Average Error: 37.1 → 15.7
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.138898032565299 \cdot 10^{-25}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon} \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}\right) \cdot \left(\frac{\log \left(e^{\tan x \cdot \sin \varepsilon}\right)}{\cos \varepsilon} + 1\right) + \left(-\tan x\right))_*\\ \mathbf{elif}\;\varepsilon \le 1.0623958280841123 \cdot 10^{-98}:\\ \;\;\;\;(\left(\varepsilon \cdot \left(x + \varepsilon\right)\right) \cdot x + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{(\left(\sin \varepsilon\right) \cdot \left(\cos x\right) + \left(\sin x \cdot \cos \varepsilon\right))_*}{\cos x \cdot \cos \varepsilon}}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target14.8
Herbie15.7
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -7.138898032565299e-25

    1. Initial program 29.1

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

    3. Applied tan-sum1.7

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

    5. Applied tan-quot1.7

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

    6. Applied associate-*r/1.7

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

    8. Applied flip--1.7

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

    9. Applied associate-/r/1.7

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

    10. Applied fma-neg1.7

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

    12. Applied add-log-exp1.8

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

    if -7.138898032565299e-25 < eps < 1.0623958280841123e-98

    1. Initial program 47.3

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

    3. Applied tan-sum47.3

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

    5. Applied tan-quot47.3

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

    6. Applied associate-*r/47.3

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

    8. Applied flip--47.3

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

    9. Applied associate-/r/47.3

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

    10. Applied fma-neg47.3

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

    11. Taylor expanded around 0 31.6

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

    12. Simplified31.5

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

    if 1.0623958280841123e-98 < eps

    1. Initial program 30.6

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

    3. Applied tan-sum6.9

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

    5. Applied tan-quot6.9

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

    6. Applied associate-*r/6.9

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

    8. Applied tan-quot7.1

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

    9. Applied tan-quot7.1

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

    10. Applied frac-add7.1

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

    11. Simplified7.1

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

  4. Final simplification15.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.138898032565299 \cdot 10^{-25}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon} \cdot \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}\right) \cdot \left(\frac{\log \left(e^{\tan x \cdot \sin \varepsilon}\right)}{\cos \varepsilon} + 1\right) + \left(-\tan x\right))_*\\ \mathbf{elif}\;\varepsilon \le 1.0623958280841123 \cdot 10^{-98}:\\ \;\;\;\;(\left(\varepsilon \cdot \left(x + \varepsilon\right)\right) \cdot x + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{(\left(\sin \varepsilon\right) \cdot \left(\cos x\right) + \left(\sin x \cdot \cos \varepsilon\right))_*}{\cos x \cdot \cos \varepsilon}}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 +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)))