Average Error: 30.3 → 0.7
Time: 7.8s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -2.76308119989467809 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \log \left(e^{\cos x + 1}\right) + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le -0.0:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -2.76308119989467809 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \log \left(e^{\cos x + 1}\right) + 1 \cdot 1}}{\sin x}\\

\mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le -0.0:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\

\end{array}
double f(double x) {
        double r66951 = 1.0;
        double r66952 = x;
        double r66953 = cos(r66952);
        double r66954 = r66951 - r66953;
        double r66955 = sin(r66952);
        double r66956 = r66954 / r66955;
        return r66956;
}
↓
double f(double x) {
        double r66957 = 1.0;
        double r66958 = x;
        double r66959 = cos(r66958);
        double r66960 = r66957 - r66959;
        double r66961 = sin(r66958);
        double r66962 = r66960 / r66961;
        double r66963 = -0.0002763081199894678;
        bool r66964 = r66962 <= r66963;
        double r66965 = 3.0;
        double r66966 = pow(r66957, r66965);
        double r66967 = pow(r66959, r66965);
        double r66968 = exp(r66967);
        double r66969 = log(r66968);
        double r66970 = r66966 - r66969;
        double r66971 = r66959 + r66957;
        double r66972 = exp(r66971);
        double r66973 = log(r66972);
        double r66974 = r66959 * r66973;
        double r66975 = r66957 * r66957;
        double r66976 = r66974 + r66975;
        double r66977 = r66970 / r66976;
        double r66978 = r66977 / r66961;
        double r66979 = -0.0;
        bool r66980 = r66962 <= r66979;
        double r66981 = 0.041666666666666664;
        double r66982 = pow(r66958, r66965);
        double r66983 = r66981 * r66982;
        double r66984 = 0.004166666666666667;
        double r66985 = 5.0;
        double r66986 = pow(r66958, r66985);
        double r66987 = r66984 * r66986;
        double r66988 = 0.5;
        double r66989 = r66988 * r66958;
        double r66990 = r66987 + r66989;
        double r66991 = r66983 + r66990;
        double r66992 = r66957 / r66961;
        double r66993 = r66959 / r66961;
        double r66994 = r66992 - r66993;
        double r66995 = r66980 ? r66991 : r66994;
        double r66996 = r66964 ? r66978 : r66995;
        return r66996;
}

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)) < -0.0002763081199894678

    1. Initial program 1.0

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

    3. Applied flip3--1.1

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

    4. Simplified1.2

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

    6. Applied add-log-exp1.2

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

    8. Applied add-log-exp1.2

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

    9. Applied add-log-exp1.2

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

    10. Applied sum-log1.2

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

    11. Simplified1.2

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

    if -0.0002763081199894678 < (/ (- 1.0 (cos x)) (sin x)) < -0.0

    1. Initial program 60.3

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

    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 -0.0 < (/ (- 1.0 (cos x)) (sin x))

    1. Initial program 1.3

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

    3. Applied div-sub1.5

      \[\leadsto \color{blue}{\frac{1}{\sin x} - \frac{\cos x}{\sin x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -2.76308119989467809 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{{1}^{3} - \log \left(e^{{\left(\cos x\right)}^{3}}\right)}{\cos x \cdot \log \left(e^{\cos x + 1}\right) + 1 \cdot 1}}{\sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le -0.0:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin x} - \frac{\cos x}{\sin x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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