Average Error: 29.6 → 0.5
Time: 7.3s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0192164982803571076 \lor \neg \left(x \le 0.0203937394598384565\right):\\ \;\;\;\;\frac{\frac{\sqrt{1}}{1}}{\frac{\sin x}{1 - \cos x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0192164982803571076 \lor \neg \left(x \le 0.0203937394598384565\right):\\
\;\;\;\;\frac{\frac{\sqrt{1}}{1}}{\frac{\sin x}{1 - \cos 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 r51934 = 1.0;
        double r51935 = x;
        double r51936 = cos(r51935);
        double r51937 = r51934 - r51936;
        double r51938 = sin(r51935);
        double r51939 = r51937 / r51938;
        return r51939;
}
↓
double f(double x) {
        double r51940 = x;
        double r51941 = -0.019216498280357108;
        bool r51942 = r51940 <= r51941;
        double r51943 = 0.020393739459838457;
        bool r51944 = r51940 <= r51943;
        double r51945 = !r51944;
        bool r51946 = r51942 || r51945;
        double r51947 = 1.0;
        double r51948 = sqrt(r51947);
        double r51949 = r51948 / r51947;
        double r51950 = sin(r51940);
        double r51951 = 1.0;
        double r51952 = cos(r51940);
        double r51953 = r51951 - r51952;
        double r51954 = r51950 / r51953;
        double r51955 = r51949 / r51954;
        double r51956 = 0.041666666666666664;
        double r51957 = 3.0;
        double r51958 = pow(r51940, r51957);
        double r51959 = r51956 * r51958;
        double r51960 = 0.004166666666666667;
        double r51961 = 5.0;
        double r51962 = pow(r51940, r51961);
        double r51963 = r51960 * r51962;
        double r51964 = 0.5;
        double r51965 = r51964 * r51940;
        double r51966 = r51963 + r51965;
        double r51967 = r51959 + r51966;
        double r51968 = r51946 ? r51955 : r51967;
        return r51968;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.6
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.019216498280357108 or 0.020393739459838457 < x

    1. Initial program 0.9

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

    3. Applied clear-num1.0

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

    5. Applied div-inv1.0

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

    6. Applied associate-/r*1.0

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

    8. Applied *-un-lft-identity1.0

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

    9. Applied add-sqr-sqrt1.0

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

    10. Applied times-frac1.0

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

    11. Applied associate-/l*1.1

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

    12. Simplified1.0

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

    if -0.019216498280357108 < x < 0.020393739459838457

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0192164982803571076 \lor \neg \left(x \le 0.0203937394598384565\right):\\ \;\;\;\;\frac{\frac{\sqrt{1}}{1}}{\frac{\sin x}{1 - \cos x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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