Average Error: 30.3 → 0.7
Time: 7.2s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.01816359460916088:\\ \;\;\;\;\frac{1}{\sin x} - \cos x \cdot \frac{1}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 6.0277248497108966 \cdot 10^{-4}:\\ \;\;\;\;0.041666666666666671 \cdot {x}^{3} + \left(0.00416666666666666661 \cdot {x}^{5} + 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{e^{\log \left(1 - \cos x\right)}}{\sin x}}\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.01816359460916088:\\
\;\;\;\;\frac{1}{\sin x} - \cos x \cdot \frac{1}{\sin x}\\

\mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 6.0277248497108966 \cdot 10^{-4}:\\
\;\;\;\;0.041666666666666671 \cdot {x}^{3} + \left(0.00416666666666666661 \cdot {x}^{5} + 0.5 \cdot x\right)\\

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

\end{array}
double f(double x) {
        double r56043 = 1.0;
        double r56044 = x;
        double r56045 = cos(r56044);
        double r56046 = r56043 - r56045;
        double r56047 = sin(r56044);
        double r56048 = r56046 / r56047;
        return r56048;
}
↓
double f(double x) {
        double r56049 = 1.0;
        double r56050 = x;
        double r56051 = cos(r56050);
        double r56052 = r56049 - r56051;
        double r56053 = sin(r56050);
        double r56054 = r56052 / r56053;
        double r56055 = -0.01816359460916088;
        bool r56056 = r56054 <= r56055;
        double r56057 = r56049 / r56053;
        double r56058 = 1.0;
        double r56059 = r56058 / r56053;
        double r56060 = r56051 * r56059;
        double r56061 = r56057 - r56060;
        double r56062 = 0.0006027724849710897;
        bool r56063 = r56054 <= r56062;
        double r56064 = 0.04166666666666667;
        double r56065 = 3.0;
        double r56066 = pow(r56050, r56065);
        double r56067 = r56064 * r56066;
        double r56068 = 0.004166666666666667;
        double r56069 = 5.0;
        double r56070 = pow(r56050, r56069);
        double r56071 = r56068 * r56070;
        double r56072 = 0.5;
        double r56073 = r56072 * r56050;
        double r56074 = r56071 + r56073;
        double r56075 = r56067 + r56074;
        double r56076 = log(r56052);
        double r56077 = exp(r56076);
        double r56078 = r56077 / r56053;
        double r56079 = exp(r56078);
        double r56080 = log(r56079);
        double r56081 = r56063 ? r56075 : r56080;
        double r56082 = r56056 ? r56061 : r56081;
        return r56082;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.3
Target0.0
Herbie0.7
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- 1.0 (cos x)) (sin x)) < -0.01816359460916088

    1. Initial program 0.8

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

    3. Applied div-sub1.0

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

    5. Applied div-inv0.9

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

    if -0.01816359460916088 < (/ (- 1.0 (cos x)) (sin x)) < 0.0006027724849710897

    1. Initial program 59.6

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

    3. Applied div-sub59.6

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

    4. Taylor expanded around 0 0.5

      \[\leadsto \color{blue}{0.041666666666666671 \cdot {x}^{3} + \left(0.00416666666666666661 \cdot {x}^{5} + 0.5 \cdot x\right)}\]

    if 0.0006027724849710897 < (/ (- 1.0 (cos x)) (sin x))

    1. Initial program 1.0

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

    3. Applied add-log-exp1.1

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

    5. Applied add-exp-log1.1

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.01816359460916088:\\ \;\;\;\;\frac{1}{\sin x} - \cos x \cdot \frac{1}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 6.0277248497108966 \cdot 10^{-4}:\\ \;\;\;\;0.041666666666666671 \cdot {x}^{3} + \left(0.00416666666666666661 \cdot {x}^{5} + 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{e^{\log \left(1 - \cos x\right)}}{\sin x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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