Average Error: 30.5 → 0.5
Time: 22.3s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02294790868899147795456627818566630594432:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \mathbf{elif}\;x \le 0.02393964535351434871901510348379815695807:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02294790868899147795456627818566630594432:\\
\;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\

\mathbf{elif}\;x \le 0.02393964535351434871901510348379815695807:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\

\end{array}
double f(double x) {
        double r59066 = 1.0;
        double r59067 = x;
        double r59068 = cos(r59067);
        double r59069 = r59066 - r59068;
        double r59070 = sin(r59067);
        double r59071 = r59069 / r59070;
        return r59071;
}
↓
double f(double x) {
        double r59072 = x;
        double r59073 = -0.022947908688991478;
        bool r59074 = r59072 <= r59073;
        double r59075 = 1.0;
        double r59076 = sin(r59072);
        double r59077 = 1.0;
        double r59078 = cos(r59072);
        double r59079 = r59077 - r59078;
        double r59080 = r59076 / r59079;
        double r59081 = r59075 / r59080;
        double r59082 = 0.02393964535351435;
        bool r59083 = r59072 <= r59082;
        double r59084 = 0.041666666666666664;
        double r59085 = 3.0;
        double r59086 = pow(r59072, r59085);
        double r59087 = r59084 * r59086;
        double r59088 = 0.004166666666666667;
        double r59089 = 5.0;
        double r59090 = pow(r59072, r59089);
        double r59091 = r59088 * r59090;
        double r59092 = 0.5;
        double r59093 = r59092 * r59072;
        double r59094 = r59091 + r59093;
        double r59095 = r59087 + r59094;
        double r59096 = r59079 / r59076;
        double r59097 = exp(r59096);
        double r59098 = log(r59097);
        double r59099 = r59083 ? r59095 : r59098;
        double r59100 = r59074 ? r59081 : r59099;
        return r59100;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -0.022947908688991478

    1. Initial program 0.9

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

    3. Applied clear-num1.0

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

    if -0.022947908688991478 < x < 0.02393964535351435

    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 0.02393964535351435 < x

    1. Initial program 0.9

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

    3. Applied add-log-exp1.1

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02294790868899147795456627818566630594432:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \mathbf{elif}\;x \le 0.02393964535351434871901510348379815695807:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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