Jmat.Real.lambertw, newton loop step

Average Error: 13.5 → 0.4

Time: 3.3m

Precision: 64

• could not determine a ground truth for program body

  1. wj = 2.7115938426145936e+256
  2. x = 2.495674666903784e+223

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj \le -5.670239909763437 \cdot 10^{-09}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{elif}\;wj \le 6.686797031083422 \cdot 10^{-09}:\\ \;\;\;\;(\left((x \cdot -2 + wj)_*\right) \cdot wj + x)_*\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array}\]

C

double f(double wj, double x) {
        double r41688136 = wj;
        double r41688137 = exp(r41688136);
        double r41688138 = r41688136 * r41688137;
        double r41688139 = x;
        double r41688140 = r41688138 - r41688139;
        double r41688141 = r41688137 + r41688138;
        double r41688142 = r41688140 / r41688141;
        double r41688143 = r41688136 - r41688142;
        return r41688143;
}

↓

double f(double wj, double x) {
        double r41688144 = wj;
        double r41688145 = -5.670239909763437e-09;
        bool r41688146 = r41688144 <= r41688145;
        double r41688147 = x;
        double r41688148 = exp(r41688144);
        double r41688149 = r41688147 / r41688148;
        double r41688150 = r41688144 - r41688149;
        double r41688151 = 1.0;
        double r41688152 = r41688144 + r41688151;
        double r41688153 = r41688150 / r41688152;
        double r41688154 = r41688144 - r41688153;
        double r41688155 = 6.686797031083422e-09;
        bool r41688156 = r41688144 <= r41688155;
        double r41688157 = -2.0;
        double r41688158 = fma(r41688147, r41688157, r41688144);
        double r41688159 = fma(r41688158, r41688144, r41688147);
        double r41688160 = r41688156 ? r41688159 : r41688154;
        double r41688161 = r41688146 ? r41688154 : r41688160;
        return r41688161;
}

Error

Bits error versus wj

Bits error versus x

Target

Original	13.5
Target	13.0
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if wj < -5.670239909763437e-09 or 6.686797031083422e-09 < wj

   1. Initial program 15.1

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

   3. Applied distribute-rgt1-in 15.1

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

   4. Applied *-un-lft-identity 15.1

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

   5. Applied times-frac 15.0

      \[\leadsto wj - \color{blue}{\frac{1}{wj + 1} \cdot \frac{wj \cdot e^{wj} - x}{e^{wj}}}\]

   6. Applied add-cube-cbrt 15.9

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

   7. Applied prod-diff 15.9

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

   8. Simplified 15.0

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

   9. Simplified 3.8

      \[\leadsto \left(wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right) + \color{blue}{0}\]

   if -5.670239909763437e-09 < wj < 6.686797031083422e-09

   1. Initial program 13.4

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

   2. Taylor expanded around 0 0.2

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

   3. Simplified 0.2

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -5.670239909763437 \cdot 10^{-09}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{elif}\;wj \le 6.686797031083422 \cdot 10^{-09}:\\ \;\;\;\;(\left((x \cdot -2 + wj)_*\right) \cdot wj + x)_*\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(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))))))