Average Error: 30.5 → 0.5
Time: 1.6m
Precision: 64
Internal Precision: 2432
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.023957373082421775:\\ \;\;\;\;\frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{\sin x}\\ \mathbf{if}\;x \le 0.006272349724070426:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{\sin x}\\ \end{array}\]

Error

Bits error versus x

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -0.023957373082421775

    1. Initial program 0.9

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

    3. Applied flip--1.4

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

    4. Applied simplify0.9

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

    if -0.023957373082421775 < x < 0.006272349724070426

    1. Initial program 59.9

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

    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 0.006272349724070426 < x

    1. Initial program 1.0

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

    3. Applied flip--1.5

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

    4. Applied simplify1.0

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

Runtime

Time bar (total: 1.6m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o reduce:binary-search
(FPCore (x)
  :name "tanhf (example 3.4)"
  :herbie-expected 1

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

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