Average Error: 37.4 → 15.3
Time: 1.6m
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -5.863232554822832 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \le 2.832219681528646 \cdot 10^{-38}:\\ \;\;\;\;(\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\cos x\right) \cdot \left(\tan x + \tan \varepsilon\right) + \left((\left(\tan x\right) \cdot \left(\tan \varepsilon\right) + -1)_* \cdot \sin x\right))_*}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.4
Target15.3
Herbie15.3
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -5.863232554822832e-17

    1. Initial program 30.2

      \[\tan \left(x + \varepsilon\right) - \tan x\]
    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 *-un-lft-identity0.8

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

    6. Applied *-un-lft-identity0.8

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

    7. Applied distribute-lft-out0.8

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

    8. Applied associate-/l*0.9

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

    if -5.863232554822832e-17 < eps < 2.832219681528646e-38

    1. Initial program 45.8

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

    3. Applied tan-sum45.8

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

    5. Applied tan-quot45.8

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

    6. Applied associate-*l/45.8

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

    8. Applied flip3--45.8

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

    9. Applied associate-/r/45.8

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

    10. Applied fma-neg45.8

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

    11. Taylor expanded around 0 31.2

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

    12. Simplified31.2

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

    if 2.832219681528646e-38 < eps

    1. Initial program 30.4

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

    3. Applied tan-quot30.2

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

    4. Applied tan-sum3.2

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

    5. Applied frac-sub3.2

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

    6. Simplified3.2

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

  4. Final simplification15.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.863232554822832 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \le 2.832219681528646 \cdot 10^{-38}:\\ \;\;\;\;(\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\cos x\right) \cdot \left(\tan x + \tan \varepsilon\right) + \left((\left(\tan x\right) \cdot \left(\tan \varepsilon\right) + -1)_* \cdot \sin x\right))_*}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes22.015.314.67.489.7%
herbie shell --seed 2018263 +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)))