Average Error: 37.1 → 15.7
Time: 44.4s
Precision: 64
Internal Precision: 2368
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.8663464846517156 \cdot 10^{-25}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}\right) \cdot \left(1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + \left(\sqrt[3]{\tan x} \cdot \tan \varepsilon\right) \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right)\right) + \left(-\tan x\right))_*\\ \mathbf{elif}\;\varepsilon \le 2.452938405587065 \cdot 10^{-37}:\\ \;\;\;\;(\left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} - \tan x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target14.6
Herbie15.7
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -2.8663464846517156e-25

    1. Initial program 29.5

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

    3. Applied tan-sum1.5

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

    5. Applied flip3--1.6

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

    6. Applied associate-/r/1.5

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

    7. Applied fma-neg1.5

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

    9. Applied add-cube-cbrt1.6

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

    10. Applied associate-*l*1.6

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

    if -2.8663464846517156e-25 < eps < 2.452938405587065e-37

    1. Initial program 45.7

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

    2. Taylor expanded around 0 31.2

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

    3. Simplified31.1

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

    if 2.452938405587065e-37 < eps

    1. Initial program 29.9

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

    3. Applied tan-sum3.0

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

    5. Applied add-log-exp3.1

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

  4. Final simplification15.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.8663464846517156 \cdot 10^{-25}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}\right) \cdot \left(1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + \left(\sqrt[3]{\tan x} \cdot \tan \varepsilon\right) \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right)\right) + \left(-\tan x\right))_*\\ \mathbf{elif}\;\varepsilon \le 2.452938405587065 \cdot 10^{-37}:\\ \;\;\;\;(\left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)} - \tan x\\ \end{array}\]

Runtime

Time bar (total: 44.4s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes22.415.714.97.589.8%
herbie shell --seed 2018340 +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)))