \tan x}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.063411921076795502138845593884088192778 \cdot 10^{-69}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)\\

\mathbf{elif}\;\varepsilon \le 7.60970569110987615975455365802284218756 \cdot 10^{-27}:\\
\;\;\;\;\mathsf{fma}\left(x, {\varepsilon}^{2}, \mathsf{fma}\left({x}^{2}, \varepsilon, \varepsilon\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \cdot \tan x}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\

\end{array}

double f(double x, double eps) {
        double r94275 = x;
        double r94276 = eps;
        double r94277 = r94275 + r94276;
        double r94278 = tan(r94277);
        double r94279 = tan(r94275);
        double r94280 = r94278 - r94279;
        return r94280;
}

↓

double f(double x, double eps) {
        double r94281 = eps;
        double r94282 = -1.0634119210767955e-69;
        bool r94283 = r94281 <= r94282;
        double r94284 = x;
        double r94285 = tan(r94284);
        double r94286 = tan(r94281);
        double r94287 = r94285 + r94286;
        double r94288 = 1.0;
        double r94289 = r94285 * r94286;
        double r94290 = sin(r94281);
        double r94291 = r94285 * r94290;
        double r94292 = cos(r94281);
        double r94293 = r94291 / r94292;
        double r94294 = r94289 * r94293;
        double r94295 = r94288 - r94294;
        double r94296 = r94287 / r94295;
        double r94297 = r94288 + r94289;
        double r94298 = -r94285;
        double r94299 = fma(r94296, r94297, r94298);
        double r94300 = 7.609705691109876e-27;
        bool r94301 = r94281 <= r94300;
        double r94302 = 2.0;
        double r94303 = pow(r94281, r94302);
        double r94304 = pow(r94284, r94302);
        double r94305 = fma(r94304, r94281, r94281);
        double r94306 = fma(r94284, r94303, r94305);
        double r94307 = r94288 - r94289;
        double r94308 = r94287 / r94307;
        double r94309 = r94308 * r94308;
        double r94310 = r94285 * r94285;
        double r94311 = r94309 - r94310;
        double r94312 = r94308 + r94285;
        double r94313 = r94311 / r94312;
        double r94314 = r94301 ? r94306 : r94313;
        double r94315 = r94283 ? r94299 : r94314;
        return r94315;
}

Original	37.1
Target	15.5
Herbie	15.1

Derivation

Split input into 3 regimes
if eps < -1.0634119210767955e-69
1. Initial program 30.6
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Using strategy rm
3. Applied tan-sum5.4
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
4. Using strategy rm
5. Applied flip--5.5
  \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 + \tan x \cdot \tan \varepsilon}}} - \tan x\]
6. Applied associate-/r/5.5
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)} - \tan x\]
7. Applied fma-neg5.5
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)}\]
8. Using strategy rm
9. Applied tan-quot5.5
  \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}\right)}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)\]
10. Applied associate-*r/5.5
  \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \color{blue}{\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)\]
if -1.0634119210767955e-69 < eps < 7.609705691109876e-27
1. Initial program 46.6
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Using strategy rm
3. Applied tan-sum46.6
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
4. Using strategy rm
5. Applied flip--46.6
  \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 + \tan x \cdot \tan \varepsilon}}} - \tan x\]
6. Applied associate-/r/46.6
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)} - \tan x\]
7. Applied fma-neg46.6
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)}\]
8. Taylor expanded around 0 30.8
  \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\varepsilon + {x}^{2} \cdot \varepsilon\right)}\]
9. Simplified30.8
  \[\leadsto \color{blue}{\mathsf{fma}\left(x, {\varepsilon}^{2}, \mathsf{fma}\left({x}^{2}, \varepsilon, \varepsilon\right)\right)}\]
if 7.609705691109876e-27 < eps
1. Initial program 30.3
  \[\tan \left(x + \varepsilon\right) - \tan x\]
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 flip--2.1
  \[\leadsto \color{blue}{\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \cdot \tan x}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}}\]
Recombined 3 regimes into one program.
Final simplification15.1
\[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.063411921076795502138845593884088192778 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \le 7.60970569110987615975455365802284218756 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(x, {\varepsilon}^{2}, \mathsf{fma}\left({x}^{2}, \varepsilon, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \cdot \tan x}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x}\\ \end{array}\]

Error

Target

Derivation

`if eps < -1.0634119210767955e-69`

`if -1.0634119210767955e-69 < eps < 7.609705691109876e-27`

`if 7.609705691109876e-27 < eps`

Reproduce