Jmat.Real.lambertw, newton loop step

Percentage Accurate: 39.3% → 99.9%

Time: 17.5s

Precision: binary64

Cost: 15817

• 25.1% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq -5.8 \cdot 10^{-6} \lor \neg \left(wj \leq 3.7 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)\\ \mathbf{else}:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot t_0\right) - x \cdot -3\right) - x \cdot 0.6666666666666666\right) + \left(\left(1 - t_0\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))

↓

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (or (<= wj -5.8e-6) (not (<= wj 3.7e-6)))
     (fma (- (/ x (exp wj)) wj) (/ 1.0 (+ wj 1.0)) wj)
     (+
      (*
       (pow wj 3.0)
       (- (- (- -1.0 (* -2.0 t_0)) (* x -3.0)) (* x 0.6666666666666666)))
      (+ (* (- 1.0 t_0) (pow wj 2.0)) (+ x (* -2.0 (* wj x))))))))

C

double code(double wj, double x) {
	return wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))));
}

↓

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if ((wj <= -5.8e-6) || !(wj <= 3.7e-6)) {
		tmp = fma(((x / exp(wj)) - wj), (1.0 / (wj + 1.0)), wj);
	} else {
		tmp = (pow(wj, 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (x * 0.6666666666666666))) + (((1.0 - t_0) * pow(wj, 2.0)) + (x + (-2.0 * (wj * x))));
	}
	return tmp;
}

Julia

function code(wj, x)
	return Float64(wj - Float64(Float64(Float64(wj * exp(wj)) - x) / Float64(exp(wj) + Float64(wj * exp(wj)))))
end

↓

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if ((wj <= -5.8e-6) || !(wj <= 3.7e-6))
		tmp = fma(Float64(Float64(x / exp(wj)) - wj), Float64(1.0 / Float64(wj + 1.0)), wj);
	else
		tmp = Float64(Float64((wj ^ 3.0) * Float64(Float64(Float64(-1.0 - Float64(-2.0 * t_0)) - Float64(x * -3.0)) - Float64(x * 0.6666666666666666))) + Float64(Float64(Float64(1.0 - t_0) * (wj ^ 2.0)) + Float64(x + Float64(-2.0 * Float64(wj * x)))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := N[(wj - N[(N[(N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[wj_, x_] := Block[{t$95$0 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[wj, -5.8e-6], N[Not[LessEqual[wj, 3.7e-6]], $MachinePrecision]], N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision], N[(N[(N[Power[wj, 3.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x * -3.0), $MachinePrecision]), $MachinePrecision] - N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 - t$95$0), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	39.3%
Target	64.1%
Herbie	99.9%

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

Derivation

1. Split input into 2 regimes

2. if wj < -5.8000000000000004e-6 or 3.7000000000000002e-6 < wj

   1. Initial program 2.8%

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

   2. Simplified 99.8%

      \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]
      Step-by-step derivation

      [Start] 2.8%	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 2.8%	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 2.8%	\[ wj + \left(-\color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [=>] 2.8%	\[ wj + \left(-\color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} + \left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]
      +-commutative [=>] 2.8%	\[ wj + \left(-\color{blue}{\left(\left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      distribute-neg-in [=>] 2.8%	\[ wj + \color{blue}{\left(\left(-\left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\right) + \left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      remove-double-neg [=>] 2.8%	\[ wj + \left(\color{blue}{\frac{x}{e^{wj} + wj \cdot e^{wj}}} + \left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right) \]
      sub-neg [<=] 2.8%	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [<=] 2.8%	\[ wj + \color{blue}{\frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} \]
      distribute-rgt1-in [=>] 49.0%	\[ wj + \frac{x - wj \cdot e^{wj}}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]
      associate-/l/ [<=] 49.0%	\[ wj + \color{blue}{\frac{\frac{x - wj \cdot e^{wj}}{e^{wj}}}{wj + 1}} \]

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)} \]
      Step-by-step derivation

      [Start] 99.8%	\[ wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \]
      +-commutative [=>] 99.8%	\[ \color{blue}{\frac{\frac{x}{e^{wj}} - wj}{wj + 1} + wj} \]
      div-inv [=>] 99.8%	\[ \color{blue}{\left(\frac{x}{e^{wj}} - wj\right) \cdot \frac{1}{wj + 1}} + wj \]
      fma-def [=>] 99.8%	\[ \color{blue}{\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)} \]

   if -5.8000000000000004e-6 < wj < 3.7000000000000002e-6

   1. Initial program 78.6%

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

   2. Simplified 78.6%

      \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]
      Step-by-step derivation

      [Start] 78.6%	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 78.6%	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 78.6%	\[ wj + \left(-\color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [=>] 78.6%	\[ wj + \left(-\color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} + \left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]
      +-commutative [=>] 78.6%	\[ wj + \left(-\color{blue}{\left(\left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      distribute-neg-in [=>] 78.6%	\[ wj + \color{blue}{\left(\left(-\left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\right) + \left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      remove-double-neg [=>] 78.6%	\[ wj + \left(\color{blue}{\frac{x}{e^{wj} + wj \cdot e^{wj}}} + \left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right) \]
      sub-neg [<=] 78.6%	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [<=] 78.6%	\[ wj + \color{blue}{\frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} \]
      distribute-rgt1-in [=>] 78.6%	\[ wj + \frac{x - wj \cdot e^{wj}}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]
      associate-/l/ [<=] 78.6%	\[ wj + \color{blue}{\frac{\frac{x - wj \cdot e^{wj}}{e^{wj}}}{wj + 1}} \]

   3. Taylor expanded in wj around 0 100.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -5.8 \cdot 10^{-6} \lor \neg \left(wj \leq 3.7 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)\\ \mathbf{else}:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot -3\right) - x \cdot 0.6666666666666666\right) + \left(\left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	15817

\[\begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq -5.8 \cdot 10^{-6} \lor \neg \left(wj \leq 3.7 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)\\ \mathbf{else}:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot t_0\right) - x \cdot -3\right) - x \cdot 0.6666666666666666\right) + \left(\left(1 - t_0\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;wj \leq -5.3 \cdot 10^{-9} \lor \neg \left(wj \leq 7.2 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, \mathsf{fma}\left(-2, wj \cdot x, x\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;wj \leq -2.65 \cdot 10^{-9} \lor \neg \left(wj \leq 5.9 \cdot 10^{-9}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, \mathsf{fma}\left(-2, wj \cdot x, x\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	99.7%
Cost	7945

\[\begin{array}{l} \mathbf{if}\;wj \leq -2.2 \cdot 10^{-8} \lor \neg \left(wj \leq 1.1 \cdot 10^{-8}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	99.8%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;wj \leq -5.1 \cdot 10^{-9} \lor \neg \left(wj \leq 5.9 \cdot 10^{-9}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \end{array} \]

Alternative 6
Accuracy	99.0%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;wj \leq -1:\\ \;\;\;\;\frac{x}{wj \cdot e^{wj}}\\ \mathbf{elif}\;wj \leq 0.0025:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \]

Alternative 7
Accuracy	99.0%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;wj \leq -1:\\ \;\;\;\;\frac{x}{wj \cdot e^{wj}}\\ \mathbf{elif}\;wj \leq 0.00034:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 8
Accuracy	82.1%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;wj \leq -1.05 \cdot 10^{+176}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq -0.95:\\ \;\;\;\;\frac{x \cdot \left(1 - wj\right)}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.0004:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 9
Accuracy	82.1%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;wj \leq -3.2 \cdot 10^{+176}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq -0.00038:\\ \;\;\;\;wj - \frac{wj + wj \cdot x}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.0022:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 10
Accuracy	82.3%
Cost	1100

\[\begin{array}{l} \mathbf{if}\;wj \leq -2.6 \cdot 10^{+175}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq -2.65 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{x \cdot \left(1 - wj\right) - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.00106:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 11
Accuracy	75.8%
Cost	976

\[\begin{array}{l} \mathbf{if}\;wj \leq -1.15 \cdot 10^{+177}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq -2.25 \cdot 10^{-17}:\\ \;\;\;\;\frac{x \cdot \left(1 - wj\right)}{wj + 1}\\ \mathbf{elif}\;wj \leq -1.5 \cdot 10^{-50}:\\ \;\;\;\;\left(wj \cdot wj\right) \cdot \left(1 - x \cdot -2.5\right)\\ \mathbf{elif}\;wj \leq 0.000102:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 12
Accuracy	76.2%
Cost	844

\[\begin{array}{l} \mathbf{if}\;wj \leq -8.2 \cdot 10^{+176}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq -0.45:\\ \;\;\;\;\frac{x \cdot \left(-wj\right)}{wj + 1}\\ \mathbf{elif}\;wj \leq 1.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 13
Accuracy	73.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;wj \leq -1.5 \cdot 10^{-50}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq 0.0026:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj\\ \end{array} \]

Alternative 14
Accuracy	74.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;wj \leq -1.5 \cdot 10^{-50}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq 2.1 \cdot 10^{-7}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 15
Accuracy	73.4%
Cost	328

\[\begin{array}{l} \mathbf{if}\;wj \leq -1.5 \cdot 10^{-50}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq 0.0026:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj\\ \end{array} \]

Alternative 16
Accuracy	67.3%
Cost	196

\[\begin{array}{l} \mathbf{if}\;wj \leq 0.0026:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj\\ \end{array} \]

Alternative 17
Accuracy	27.0%
Cost	64

\[wj \]

Reproduce

herbie shell --seed 2023263 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

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