Average Error: 30.2 → 0.5
Time: 1.2m
Precision: 64
Internal Precision: 2432
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.023802979002913602:\\ \;\;\;\;\frac{1}{\frac{\sin x}{1 - \cos x}}\\ \mathbf{if}\;x \le 2.1789124480322282 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{\sin x}\\ \end{array}\]

Error

Bits error versus x

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -0.023802979002913602

    1. Initial program 1.0

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

    3. Applied clear-num1.0

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

    if -0.023802979002913602 < x < 2.1789124480322282e-26

    1. Initial program 60.2

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

    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 2.1789124480322282e-26 < x

    1. Initial program 4.7

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

    3. Applied flip--5.1

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

    4. Applied simplify0.9

      \[\leadsto \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}{\sin x}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1063313015 2771194459 1594909340 1344785158 2223560818 546365448)' 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 1

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

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