Average Error: 13.9 → 0.6
Time: 42.7s
Precision: 64
Internal Precision: 128
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 3.052356521191451 \cdot 10^{-06}:\\ \;\;\;\;\left(\left(\frac{5}{2} \cdot wj + -2\right) \cdot \left(x \cdot wj\right) + x\right) + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{wj \cdot wj - \frac{wj}{wj + 1} \cdot \frac{wj}{wj + 1}}{\frac{wj}{wj + 1} + wj} + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.9
Target13.3
Herbie0.6
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 3.052356521191451e-06

    1. Initial program 13.6

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]

    2. Initial simplification7.0

      \[\leadsto \left(wj - \frac{wj}{wj + 1}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\]

    3. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{\left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right)} + \frac{\frac{x}{e^{wj}}}{wj + 1}\]

    4. Taylor expanded around 0 0.6

      \[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \color{blue}{\left(\left(x + \frac{5}{2} \cdot \left(x \cdot {wj}^{2}\right)\right) - 2 \cdot \left(x \cdot wj\right)\right)}\]

    5. Simplified0.6

      \[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \color{blue}{\left(\left(wj \cdot \frac{5}{2} + -2\right) \cdot \left(x \cdot wj\right) + x\right)}\]

    if 3.052356521191451e-06 < wj

    1. Initial program 25.8

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]

    2. Initial simplification1.7

      \[\leadsto \left(wj - \frac{wj}{wj + 1}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\]
    3. Using strategy rm

    4. Applied flip--1.8

      \[\leadsto \color{blue}{\frac{wj \cdot wj - \frac{wj}{wj + 1} \cdot \frac{wj}{wj + 1}}{wj + \frac{wj}{wj + 1}}} + \frac{\frac{x}{e^{wj}}}{wj + 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 3.052356521191451 \cdot 10^{-06}:\\ \;\;\;\;\left(\left(\frac{5}{2} \cdot wj + -2\right) \cdot \left(x \cdot wj\right) + x\right) + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{wj \cdot wj - \frac{wj}{wj + 1} \cdot \frac{wj}{wj + 1}}{\frac{wj}{wj + 1} + wj} + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \end{array}\]

Runtime

Time bar (total: 42.7s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes1.10.60.11.046.7%
herbie shell --seed 2018353 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))