Average Error: 37.5 → 14.1
Time: 1.6m
Precision: 64
Internal Precision: 2432
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\log \left(e^{\tan \varepsilon + \tan x}\right)}{1 - \tan \varepsilon \cdot \tan x} - \tan x \le -1.1259439561261964 \cdot 10^{-16}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{{1}^{3} - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}}\right) \cdot \left(\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \tan \varepsilon \cdot \tan x\right) + 1\right) + \left(-\tan x\right))_*\\ \mathbf{if}\;\frac{\log \left(e^{\tan \varepsilon + \tan x}\right)}{1 - \tan \varepsilon \cdot \tan x} - \tan x \le 3.142285770032561 \cdot 10^{-17}:\\ \;\;\;\;(\left(\varepsilon \cdot \varepsilon\right) \cdot \left((x \cdot \left(x \cdot \varepsilon\right) + x)_*\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\tan \varepsilon + \tan x}{(\left(\tan x\right) \cdot \left(\tan \varepsilon\right) + \left(-1\right))_*}\right) - \tan x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.5
Target15.4
Herbie14.1
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (/ (log (exp (+ (tan x) (tan eps)))) (- 1 (* (tan x) (tan eps)))) (tan x)) < -1.1259439561261964e-16

    1. Initial program 29.9

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

    3. Applied tan-sum1.0

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

    5. Applied flip3--1.1

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

    6. Applied associate-/r/1.1

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

    7. Applied fma-neg1.0

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

    if -1.1259439561261964e-16 < (- (/ (log (exp (+ (tan x) (tan eps)))) (- 1 (* (tan x) (tan eps)))) (tan x)) < 3.142285770032561e-17

    1. Initial program 44.5

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

    2. Taylor expanded around 0 27.9

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

    3. Applied simplify26.7

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

    if 3.142285770032561e-17 < (- (/ (log (exp (+ (tan x) (tan eps)))) (- 1 (* (tan x) (tan eps)))) (tan x))

    1. Initial program 33.8

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

    3. Applied tan-sum7.1

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

    5. Applied frac-2neg7.1

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

    6. Applied simplify7.1

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

  4. Applied simplify14.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\log \left(e^{\tan \varepsilon + \tan x}\right)}{1 - \tan \varepsilon \cdot \tan x} - \tan x \le -1.1259439561261964 \cdot 10^{-16}:\\ \;\;\;\;(\left(\frac{\tan \varepsilon + \tan x}{{1}^{3} - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}}\right) \cdot \left(\left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \tan \varepsilon \cdot \tan x\right) + 1\right) + \left(-\tan x\right))_*\\ \mathbf{if}\;\frac{\log \left(e^{\tan \varepsilon + \tan x}\right)}{1 - \tan \varepsilon \cdot \tan x} - \tan x \le 3.142285770032561 \cdot 10^{-17}:\\ \;\;\;\;(\left(\varepsilon \cdot \varepsilon\right) \cdot \left((x \cdot \left(x \cdot \varepsilon\right) + x)_*\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\tan \varepsilon + \tan x}{(\left(\tan x\right) \cdot \left(\tan \varepsilon\right) + \left(-1\right))_*}\right) - \tan x\\ \end{array}}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :herbie-expected 28

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))