Average Error: 36.9 → 14.2
Time: 1.4m
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le -2.474724364787046 \cdot 10^{-18} \lor \neg \left(\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le 9.979318880227918 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
Target15.3
Herbie14.2
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x))) < -2.474724364787046e-18 or 9.979318880227918e-34 < (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x)))

    1. Initial program 35.7

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

    3. Applied tan-sum13.1

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

    5. Applied add-cbrt-cube13.1

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

    6. Applied add-cbrt-cube13.1

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

    7. Applied cbrt-unprod13.1

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

    8. Applied simplify13.1

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

    if -2.474724364787046e-18 < (+ eps (+ (* (pow eps 3) (pow x 2)) (* (pow eps 2) x))) < 9.979318880227918e-34

    1. Initial program 39.3

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

    2. Taylor expanded around 0 16.6

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

  4. Applied simplify14.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le -2.474724364787046 \cdot 10^{-18} \lor \neg \left(\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon \le 9.979318880227918 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot {\varepsilon}^{2} + {\varepsilon}^{3} \cdot {x}^{2}\right) + \varepsilon\\ \end{array}}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018195 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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