Average Error: 30.2 → 7.8
Time: 18.9s
Precision: 64
$\tan \left(\tan^{-1} x + z\right)$
$\begin{array}{l} \mathbf{if}\;z \le -5.127387873861141807934827997447987211596 \cdot 10^{-23} \lor \neg \left(z \le 4.291367477449918134219884650765870230149 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{\tan \left(\tan^{-1} x\right) + \tan z}{1 - \left(\tan \left(\tan^{-1} x\right) \cdot \tan z\right) \cdot \left(\tan \left(\tan^{-1} x\right) \cdot \tan z\right)} \cdot \left(1 + \tan \left(\tan^{-1} x\right) \cdot \tan z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \tan z}{1 - \tan \left(\tan^{-1} x\right) \cdot \tan z}\\ \end{array}$
\tan \left(\tan^{-1} x + z\right)
\begin{array}{l}
\mathbf{if}\;z \le -5.127387873861141807934827997447987211596 \cdot 10^{-23} \lor \neg \left(z \le 4.291367477449918134219884650765870230149 \cdot 10^{-45}\right):\\
\;\;\;\;\frac{\tan \left(\tan^{-1} x\right) + \tan z}{1 - \left(\tan \left(\tan^{-1} x\right) \cdot \tan z\right) \cdot \left(\tan \left(\tan^{-1} x\right) \cdot \tan z\right)} \cdot \left(1 + \tan \left(\tan^{-1} x\right) \cdot \tan z\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \tan z}{1 - \tan \left(\tan^{-1} x\right) \cdot \tan z}\\

\end{array}
double f(double x, double z) {
double r1263490 = x;
double r1263491 = atan(r1263490);
double r1263492 = z;
double r1263493 = r1263491 + r1263492;
double r1263494 = tan(r1263493);
return r1263494;
}


double f(double x, double z) {
double r1263495 = z;
double r1263496 = -5.127387873861142e-23;
bool r1263497 = r1263495 <= r1263496;
double r1263498 = 4.291367477449918e-45;
bool r1263499 = r1263495 <= r1263498;
double r1263500 = !r1263499;
bool r1263501 = r1263497 || r1263500;
double r1263502 = x;
double r1263503 = atan(r1263502);
double r1263504 = tan(r1263503);
double r1263505 = tan(r1263495);
double r1263506 = r1263504 + r1263505;
double r1263507 = 1.0;
double r1263508 = r1263504 * r1263505;
double r1263509 = r1263508 * r1263508;
double r1263510 = r1263507 - r1263509;
double r1263511 = r1263506 / r1263510;
double r1263512 = r1263507 + r1263508;
double r1263513 = r1263511 * r1263512;
double r1263514 = r1263502 + r1263505;
double r1263515 = r1263507 - r1263508;
double r1263516 = r1263514 / r1263515;
double r1263517 = r1263501 ? r1263513 : r1263516;
return r1263517;
}



# Try it out

Results

 In Out
Enter valid numbers for all inputs

# Derivation

1. Split input into 2 regimes
2. ## if z < -5.127387873861142e-23 or 4.291367477449918e-45 < z

1. Initial program 30.4

$\tan \left(\tan^{-1} x + z\right)$
2. Using strategy rm
3. Applied tan-sum2.4

$\leadsto \color{blue}{\frac{\tan \left(\tan^{-1} x\right) + \tan z}{1 - \tan \left(\tan^{-1} x\right) \cdot \tan z}}$
4. Using strategy rm
5. Applied flip--2.4

$\leadsto \frac{\tan \left(\tan^{-1} x\right) + \tan z}{\color{blue}{\frac{1 \cdot 1 - \left(\tan \left(\tan^{-1} x\right) \cdot \tan z\right) \cdot \left(\tan \left(\tan^{-1} x\right) \cdot \tan z\right)}{1 + \tan \left(\tan^{-1} x\right) \cdot \tan z}}}$
6. Applied associate-/r/2.4

$\leadsto \color{blue}{\frac{\tan \left(\tan^{-1} x\right) + \tan z}{1 \cdot 1 - \left(\tan \left(\tan^{-1} x\right) \cdot \tan z\right) \cdot \left(\tan \left(\tan^{-1} x\right) \cdot \tan z\right)} \cdot \left(1 + \tan \left(\tan^{-1} x\right) \cdot \tan z\right)}$
7. Simplified2.4

$\leadsto \color{blue}{\frac{\tan \left(\tan^{-1} x\right) + \tan z}{1 - \left(\tan \left(\tan^{-1} x\right) \cdot \tan z\right) \cdot \left(\tan \left(\tan^{-1} x\right) \cdot \tan z\right)}} \cdot \left(1 + \tan \left(\tan^{-1} x\right) \cdot \tan z\right)$

## if -5.127387873861142e-23 < z < 4.291367477449918e-45

1. Initial program 30.1

$\tan \left(\tan^{-1} x + z\right)$
2. Using strategy rm
3. Applied tan-sum30.1

$\leadsto \color{blue}{\frac{\tan \left(\tan^{-1} x\right) + \tan z}{1 - \tan \left(\tan^{-1} x\right) \cdot \tan z}}$
4. Using strategy rm
5. Applied tan-atan14.3

$\leadsto \frac{\color{blue}{x} + \tan z}{1 - \tan \left(\tan^{-1} x\right) \cdot \tan z}$
3. Recombined 2 regimes into one program.
4. Final simplification7.8

$\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.127387873861141807934827997447987211596 \cdot 10^{-23} \lor \neg \left(z \le 4.291367477449918134219884650765870230149 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{\tan \left(\tan^{-1} x\right) + \tan z}{1 - \left(\tan \left(\tan^{-1} x\right) \cdot \tan z\right) \cdot \left(\tan \left(\tan^{-1} x\right) \cdot \tan z\right)} \cdot \left(1 + \tan \left(\tan^{-1} x\right) \cdot \tan z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \tan z}{1 - \tan \left(\tan^{-1} x\right) \cdot \tan z}\\ \end{array}$

# Reproduce

herbie shell --seed 1
(FPCore (x z)
:name "tan(atan(x)+z)"
:precision binary64
(tan (+ (atan x) z)))