Average Error: 13.4 → 1.0
Time: 13.5s
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 r472460 = wj;
        double r472461 = exp(r472460);
        double r472462 = r472460 * r472461;
        double r472463 = x;
        double r472464 = r472462 - r472463;
        double r472465 = r472461 + r472462;
        double r472466 = r472464 / r472465;
        double r472467 = r472460 - r472466;
        return r472467;
}
↓
double f(double wj, double x) {
        double r472468 = 1.0;
        double r472469 = wj;
        double r472470 = r472468 - r472469;
        double r472471 = r472469 * r472469;
        double r472472 = r472470 + r472471;
        double r472473 = x;
        double r472474 = exp(r472469);
        double r472475 = r472473 / r472474;
        double r472476 = 3.0;
        double r472477 = pow(r472469, r472476);
        double r472478 = r472468 + r472477;
        double r472479 = r472475 / r472478;
        double r472480 = r472472 * r472479;
        double r472481 = 4.0;
        double r472482 = pow(r472469, r472481);
        double r472483 = 2.0;
        double r472484 = pow(r472469, r472483);
        double r472485 = r472482 + r472484;
        double r472486 = r472477 - r472485;
        double r472487 = r472480 - r472486;
        return r472487;
}

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

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))))))