Average Error: 30.0 → 0.7
Time: 7.8s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.008959247045566699763075035889414721168578:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 0.0:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{1 - \cos x}{\sin x}\right)}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.008959247045566699763075035889414721168578:\\
\;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1\right) \cdot \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}:\\
\;\;\;\;e^{\log \left(\frac{1 - \cos x}{\sin x}\right)}\\

\end{array}
double f(double x) {
        double r51057 = 1.0;
        double r51058 = x;
        double r51059 = cos(r51058);
        double r51060 = r51057 - r51059;
        double r51061 = sin(r51058);
        double r51062 = r51060 / r51061;
        return r51062;
}
↓
double f(double x) {
        double r51063 = 1.0;
        double r51064 = x;
        double r51065 = cos(r51064);
        double r51066 = r51063 - r51065;
        double r51067 = sin(r51064);
        double r51068 = r51066 / r51067;
        double r51069 = -0.0089592470455667;
        bool r51070 = r51068 <= r51069;
        double r51071 = 3.0;
        double r51072 = pow(r51063, r51071);
        double r51073 = pow(r51065, r51071);
        double r51074 = r51072 - r51073;
        double r51075 = 2.0;
        double r51076 = pow(r51065, r51075);
        double r51077 = r51063 * r51063;
        double r51078 = r51076 - r51077;
        double r51079 = r51065 - r51063;
        double r51080 = r51078 / r51079;
        double r51081 = r51065 * r51080;
        double r51082 = r51081 + r51077;
        double r51083 = r51082 * r51067;
        double r51084 = r51074 / r51083;
        double r51085 = 0.0;
        bool r51086 = r51068 <= r51085;
        double r51087 = 0.041666666666666664;
        double r51088 = pow(r51064, r51071);
        double r51089 = r51087 * r51088;
        double r51090 = 0.004166666666666667;
        double r51091 = 5.0;
        double r51092 = pow(r51064, r51091);
        double r51093 = r51090 * r51092;
        double r51094 = 0.5;
        double r51095 = r51094 * r51064;
        double r51096 = r51093 + r51095;
        double r51097 = r51089 + r51096;
        double r51098 = log(r51068);
        double r51099 = exp(r51098);
        double r51100 = r51086 ? r51097 : r51099;
        double r51101 = r51070 ? r51084 : r51100;
        return r51101;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.0
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)) < -0.0089592470455667

    1. Initial program 0.8

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

    3. Applied flip3--0.9

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

    4. Applied associate-/l/1.0

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

    5. Simplified1.0

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

    7. Applied flip-+1.0

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

    8. Simplified1.0

      \[\leadsto \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \frac{\color{blue}{{\left(\cos x\right)}^{2} - 1 \cdot 1}}{\cos x - 1} + 1 \cdot 1\right) \cdot \sin x}\]

    if -0.0089592470455667 < (/ (- 1.0 (cos x)) (sin x)) < 0.0

    1. Initial program 59.9

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

    2. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)}\]

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

    1. Initial program 29.8

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

    3. Applied add-exp-log31.5

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

    4. Applied add-exp-log31.5

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

    5. Applied div-exp31.6

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

    6. Simplified29.9

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - \cos x}{\sin x} \le -0.008959247045566699763075035889414721168578:\\ \;\;\;\;\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\cos x \cdot \frac{{\left(\cos x\right)}^{2} - 1 \cdot 1}{\cos x - 1} + 1 \cdot 1\right) \cdot \sin x}\\ \mathbf{elif}\;\frac{1 - \cos x}{\sin x} \le 0.0:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{1 - \cos x}{\sin x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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