Average Error: 13.4 → 1.0
Time: 13.7s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(\left(1 - wj\right) + wj \cdot wj\right) \cdot \frac{\frac{x}{e^{wj}}}{1 + {wj}^{3}} - \left({wj}^{3} - \left({wj}^{4} + {wj}^{2}\right)\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(\left(1 - wj\right) + wj \cdot wj\right) \cdot \frac{\frac{x}{e^{wj}}}{1 + {wj}^{3}} - \left({wj}^{3} - \left({wj}^{4} + {wj}^{2}\right)\right)
double f(double wj, double x) {
        double r592105 = wj;
        double r592106 = exp(r592105);
        double r592107 = r592105 * r592106;
        double r592108 = x;
        double r592109 = r592107 - r592108;
        double r592110 = r592106 + r592107;
        double r592111 = r592109 / r592110;
        double r592112 = r592105 - r592111;
        return r592112;
}
↓
double f(double wj, double x) {
        double r592113 = 1.0;
        double r592114 = wj;
        double r592115 = r592113 - r592114;
        double r592116 = r592114 * r592114;
        double r592117 = r592115 + r592116;
        double r592118 = x;
        double r592119 = exp(r592114);
        double r592120 = r592118 / r592119;
        double r592121 = 3.0;
        double r592122 = pow(r592114, r592121);
        double r592123 = r592113 + r592122;
        double r592124 = r592120 / r592123;
        double r592125 = r592117 * r592124;
        double r592126 = 4.0;
        double r592127 = pow(r592114, r592126);
        double r592128 = 2.0;
        double r592129 = pow(r592114, r592128);
        double r592130 = r592127 + r592129;
        double r592131 = r592122 - r592130;
        double r592132 = r592125 - r592131;
        return r592132;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.4
Target12.8
Herbie1.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.4

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

  2. Simplified12.9

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

  4. Applied div-sub12.8

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

  5. Applied associate-+l-6.7

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

  6. Taylor expanded around 0 1.0

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

  8. Applied flip3-+1.0

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

  9. Applied associate-/r/1.0

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

  10. Simplified1.0

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

  11. Final simplification1.0

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

Reproduce

herbie shell --seed 2020042 
(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))))))