Average Error: 29.8 → 0.5
Time: 38.7s
Precision: 64
Internal precision: 2432
\[\frac{1 - \cos x}{\sin x}\]
⬇
\[\begin{array}{l}
\mathbf{if}\;x \le -0.0007664512124618743:\\
\;\;\;\;\frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{\sin x}\\
\mathbf{if}\;x \le 6.683176783612011 \cdot 10^{-65}:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{\sin x}\\
\end{array}\]
Target
| Original | 29.8 |
|---|
| Comparison | 0.0 |
|---|
| Herbie | 0.5 |
|---|
\[ \tan \left(\frac{x}{2}\right) \]
Derivation
- Split input into 3 regimes.
-
if x < -0.0007664512124618743
Initial program 1.0
\[\frac{1 - \cos x}{\sin x}\]
- Using strategy
rm
Applied flip-- 1.4
\[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{\sin x}\]
Applied simplify 1.0
\[\leadsto \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}{\sin x}\]
if -0.0007664512124618743 < x < 6.683176783612011e-65
Initial program 60.2
\[\frac{1 - \cos x}{\sin x}\]
Applied taylor 0.0
\[\leadsto \frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\]
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 6.683176783612011e-65 < x
Initial program 10.4
\[\frac{1 - \cos x}{\sin x}\]
- Using strategy
rm
Applied flip-- 10.8
\[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{\sin x}\]
Applied simplify 0.9
\[\leadsto \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}{\sin x}\]
- Recombined 3 regimes into one program.
- Removed slow pow expressions
Runtime
Please include this information when filing a bug report:
herbie shell --seed '#(1068028399 4028058041 2917032441 2563479541 765645300 1132738916)'
(FPCore (x)
:name "tanhf (example 3.4)"
:herbie-expected 1
:target
(tan (/ x 2))
(/ (- 1 (cos x)) (sin x)))
