Average Error: 36.2 → 10.1
Time: 1.2m
Precision: 64
Internal precision: 128
\[\tan \left(x + \varepsilon\right) - \tan x\]
⬇
\[\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.939839550679413 \cdot 10^{-37}:\\
\;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}} - \tan x\\
\mathbf{if}\;\varepsilon \le 7.011714412203457 \cdot 10^{-69}:\\
\;\;\;\;\left(\varepsilon + {\varepsilon}^{4} \cdot {x}^3\right) + {\varepsilon}^3 \cdot {x}^2\\
\mathbf{else}:\\
\;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \sin x}{\left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos x}\\
\end{array}\]
Target
| Original | 36.2 |
|---|
| Comparison | 25.8 |
|---|
| Herbie | 10.1 |
|---|
\[ \frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]
Derivation
- Split input into 3 regimes.
-
if eps < -1.939839550679413e-37
Initial program 30.1
\[\tan \left(x + \varepsilon\right) - \tan x\]
- Using strategy
rm
Applied tan-sum 2.8
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
- Using strategy
rm
Applied tan-quot 2.8
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}} - \tan x\]
Applied tan-quot 2.8
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\cos x}} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \tan x\]
Applied frac-times 2.8
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}} - \tan x\]
if -1.939839550679413e-37 < eps < 7.011714412203457e-69
Initial program 45.6
\[\tan \left(x + \varepsilon\right) - \tan x\]
Applied taylor 19.0
\[\leadsto \varepsilon + \left({\varepsilon}^{4} \cdot {x}^{3} + {\varepsilon}^{3} \cdot {x}^2\right)\]
Taylor expanded around 0 19.0
\[\leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{4} \cdot {x}^{3} + {\varepsilon}^{3} \cdot {x}^2\right)}\]
Applied simplify 19.0
\[\leadsto \color{blue}{\left(\varepsilon + {\varepsilon}^{4} \cdot {x}^3\right) + \left(x \cdot x\right) \cdot {\varepsilon}^3}\]
Applied simplify 19.0
\[\leadsto \left(\varepsilon + {\varepsilon}^{4} \cdot {x}^3\right) + \color{blue}{{\varepsilon}^3 \cdot {x}^2}\]
if 7.011714412203457e-69 < eps
Initial program 29.6
\[\tan \left(x + \varepsilon\right) - \tan x\]
- Using strategy
rm
Applied tan-sum 4.9
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
- Using strategy
rm
Applied tan-quot 4.9
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}} - \tan x\]
Applied tan-quot 4.9
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x}{\cos x}} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \tan x\]
Applied frac-times 4.9
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}} - \tan x\]
- Using strategy
rm
Applied tan-quot 4.9
\[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}} - \color{blue}{\frac{\sin x}{\cos x}}\]
Applied frac-sub 5.0
\[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \sin x}{\left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos x}}\]
- Recombined 3 regimes into one program.
- Removed slow pow expressions
Runtime
Please include this information when filing a bug report:
herbie --seed '#(3276454670 1596714944 2783243138 4136209721 1268985944 2620999200)'
(FPCore (x eps)
:name "2tan (problem 3.3.2)"
:herbie-expected 28
:target
(/ (sin eps) (* (cos x) (cos (+ x eps))))
(- (tan (+ x eps)) (tan x)))
