Average Error: 37.4 → 13.9
Time: 2.5m
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\tan x + \tan \varepsilon}{1 - (e^{\log_* (1 + \tan x \cdot \tan \varepsilon)} - 1)^*} - \tan x \le -3.295028045312774 \cdot 10^{-13}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - (e^{\log_* (1 + \tan x \cdot \tan \varepsilon)} - 1)^*} - \tan x\\ \mathbf{if}\;\frac{\tan x + \tan \varepsilon}{1 - (e^{\log_* (1 + \tan x \cdot \tan \varepsilon)} - 1)^*} - \tan x \le 3.9954989461759 \cdot 10^{-310}:\\ \;\;\;\;(\varepsilon \cdot \left((\left(\varepsilon \cdot x\right) \cdot \left(\varepsilon \cdot x\right) + \left(\varepsilon \cdot x\right))_*\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan \varepsilon \cdot \tan x\right)}^{3}}} - \tan x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.4
Target15.2
Herbie13.9
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (/ (+ (tan x) (tan eps)) (- 1 (expm1 (log1p (* (tan x) (tan eps)))))) (tan x)) < -3.295028045312774e-13

    1. Initial program 27.1

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

    3. Applied tan-sum0.8

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

    5. Applied expm1-log1p-u0.8

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{(e^{\log_* (1 + \tan x \cdot \tan \varepsilon)} - 1)^*}} - \tan x\]

    if -3.295028045312774e-13 < (- (/ (+ (tan x) (tan eps)) (- 1 (expm1 (log1p (* (tan x) (tan eps)))))) (tan x)) < 3.9954989461759e-310

    1. Initial program 52.0

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

    2. Taylor expanded around 0 33.6

      \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left({x}^{2} \cdot {\varepsilon}^{3} + \varepsilon\right)}\]

    3. Applied simplify32.3

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

    if 3.9954989461759e-310 < (- (/ (+ (tan x) (tan eps)) (- 1 (expm1 (log1p (* (tan x) (tan eps)))))) (tan x))

    1. Initial program 27.0

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

    3. Applied tan-sum1.1

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

    5. Applied add-cbrt-cube1.1

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

    6. Applied add-cbrt-cube1.2

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

    7. Applied cbrt-unprod1.1

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

    8. Applied simplify1.1

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{\color{blue}{{\left(\tan \varepsilon \cdot \tan x\right)}^{3}}}} - \tan x\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2020178 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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