2tan (problem 3.3.2)

Average Error: 37.0 → 15.4

Time: 6.4s

Precision: binary64

Math

\[\tan \left(x + \varepsilon\right) - \tan x\]

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.61097321673295567 \cdot 10^{-34} \lor \neg \left(\varepsilon \le 6.04579305488383327 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{\left(\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x\right) - \left(-\sin x \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}{\cos x + \left(-\cos x \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original	37.0
Target	15.0
Herbie	15.4

\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

1. Split input into 2 regimes

2. if eps < -7.61097321673295567e-34 or 6.04579305488383327e-71 < eps

   1. Initial program 30.0

      \[\tan \left(x + \varepsilon\right) - \tan x\]
   2. Using strategy rm

   3. Applied tan-sum 4.0

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 4.1

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\left(\sqrt[3]{\tan x \cdot \tan \varepsilon} \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}\right) \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}}} - \tan x\]
   6. Using strategy rm

   7. Applied tan-quot 4.2

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \left(\sqrt[3]{\tan x \cdot \tan \varepsilon} \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}\right) \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}} - \color{blue}{\frac{\sin x}{\cos x}}\]

   8. Applied frac-sub 4.2

      \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \left(\sqrt[3]{\tan x \cdot \tan \varepsilon} \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}\right) \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}\right) \cdot \sin x}{\left(1 - \left(\sqrt[3]{\tan x \cdot \tan \varepsilon} \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}\right) \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}\right) \cdot \cos x}}\]

   9. Simplified 4.0

      \[\leadsto \frac{\color{blue}{\left(\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x\right) - \left(-\sin x \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}{\left(1 - \left(\sqrt[3]{\tan x \cdot \tan \varepsilon} \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}\right) \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}\right) \cdot \cos x}\]

   10. Simplified 3.7

       \[\leadsto \frac{\left(\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x\right) - \left(-\sin x \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}{\color{blue}{\cos x + \left(-\cos x \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}\]

   if -7.61097321673295567e-34 < eps < 6.04579305488383327e-71

   1. Initial program 46.8

      \[\tan \left(x + \varepsilon\right) - \tan x\]

   2. Taylor expanded around 0 31.8

      \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\varepsilon + {x}^{2} \cdot \varepsilon\right)}\]

   3. Simplified 31.6

      \[\leadsto \color{blue}{\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon}\]
3. Recombined 2 regimes into one program.

4. Final simplification 15.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.61097321673295567 \cdot 10^{-34} \lor \neg \left(\varepsilon \le 6.04579305488383327 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{\left(\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \sin x\right) - \left(-\sin x \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}{\cos x + \left(-\cos x \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))