Average Error: 29.7 → 0.5
Time: 7.3s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0133795329351141800700020212389063090086:\\ \;\;\;\;\frac{\sin x \cdot \left(1 - \cos x\right)}{\sin x \cdot \sin x}\\ \mathbf{elif}\;x \le 0.02120826393528481637607008281065645860508:\\ \;\;\;\;0.04166666666666667129259593593815225176513 \cdot {x}^{3} + \left(0.004166666666666666608842550800773096852936 \cdot {x}^{5} + 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{1}{\sin x} - \cos x \cdot \frac{1}{\sin x}}\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0133795329351141800700020212389063090086:\\
\;\;\;\;\frac{\sin x \cdot \left(1 - \cos x\right)}{\sin x \cdot \sin x}\\

\mathbf{elif}\;x \le 0.02120826393528481637607008281065645860508:\\
\;\;\;\;0.04166666666666667129259593593815225176513 \cdot {x}^{3} + \left(0.004166666666666666608842550800773096852936 \cdot {x}^{5} + 0.5 \cdot x\right)\\

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

\end{array}
double f(double x) {
        double r59031 = 1.0;
        double r59032 = x;
        double r59033 = cos(r59032);
        double r59034 = r59031 - r59033;
        double r59035 = sin(r59032);
        double r59036 = r59034 / r59035;
        return r59036;
}
↓
double f(double x) {
        double r59037 = x;
        double r59038 = -0.01337953293511418;
        bool r59039 = r59037 <= r59038;
        double r59040 = sin(r59037);
        double r59041 = 1.0;
        double r59042 = cos(r59037);
        double r59043 = r59041 - r59042;
        double r59044 = r59040 * r59043;
        double r59045 = r59040 * r59040;
        double r59046 = r59044 / r59045;
        double r59047 = 0.021208263935284816;
        bool r59048 = r59037 <= r59047;
        double r59049 = 0.04166666666666667;
        double r59050 = 3.0;
        double r59051 = pow(r59037, r59050);
        double r59052 = r59049 * r59051;
        double r59053 = 0.004166666666666667;
        double r59054 = 5.0;
        double r59055 = pow(r59037, r59054);
        double r59056 = r59053 * r59055;
        double r59057 = 0.5;
        double r59058 = r59057 * r59037;
        double r59059 = r59056 + r59058;
        double r59060 = r59052 + r59059;
        double r59061 = r59041 / r59040;
        double r59062 = 1.0;
        double r59063 = r59062 / r59040;
        double r59064 = r59042 * r59063;
        double r59065 = r59061 - r59064;
        double r59066 = exp(r59065);
        double r59067 = log(r59066);
        double r59068 = r59048 ? r59060 : r59067;
        double r59069 = r59039 ? r59046 : r59068;
        return r59069;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.01337953293511418

    1. Initial program 0.9

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

    3. Applied div-sub1.1

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

    5. Applied frac-sub1.1

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

    6. Simplified1.0

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

    if -0.01337953293511418 < x < 0.021208263935284816

    1. Initial program 59.7

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

    3. Applied div-sub59.8

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

    4. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{0.04166666666666667129259593593815225176513 \cdot {x}^{3} + \left(0.004166666666666666608842550800773096852936 \cdot {x}^{5} + 0.5 \cdot x\right)}\]

    if 0.021208263935284816 < x

    1. Initial program 0.9

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

    3. Applied div-sub1.1

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

    5. Applied add-log-exp1.3

      \[\leadsto \frac{1}{\sin x} - \color{blue}{\log \left(e^{\frac{\cos x}{\sin x}}\right)}\]

    6. Applied add-log-exp1.3

      \[\leadsto \color{blue}{\log \left(e^{\frac{1}{\sin x}}\right)} - \log \left(e^{\frac{\cos x}{\sin x}}\right)\]

    7. Applied diff-log1.3

      \[\leadsto \color{blue}{\log \left(\frac{e^{\frac{1}{\sin x}}}{e^{\frac{\cos x}{\sin x}}}\right)}\]

    8. Simplified1.2

      \[\leadsto \log \color{blue}{\left(e^{\frac{1}{\sin x} - \frac{\cos x}{\sin x}}\right)}\]
    9. Using strategy rm

    10. Applied div-inv1.2

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0133795329351141800700020212389063090086:\\ \;\;\;\;\frac{\sin x \cdot \left(1 - \cos x\right)}{\sin x \cdot \sin x}\\ \mathbf{elif}\;x \le 0.02120826393528481637607008281065645860508:\\ \;\;\;\;0.04166666666666667129259593593815225176513 \cdot {x}^{3} + \left(0.004166666666666666608842550800773096852936 \cdot {x}^{5} + 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{1}{\sin x} - \cos x \cdot \frac{1}{\sin x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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