Average Error: 29.9 → 0.6
Time: 20.8s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.01945573319338761431041007199382875114679:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;x \le 0.02213022324220578415654436810200422769412:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{1}^{3} - {\left(\cos x\right)}^{3}}}{\frac{\sin x}{\sqrt{1 - \cos x}} \cdot \sqrt{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.01945573319338761431041007199382875114679:\\
\;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\

\mathbf{elif}\;x \le 0.02213022324220578415654436810200422769412:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{{1}^{3} - {\left(\cos x\right)}^{3}}}{\frac{\sin x}{\sqrt{1 - \cos x}} \cdot \sqrt{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}\\

\end{array}
double f(double x) {
        double r32497 = 1.0;
        double r32498 = x;
        double r32499 = cos(r32498);
        double r32500 = r32497 - r32499;
        double r32501 = sin(r32498);
        double r32502 = r32500 / r32501;
        return r32502;
}
↓
double f(double x) {
        double r32503 = x;
        double r32504 = -0.019455733193387614;
        bool r32505 = r32503 <= r32504;
        double r32506 = 1.0;
        double r32507 = cos(r32503);
        double r32508 = r32506 - r32507;
        double r32509 = sin(r32503);
        double r32510 = r32508 / r32509;
        double r32511 = exp(r32510);
        double r32512 = log(r32511);
        double r32513 = 0.022130223242205784;
        bool r32514 = r32503 <= r32513;
        double r32515 = 0.041666666666666664;
        double r32516 = 3.0;
        double r32517 = pow(r32503, r32516);
        double r32518 = r32515 * r32517;
        double r32519 = 0.004166666666666667;
        double r32520 = 5.0;
        double r32521 = pow(r32503, r32520);
        double r32522 = r32519 * r32521;
        double r32523 = 0.5;
        double r32524 = r32523 * r32503;
        double r32525 = r32522 + r32524;
        double r32526 = r32518 + r32525;
        double r32527 = pow(r32506, r32516);
        double r32528 = pow(r32507, r32516);
        double r32529 = r32527 - r32528;
        double r32530 = sqrt(r32529);
        double r32531 = sqrt(r32508);
        double r32532 = r32509 / r32531;
        double r32533 = r32506 * r32506;
        double r32534 = r32507 * r32507;
        double r32535 = r32506 * r32507;
        double r32536 = r32534 + r32535;
        double r32537 = r32533 + r32536;
        double r32538 = sqrt(r32537);
        double r32539 = r32532 * r32538;
        double r32540 = r32530 / r32539;
        double r32541 = r32514 ? r32526 : r32540;
        double r32542 = r32505 ? r32512 : r32541;
        return r32542;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.019455733193387614

    1. Initial program 0.9

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

    3. Applied add-log-exp1.1

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

    if -0.019455733193387614 < x < 0.022130223242205784

    1. Initial program 60.0

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

    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 0.022130223242205784 < x

    1. Initial program 0.9

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

    3. Applied add-sqr-sqrt1.1

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

    4. Applied associate-/l*1.1

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

    6. Applied flip3--1.1

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

    7. Applied sqrt-div1.1

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

    8. Applied associate-/l/1.1

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

    10. Applied add-log-exp1.2

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

    12. Applied rem-log-exp1.1

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.01945573319338761431041007199382875114679:\\ \;\;\;\;\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)\\ \mathbf{elif}\;x \le 0.02213022324220578415654436810200422769412:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{1}^{3} - {\left(\cos x\right)}^{3}}}{\frac{\sin x}{\sqrt{1 - \cos x}} \cdot \sqrt{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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