Average Error: 14.1 → 0.1
Time: 24.4s
Precision: 64
Internal Precision: 832
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le -2.1942302806180627 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{1 + wj} + \frac{{wj}^{3} - {\left(\frac{wj}{1 + wj}\right)}^{3}}{\left(\frac{wj}{1 + wj} \cdot wj + \frac{wj}{1 + wj} \cdot \frac{wj}{1 + wj}\right) + wj \cdot wj}\\ \mathbf{elif}\;wj \le 2.1202725619686673 \cdot 10^{-05}:\\ \;\;\;\;\left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \left(x + \left(\frac{5}{2} \cdot wj + -2\right) \cdot \left(wj \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{1 + wj}\right) + \frac{\frac{\frac{x}{\sqrt{e^{wj}}}}{\sqrt{e^{wj}}}}{1 + wj}\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.1
Target13.5
Herbie0.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if wj < -2.1942302806180627e-05

    1. Initial program 2.2

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

    2. Initial simplification2.0

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

    4. Applied flip3--2.1

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

    if -2.1942302806180627e-05 < wj < 2.1202725619686673e-05

    1. Initial program 13.9

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

    2. Initial simplification7.2

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

    3. Taylor expanded around 0 0.0

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

    4. Taylor expanded around 0 0.0

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

    5. Simplified0.0

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

    if 2.1202725619686673e-05 < wj

    1. Initial program 31.1

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

    2. Initial simplification1.2

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

    4. Applied add-sqr-sqrt1.2

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

    5. Applied associate-/r*1.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -2.1942302806180627 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{1 + wj} + \frac{{wj}^{3} - {\left(\frac{wj}{1 + wj}\right)}^{3}}{\left(\frac{wj}{1 + wj} \cdot wj + \frac{wj}{1 + wj} \cdot \frac{wj}{1 + wj}\right) + wj \cdot wj}\\ \mathbf{elif}\;wj \le 2.1202725619686673 \cdot 10^{-05}:\\ \;\;\;\;\left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \left(x + \left(\frac{5}{2} \cdot wj + -2\right) \cdot \left(wj \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{1 + wj}\right) + \frac{\frac{\frac{x}{\sqrt{e^{wj}}}}{\sqrt{e^{wj}}}}{1 + wj}\\ \end{array}\]

Runtime

Time bar (total: 24.4s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes1.50.10.11.599.2%
herbie shell --seed 2018339 
(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))))))