Average Error: 30.0 → 0.9
Time: 13.8s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le \frac{-3517324383654103}{72057594037927936}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le \frac{4505605101079521}{590295810358705651712}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sin x} \cdot \left(\sqrt[3]{1} \cdot \left(1 - \cos x\right)\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le \frac{-3517324383654103}{72057594037927936}:\\
\;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{\sin x}\\

\mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le \frac{4505605101079521}{590295810358705651712}:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sin x} \cdot \left(\sqrt[3]{1} \cdot \left(1 - \cos x\right)\right)\\

\end{array}
double f(double x) {
        double r47295 = 1.0;
        double r47296 = x;
        double r47297 = cos(r47296);
        double r47298 = r47295 - r47297;
        double r47299 = sin(r47296);
        double r47300 = r47298 / r47299;
        return r47300;
}
↓
double f(double x) {
        double r47301 = 1.0;
        double r47302 = x;
        double r47303 = cos(r47302);
        double r47304 = r47301 - r47303;
        double r47305 = sin(r47302);
        double r47306 = r47304 / r47305;
        double r47307 = -3517324383654103.0;
        double r47308 = 7.205759403792794e+16;
        double r47309 = r47307 / r47308;
        bool r47310 = r47306 <= r47309;
        double r47311 = 3.0;
        double r47312 = pow(r47301, r47311);
        double r47313 = pow(r47303, r47311);
        double r47314 = r47312 - r47313;
        double r47315 = r47303 + r47301;
        double r47316 = r47303 * r47315;
        double r47317 = r47301 * r47301;
        double r47318 = r47316 + r47317;
        double r47319 = r47314 / r47318;
        double r47320 = r47319 / r47305;
        double r47321 = 4505605101079521.0;
        double r47322 = 5.902958103587057e+20;
        double r47323 = r47321 / r47322;
        bool r47324 = r47306 <= r47323;
        double r47325 = 0.041666666666666664;
        double r47326 = pow(r47302, r47311);
        double r47327 = r47325 * r47326;
        double r47328 = 0.004166666666666667;
        double r47329 = 5.0;
        double r47330 = pow(r47302, r47329);
        double r47331 = r47328 * r47330;
        double r47332 = 0.5;
        double r47333 = r47332 * r47302;
        double r47334 = r47331 + r47333;
        double r47335 = r47327 + r47334;
        double r47336 = 1.0;
        double r47337 = cbrt(r47336);
        double r47338 = r47337 * r47337;
        double r47339 = r47338 / r47305;
        double r47340 = r47337 * r47304;
        double r47341 = r47339 * r47340;
        double r47342 = r47324 ? r47335 : r47341;
        double r47343 = r47310 ? r47320 : r47342;
        return r47343;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.0
Target0
Herbie0.9
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.04881268144760341

    1. Initial program 0.7

      \[\frac{1 - \cos x}{\sin x}\]
    2. Using strategy rm

    3. Applied flip3--0.8

      \[\leadsto \frac{\color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}}{\sin x}\]

    4. Simplified0.8

      \[\leadsto \frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\color{blue}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}}{\sin x}\]

    if -0.04881268144760341 < (/ (- 1.0 (cos x)) (sin x)) < 7.632791935862793e-06

    1. Initial program 59.3

      \[\frac{1 - \cos x}{\sin x}\]

    2. Taylor expanded around 0 0.9

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)}\]

    if 7.632791935862793e-06 < (/ (- 1.0 (cos x)) (sin x))

    1. Initial program 1.1

      \[\frac{1 - \cos x}{\sin x}\]
    2. Using strategy rm

    3. Applied clear-num1.1

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{1 - \cos x}}}\]
    4. Using strategy rm

    5. Applied div-inv1.2

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

    6. Applied add-cube-cbrt1.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sin x \cdot \frac{1}{1 - \cos x}}\]

    7. Applied times-frac1.2

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sin x} \cdot \frac{\sqrt[3]{1}}{\frac{1}{1 - \cos x}}}\]

    8. Simplified1.2

      \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sin x} \cdot \color{blue}{\left(\sqrt[3]{1} \cdot \left(1 - \cos x\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le \frac{-3517324383654103}{72057594037927936}:\\ \;\;\;\;\frac{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le \frac{4505605101079521}{590295810358705651712}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sin x} \cdot \left(\sqrt[3]{1} \cdot \left(1 - \cos x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 197574269 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2))

  (/ (- 1 (cos x)) (sin x)))