Average Error: 29.9 → 0.5
Time: 20.7s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02184956452552735267635952709497360046953:\\ \;\;\;\;\frac{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \left(1 - \cos x\right)}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\ \mathbf{elif}\;x \le 0.02174222265971261988659612995888892328367:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - \cos x \cdot {\left(\cos x\right)}^{2}}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02184956452552735267635952709497360046953:\\
\;\;\;\;\frac{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \left(1 - \cos x\right)}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\

\mathbf{elif}\;x \le 0.02174222265971261988659612995888892328367:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{{1}^{3} - \cos x \cdot {\left(\cos x\right)}^{2}}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\

\end{array}
double f(double x) {
        double r54185 = 1.0;
        double r54186 = x;
        double r54187 = cos(r54186);
        double r54188 = r54185 - r54187;
        double r54189 = sin(r54186);
        double r54190 = r54188 / r54189;
        return r54190;
}
↓
double f(double x) {
        double r54191 = x;
        double r54192 = -0.021849564525527353;
        bool r54193 = r54191 <= r54192;
        double r54194 = cos(r54191);
        double r54195 = 1.0;
        double r54196 = r54194 + r54195;
        double r54197 = r54194 * r54196;
        double r54198 = r54195 * r54195;
        double r54199 = r54197 + r54198;
        double r54200 = r54195 - r54194;
        double r54201 = r54199 * r54200;
        double r54202 = sin(r54191);
        double r54203 = r54195 + r54194;
        double r54204 = r54194 * r54203;
        double r54205 = r54204 + r54198;
        double r54206 = r54202 * r54205;
        double r54207 = r54201 / r54206;
        double r54208 = 0.02174222265971262;
        bool r54209 = r54191 <= r54208;
        double r54210 = 0.041666666666666664;
        double r54211 = 3.0;
        double r54212 = pow(r54191, r54211);
        double r54213 = r54210 * r54212;
        double r54214 = 0.004166666666666667;
        double r54215 = 5.0;
        double r54216 = pow(r54191, r54215);
        double r54217 = r54214 * r54216;
        double r54218 = 0.5;
        double r54219 = r54218 * r54191;
        double r54220 = r54217 + r54219;
        double r54221 = r54213 + r54220;
        double r54222 = pow(r54195, r54211);
        double r54223 = 2.0;
        double r54224 = pow(r54194, r54223);
        double r54225 = r54194 * r54224;
        double r54226 = r54222 - r54225;
        double r54227 = r54226 / r54206;
        double r54228 = r54209 ? r54221 : r54227;
        double r54229 = r54193 ? r54207 : r54228;
        return r54229;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.021849564525527353

    1. Initial program 0.9

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

    3. Applied flip3--1.0

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

    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

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

    7. Applied difference-cubes1.1

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

    8. Simplified1.0

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

    if -0.021849564525527353 < x < 0.02174222265971262

    1. Initial program 59.9

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

    2. Taylor expanded around 0 0.0

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

    if 0.02174222265971262 < x

    1. Initial program 0.9

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

    3. Applied flip3--1.0

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

    4. Applied associate-/l/1.0

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

    5. Simplified1.0

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

    7. Applied cube-mult1.0

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

    8. Simplified1.0

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02184956452552735267635952709497360046953:\\ \;\;\;\;\frac{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot \left(1 - \cos x\right)}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\ \mathbf{elif}\;x \le 0.02174222265971261988659612995888892328367:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{1}^{3} - \cos x \cdot {\left(\cos x\right)}^{2}}{\sin x \cdot \left(\cos x \cdot \left(1 + \cos x\right) + 1 \cdot 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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