Average Error: 37.2 → 14.2
Time: 1.4m
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(x \cdot \varepsilon\right) \cdot \left((\left(x \cdot \varepsilon\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_* \le -7.551578005361442 \cdot 10^{-38}:\\ \;\;\;\;(\left(\tan x + \tan \varepsilon\right) \cdot \left(\frac{1}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)}\right) + \left(-\tan x\right))_*\\ \mathbf{if}\;(\left(x \cdot \varepsilon\right) \cdot \left((\left(x \cdot \varepsilon\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_* \le 1.1994857179829721 \cdot 10^{-32}:\\ \;\;\;\;(\left(x \cdot \varepsilon\right) \cdot \left((\left(x \cdot \varepsilon\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(\tan x + \tan \varepsilon\right)}{(\left(\tan x\right) \cdot \left(\tan \varepsilon\right) + \left(-1\right))_*} - \tan x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.2
Target14.9
Herbie14.2
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (fma (* x eps) (fma (* x eps) eps eps) eps) < -7.551578005361442e-38

    1. Initial program 33.4

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

    3. Applied tan-sum9.5

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

    5. Applied div-inv9.5

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

    6. Applied fma-neg9.4

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

    8. Applied add-log-exp9.6

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

    if -7.551578005361442e-38 < (fma (* x eps) (fma (* x eps) eps eps) eps) < 1.1994857179829721e-32

    1. Initial program 43.4

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

    2. Taylor expanded around 0 23.5

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

    3. Applied simplify22.2

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

    if 1.1994857179829721e-32 < (fma (* x eps) (fma (* x eps) eps eps) eps)

    1. Initial program 33.4

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

    3. Applied tan-sum9.0

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

    5. Applied frac-2neg9.0

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

    6. Applied simplify9.0

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018201 +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)))