Average Error: 13.5 → 1.1
Time: 4.1s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)
double f(double wj, double x) {
        double r209192 = wj;
        double r209193 = exp(r209192);
        double r209194 = r209192 * r209193;
        double r209195 = x;
        double r209196 = r209194 - r209195;
        double r209197 = r209193 + r209194;
        double r209198 = r209196 / r209197;
        double r209199 = r209192 - r209198;
        return r209199;
}
↓
double f(double wj, double x) {
        double r209200 = x;
        double r209201 = wj;
        double r209202 = 1.0;
        double r209203 = r209201 + r209202;
        double r209204 = r209200 / r209203;
        double r209205 = exp(r209201);
        double r209206 = r209204 / r209205;
        double r209207 = 4.0;
        double r209208 = pow(r209201, r209207);
        double r209209 = 2.0;
        double r209210 = pow(r209201, r209209);
        double r209211 = r209208 + r209210;
        double r209212 = 3.0;
        double r209213 = pow(r209201, r209212);
        double r209214 = r209211 - r209213;
        double r209215 = r209206 + r209214;
        return r209215;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.5
Target12.8
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.5

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

  2. Simplified12.8

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

  4. Applied associate--l+6.6

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

  5. Taylor expanded around 0 1.1

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

  6. Final simplification1.1

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

Reproduce

herbie shell --seed 2020060 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

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

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