Average Error: 36.9 → 13.7
Time: 2.2m
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(\varepsilon \cdot x\right) \cdot \left((\left(\varepsilon \cdot x\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_* \le -1.211512109798028 \cdot 10^{-19}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{{1}^{3} - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}}\right) \cdot \left(\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \tan \varepsilon \cdot \tan x\right) + 1\right) + \left(-\tan x\right))_*\\ \mathbf{if}\;(\left(\varepsilon \cdot x\right) \cdot \left((\left(\varepsilon \cdot x\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_* \le 3.8510963537793776 \cdot 10^{-55}:\\ \;\;\;\;(\left(\varepsilon \cdot x\right) \cdot \left((\left(\varepsilon \cdot x\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan \varepsilon + \tan x\right) \cdot \cos x - \sin x \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.9
Target15.2
Herbie13.7
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (fma (* x eps) (fma (* x eps) eps eps) eps) < -1.211512109798028e-19

    1. Initial program 33.8

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

    3. Applied tan-sum8.7

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

    5. Applied flip3--8.7

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

    6. Applied associate-/r/8.7

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

    7. Applied fma-neg8.7

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

    if -1.211512109798028e-19 < (fma (* x eps) (fma (* x eps) eps eps) eps) < 3.8510963537793776e-55

    1. Initial program 43.0

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

    2. Taylor expanded around 0 22.9

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

    3. Applied simplify21.8

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

    if 3.8510963537793776e-55 < (fma (* x eps) (fma (* x eps) eps eps) eps)

    1. Initial program 32.7

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

    3. Applied tan-quot32.5

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

    4. Applied tan-sum9.0

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \frac{\sin x}{\cos x}\]

    5. Applied frac-sub9.0

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

  4. Applied simplify13.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;(\left(\varepsilon \cdot x\right) \cdot \left((\left(\varepsilon \cdot x\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_* \le -1.211512109798028 \cdot 10^{-19}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{{1}^{3} - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}}\right) \cdot \left(\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \tan \varepsilon \cdot \tan x\right) + 1\right) + \left(-\tan x\right))_*\\ \mathbf{if}\;(\left(\varepsilon \cdot x\right) \cdot \left((\left(\varepsilon \cdot x\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_* \le 3.8510963537793776 \cdot 10^{-55}:\\ \;\;\;\;(\left(\varepsilon \cdot x\right) \cdot \left((\left(\varepsilon \cdot x\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan \varepsilon + \tan x\right) \cdot \cos x - \sin x \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\\ \end{array}}\]

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed '#(1071215679 2002590028 935158157 1944352234 2656991306 2955288481)' +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)))