Average Error: 13.5 → 0.9
Time: 20.4s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(\left(\left({wj}^{2} + 1\right) - wj\right) \cdot \frac{x}{e^{wj} \cdot \left(1 + {wj}^{3}\right)} + \left({wj}^{2} - {wj}^{3}\right)\right) + {wj}^{4}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(\left(\left({wj}^{2} + 1\right) - wj\right) \cdot \frac{x}{e^{wj} \cdot \left(1 + {wj}^{3}\right)} + \left({wj}^{2} - {wj}^{3}\right)\right) + {wj}^{4}
double f(double wj, double x) {
        double r111234 = wj;
        double r111235 = exp(r111234);
        double r111236 = r111234 * r111235;
        double r111237 = x;
        double r111238 = r111236 - r111237;
        double r111239 = r111235 + r111236;
        double r111240 = r111238 / r111239;
        double r111241 = r111234 - r111240;
        return r111241;
}
↓
double f(double wj, double x) {
        double r111242 = wj;
        double r111243 = 2.0;
        double r111244 = pow(r111242, r111243);
        double r111245 = 1.0;
        double r111246 = r111244 + r111245;
        double r111247 = r111246 - r111242;
        double r111248 = x;
        double r111249 = exp(r111242);
        double r111250 = 3.0;
        double r111251 = pow(r111242, r111250);
        double r111252 = r111245 + r111251;
        double r111253 = r111249 * r111252;
        double r111254 = r111248 / r111253;
        double r111255 = r111247 * r111254;
        double r111256 = r111244 - r111251;
        double r111257 = r111255 + r111256;
        double r111258 = 4.0;
        double r111259 = pow(r111242, r111258);
        double r111260 = r111257 + r111259;
        return r111260;
}

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
Target13.0
Herbie0.9
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.5

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

  2. Simplified13.0

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

  4. Applied associate--l+7.0

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

  5. Taylor expanded around 0 0.9

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

  7. Applied flip3-+0.9

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

  8. Applied associate-/r/0.9

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

  9. Applied associate-/l*0.9

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

  10. Simplified0.9

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

  11. Final simplification0.9

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

Reproduce

herbie shell --seed 2019291 
(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))))))