Average Error: 30.3 → 0.5
Time: 17.5s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02187035372515749864774647903686854988337 \lor \neg \left(x \le 0.02243777298164771269450845636583835585043\right):\\ \;\;\;\;\left(\left(\left(1 + \cos x\right) \cdot \left(1 - \cos x\right)\right) \cdot \frac{1}{\sin x}\right) \cdot \frac{1}{1 + \cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(x \cdot \frac{1}{2} + {x}^{5} \cdot \frac{1}{240}\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02187035372515749864774647903686854988337 \lor \neg \left(x \le 0.02243777298164771269450845636583835585043\right):\\
\;\;\;\;\left(\left(\left(1 + \cos x\right) \cdot \left(1 - \cos x\right)\right) \cdot \frac{1}{\sin x}\right) \cdot \frac{1}{1 + \cos x}\\

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

\end{array}
double f(double x) {
        double r33365 = 1.0;
        double r33366 = x;
        double r33367 = cos(r33366);
        double r33368 = r33365 - r33367;
        double r33369 = sin(r33366);
        double r33370 = r33368 / r33369;
        return r33370;
}
↓
double f(double x) {
        double r33371 = x;
        double r33372 = -0.0218703537251575;
        bool r33373 = r33371 <= r33372;
        double r33374 = 0.022437772981647713;
        bool r33375 = r33371 <= r33374;
        double r33376 = !r33375;
        bool r33377 = r33373 || r33376;
        double r33378 = 1.0;
        double r33379 = cos(r33371);
        double r33380 = r33378 + r33379;
        double r33381 = r33378 - r33379;
        double r33382 = r33380 * r33381;
        double r33383 = 1.0;
        double r33384 = sin(r33371);
        double r33385 = r33383 / r33384;
        double r33386 = r33382 * r33385;
        double r33387 = r33383 / r33380;
        double r33388 = r33386 * r33387;
        double r33389 = 0.041666666666666664;
        double r33390 = 3.0;
        double r33391 = pow(r33371, r33390);
        double r33392 = r33389 * r33391;
        double r33393 = 0.5;
        double r33394 = r33371 * r33393;
        double r33395 = 5.0;
        double r33396 = pow(r33371, r33395);
        double r33397 = 0.004166666666666667;
        double r33398 = r33396 * r33397;
        double r33399 = r33394 + r33398;
        double r33400 = r33392 + r33399;
        double r33401 = r33377 ? r33388 : r33400;
        return r33401;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.3
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0218703537251575 or 0.022437772981647713 < x

    1. Initial program 0.9

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

    3. Applied clear-num1.0

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

    5. Applied flip--1.4

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

    6. Applied associate-/r/1.4

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

    7. Applied add-cube-cbrt1.4

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

    8. Applied times-frac1.4

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\sin x}{1 \cdot 1 - \cos x \cdot \cos x}} \cdot \frac{\sqrt[3]{1}}{1 + \cos x}}\]

    9. Simplified1.0

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

    10. Simplified1.0

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

    12. Applied div-inv1.1

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

    if -0.0218703537251575 < x < 0.022437772981647713

    1. Initial program 59.9

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

    3. Applied clear-num59.9

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

    4. Taylor expanded around 0 0.0

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

    5. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x + {x}^{5} \cdot \frac{1}{240}\right) + \frac{1}{24} \cdot {x}^{3}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02187035372515749864774647903686854988337 \lor \neg \left(x \le 0.02243777298164771269450845636583835585043\right):\\ \;\;\;\;\left(\left(\left(1 + \cos x\right) \cdot \left(1 - \cos x\right)\right) \cdot \frac{1}{\sin x}\right) \cdot \frac{1}{1 + \cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(x \cdot \frac{1}{2} + {x}^{5} \cdot \frac{1}{240}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2.0))

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