Average Error: 30.4 → 0.5
Time: 11.6s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.020365697215685236:\\ \;\;\;\;\left(\frac{1 - \cos x}{\sin x} \cdot \left(1 + \cos x\right)\right) \cdot \frac{1}{1 + \cos x}\\ \mathbf{elif}\;x \le 0.0228384202901172999:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \cos x\right) \cdot \frac{\cos x + 1}{\sin x}\right) \cdot \frac{1}{1 + \cos x}\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.020365697215685236:\\
\;\;\;\;\left(\frac{1 - \cos x}{\sin x} \cdot \left(1 + \cos x\right)\right) \cdot \frac{1}{1 + \cos x}\\

\mathbf{elif}\;x \le 0.0228384202901172999:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

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

\end{array}
double f(double x) {
        double r41641 = 1.0;
        double r41642 = x;
        double r41643 = cos(r41642);
        double r41644 = r41641 - r41643;
        double r41645 = sin(r41642);
        double r41646 = r41644 / r41645;
        return r41646;
}
↓
double f(double x) {
        double r41647 = x;
        double r41648 = -0.020365697215685236;
        bool r41649 = r41647 <= r41648;
        double r41650 = 1.0;
        double r41651 = cos(r41647);
        double r41652 = r41650 - r41651;
        double r41653 = sin(r41647);
        double r41654 = r41652 / r41653;
        double r41655 = r41650 + r41651;
        double r41656 = r41654 * r41655;
        double r41657 = 1.0;
        double r41658 = r41657 / r41655;
        double r41659 = r41656 * r41658;
        double r41660 = 0.0228384202901173;
        bool r41661 = r41647 <= r41660;
        double r41662 = 0.041666666666666664;
        double r41663 = 3.0;
        double r41664 = pow(r41647, r41663);
        double r41665 = r41662 * r41664;
        double r41666 = 0.004166666666666667;
        double r41667 = 5.0;
        double r41668 = pow(r41647, r41667);
        double r41669 = r41666 * r41668;
        double r41670 = 0.5;
        double r41671 = r41670 * r41647;
        double r41672 = r41669 + r41671;
        double r41673 = r41665 + r41672;
        double r41674 = r41651 + r41650;
        double r41675 = r41674 / r41653;
        double r41676 = r41652 * r41675;
        double r41677 = r41676 * r41658;
        double r41678 = r41661 ? r41673 : r41677;
        double r41679 = r41649 ? r41659 : r41678;
        return r41679;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.4
Target0.0
Herbie0.5
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.020365697215685236

    1. Initial program 1.0

      \[\frac{1 - \cos x}{\sin x}\]
    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-frac1.5

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

    9. Simplified1.1

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

    10. Simplified1.1

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

    if -0.020365697215685236 < x < 0.0228384202901173

    1. Initial program 59.7

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

    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 0.0228384202901173 < x

    1. Initial program 0.8

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

    3. Applied clear-num0.9

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{1 - \cos x}}}\]
    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. Simplified0.9

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

    10. Simplified0.9

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

    12. Applied div-inv1.0

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

    13. Applied associate-*l*1.0

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

    14. Simplified1.0

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.020365697215685236:\\ \;\;\;\;\left(\frac{1 - \cos x}{\sin x} \cdot \left(1 + \cos x\right)\right) \cdot \frac{1}{1 + \cos x}\\ \mathbf{elif}\;x \le 0.0228384202901172999:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - \cos x\right) \cdot \frac{\cos x + 1}{\sin x}\right) \cdot \frac{1}{1 + \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

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

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