Average Error: 30.2 → 0.5
Time: 39.4s
Precision: 64
Internal precision: 128
\[\frac{1 - \cos x}{\sin x}\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -5.458497096453193 \cdot 10^{-06}:\\ \;\;\;\;\frac{\frac{{\left(\sin x\right)}^2}{1 + \cos x}}{\sin x}\\ \mathbf{if}\;x \le 85.36504497937187:\\ \;\;\;\;\frac{1}{24} \cdot {x}^3 + \left({x}^{5} \cdot \frac{1}{240} + x \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{{\left(\frac{{\left(\sin x\right)}^2}{1 + \cos x}\right)}^3}{{\left(\sin x\right)}^3}}\\ \end{array}\]

Error

Bits error versus x

Target

Original30.2
Comparison0.1
Herbie0.5
\[ \tan \left(\frac{x}{2}\right) \]

Derivation

  1. Split input into 3 regimes.

  2. if x < -5.458497096453193e-06

    1. Initial program 1.2

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

    3. Applied flip-- 1.6

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

    4. Applied simplify 0.9

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

    if -5.458497096453193e-06 < x < 85.36504497937187

    1. Initial program 59.9

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

    2. Applied taylor 0.0

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

    3. Taylor expanded around 0 0.0

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

    4. Applied simplify 0.0

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

    if 85.36504497937187 < x

    1. Initial program 0.9

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

    3. Applied flip-- 1.3

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

    4. Applied simplify 0.9

      \[\leadsto \frac{\frac{\color{blue}{{\left(\sin x\right)}^2}}{1 + \cos x}}{\sin x}\]
    5. Using strategy rm

    6. Applied add-cbrt-cube 1.1

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

    7. Applied add-cbrt-cube 1.2

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

    8. Applied cbrt-undiv 1.1

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{\left(\frac{{\left(\sin x\right)}^2}{1 + \cos x}\right)}^3}{{\left(\sin x\right)}^3}}}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 39.4s) Debug log

Please include this information when filing a bug report:

herbie --seed '#(1503968389 4000613028 1192568923 132757939 1804726043 409631676)'
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 1

  :target
  (tan (/ x 2))

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