Average Error: 30.2 → 0.5
Time: 18.2s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02175733191988275458395030170777317835018 \lor \neg \left(x \le 0.02301026064809699039903634343318117316812\right):\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \frac{\cos x \cdot \cos x - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02175733191988275458395030170777317835018 \lor \neg \left(x \le 0.02301026064809699039903634343318117316812\right):\\
\;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \frac{\cos x \cdot \cos x - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1\right) \cdot \sin x}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

\end{array}
double f(double x) {
        double r35222 = 1.0;
        double r35223 = x;
        double r35224 = cos(r35223);
        double r35225 = r35222 - r35224;
        double r35226 = sin(r35223);
        double r35227 = r35225 / r35226;
        return r35227;
}
↓
double f(double x) {
        double r35228 = x;
        double r35229 = -0.021757331919882755;
        bool r35230 = r35228 <= r35229;
        double r35231 = 0.02301026064809699;
        bool r35232 = r35228 <= r35231;
        double r35233 = !r35232;
        bool r35234 = r35230 || r35233;
        double r35235 = 1.0;
        double r35236 = 3.0;
        double r35237 = pow(r35235, r35236);
        double r35238 = cos(r35228);
        double r35239 = pow(r35238, r35236);
        double r35240 = r35237 - r35239;
        double r35241 = r35238 * r35238;
        double r35242 = r35235 * r35235;
        double r35243 = r35241 - r35242;
        double r35244 = r35238 - r35235;
        double r35245 = r35243 / r35244;
        double r35246 = r35238 * r35245;
        double r35247 = r35246 + r35242;
        double r35248 = sin(r35228);
        double r35249 = r35247 * r35248;
        double r35250 = r35240 / r35249;
        double r35251 = 0.041666666666666664;
        double r35252 = pow(r35228, r35236);
        double r35253 = r35251 * r35252;
        double r35254 = 0.004166666666666667;
        double r35255 = 5.0;
        double r35256 = pow(r35228, r35255);
        double r35257 = r35254 * r35256;
        double r35258 = 0.5;
        double r35259 = r35258 * r35228;
        double r35260 = r35257 + r35259;
        double r35261 = r35253 + r35260;
        double r35262 = r35234 ? r35250 : r35261;
        return r35262;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.2
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.021757331919882755 or 0.02301026064809699 < x

    1. Initial program 0.9

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

    3. Applied flip3--1.1

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

    4. Applied associate-/l/1.1

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

    5. Simplified1.1

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

    7. Applied add-sqr-sqrt1.1

      \[\leadsto \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \color{blue}{\left(\sqrt{\cos x + 1} \cdot \sqrt{\cos x + 1}\right)} + 1 \cdot 1\right) \cdot \sin x}\]
    8. Using strategy rm

    9. Applied flip-+1.1

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

    10. Applied sqrt-div64.0

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

    11. Applied flip-+64.0

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

    12. Applied sqrt-div64.0

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

    13. Applied frac-times64.0

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

    14. Simplified64.0

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

    15. Simplified1.1

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

    if -0.021757331919882755 < x < 0.02301026064809699

    1. Initial program 59.9

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02175733191988275458395030170777317835018 \lor \neg \left(x \le 0.02301026064809699039903634343318117316812\right):\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \frac{\cos x \cdot \cos x - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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