Average Error: 59.8 → 0.5
Time: 49.6s
Precision: 64
Internal Precision: 2432
\[\frac{1}{x} - \frac{1}{\tan x}\]
↓
\[\log \left(e^{\frac{1}{45} \cdot {x}^{3}}\right) + \left(\frac{2}{945} \cdot {x}^{5} + \frac{1}{3} \cdot x\right)\]

Error

Bits error versus x

Target

Original59.8
Target0.1
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \lt 0.026:\\ \;\;\;\;\frac{x}{3} \cdot \left(1 + \frac{x \cdot x}{15}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{1}{\tan x}\\ \end{array}\]

Derivation

  1. Initial program 59.8

    \[\frac{1}{x} - \frac{1}{\tan x}\]

  2. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{\frac{1}{45} \cdot {x}^{3} + \left(\frac{2}{945} \cdot {x}^{5} + \frac{1}{3} \cdot x\right)}\]
  3. Using strategy rm

  4. Applied add-log-exp0.5

    \[\leadsto \color{blue}{\log \left(e^{\frac{1}{45} \cdot {x}^{3}}\right)} + \left(\frac{2}{945} \cdot {x}^{5} + \frac{1}{3} \cdot x\right)\]
  5. Removed slow pow expressions.

Runtime

Time bar (total: 49.6s)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o reduce:binary-search
(FPCore (x)
  :name "invcot (example 3.9)"
  :pre (and (< -0.026 x) (< x 0.026))

  :herbie-target
  (if (< (fabs x) 0.026) (* (/ x 3) (+ 1 (/ (* x x) 15))) (- (/ 1 x) (/ 1 (tan x))))

  (- (/ 1 x) (/ 1 (tan x))))