Average Error: 13.8 → 0.7
Time: 1.0m
Precision: 64
Internal Precision: 896
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\sqrt[3]{{\left(\frac{1}{wj}\right)}^{-2}} \cdot \sqrt[3]{{\left(\frac{1}{wj}\right)}^{-2}}\right) \cdot \log \left(e^{\sqrt[3]{{\left(\frac{1}{wj}\right)}^{-2}}}\right) + x\right) - 2 \cdot \left(wj \cdot x\right) \le 4.478911691085116 \cdot 10^{-11}:\\ \;\;\;\;\left({wj}^{2} + x\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{1 + wj}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

Target

Original13.8
Target13.1
Herbie0.7
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (- (+ (* (* (cbrt (pow (/ 1 wj) -2)) (cbrt (pow (/ 1 wj) -2))) (log (exp (cbrt (pow (/ 1 wj) -2))))) x) (* 2 (* wj x))) < 4.478911691085116e-11

    1. Initial program 18.1

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

    2. Taylor expanded around 0 0.9

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

    if 4.478911691085116e-11 < (- (+ (* (* (cbrt (pow (/ 1 wj) -2)) (cbrt (pow (/ 1 wj) -2))) (log (exp (cbrt (pow (/ 1 wj) -2))))) x) (* 2 (* wj x)))

    1. Initial program 2.4

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
    2. Using strategy rm

    3. Applied div-sub2.4

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

    4. Applied associate--r-2.4

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

    5. Applied simplify0.2

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' 
(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))))))