Jmat.Real.lambertw, newton loop step

Average Error: 14.1 → 0.9

Time: 15.0s

Precision: binary64

Cost: 13896

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj \leq -3.3 \cdot 10^{-7}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\\ \mathbf{elif}\;wj \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\left(x + wj \cdot \left(x \cdot -2\right)\right) + \left({wj}^{2} - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(1 + \left(-\frac{1 + \frac{x}{e^{wj}}}{wj}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (if (<= wj -3.3e-7)
   (- wj (/ (- (* wj (exp wj)) x) (* (+ wj 1.0) (exp wj))))
   (if (<= wj 5.8e-7)
     (+ (+ x (* wj (* x -2.0))) (- (pow wj 2.0) (pow wj 3.0)))
     (- wj (+ 1.0 (- (/ (+ 1.0 (/ x (exp wj))) wj)))))))

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 tmp;
	if (wj <= -3.3e-7) {
		tmp = wj - (((wj * exp(wj)) - x) / ((wj + 1.0) * exp(wj)));
	} else if (wj <= 5.8e-7) {
		tmp = (x + (wj * (x * -2.0))) + (pow(wj, 2.0) - pow(wj, 3.0));
	} else {
		tmp = wj - (1.0 + -((1.0 + (x / exp(wj))) / wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))))
end function

↓

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-3.3d-7)) then
        tmp = wj - (((wj * exp(wj)) - x) / ((wj + 1.0d0) * exp(wj)))
    else if (wj <= 5.8d-7) then
        tmp = (x + (wj * (x * (-2.0d0)))) + ((wj ** 2.0d0) - (wj ** 3.0d0))
    else
        tmp = wj - (1.0d0 + -((1.0d0 + (x / exp(wj))) / wj))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -3.3e-7) {
		tmp = wj - (((wj * Math.exp(wj)) - x) / ((wj + 1.0) * Math.exp(wj)));
	} else if (wj <= 5.8e-7) {
		tmp = (x + (wj * (x * -2.0))) + (Math.pow(wj, 2.0) - Math.pow(wj, 3.0));
	} else {
		tmp = wj - (1.0 + -((1.0 + (x / Math.exp(wj))) / wj));
	}
	return tmp;
}

Python

def code(wj, x):
	return wj - (((wj * math.exp(wj)) - x) / (math.exp(wj) + (wj * math.exp(wj))))

↓

def code(wj, x):
	tmp = 0
	if wj <= -3.3e-7:
		tmp = wj - (((wj * math.exp(wj)) - x) / ((wj + 1.0) * math.exp(wj)))
	elif wj <= 5.8e-7:
		tmp = (x + (wj * (x * -2.0))) + (math.pow(wj, 2.0) - math.pow(wj, 3.0))
	else:
		tmp = wj - (1.0 + -((1.0 + (x / math.exp(wj))) / wj))
	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)
	tmp = 0.0
	if (wj <= -3.3e-7)
		tmp = Float64(wj - Float64(Float64(Float64(wj * exp(wj)) - x) / Float64(Float64(wj + 1.0) * exp(wj))));
	elseif (wj <= 5.8e-7)
		tmp = Float64(Float64(x + Float64(wj * Float64(x * -2.0))) + Float64((wj ^ 2.0) - (wj ^ 3.0)));
	else
		tmp = Float64(wj - Float64(1.0 + Float64(-Float64(Float64(1.0 + Float64(x / exp(wj))) / wj))));
	end
	return tmp
end

MATLAB

function tmp = code(wj, x)
	tmp = wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))));
end

↓

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -3.3e-7)
		tmp = wj - (((wj * exp(wj)) - x) / ((wj + 1.0) * exp(wj)));
	elseif (wj <= 5.8e-7)
		tmp = (x + (wj * (x * -2.0))) + ((wj ^ 2.0) - (wj ^ 3.0));
	else
		tmp = wj - (1.0 + -((1.0 + (x / exp(wj))) / wj));
	end
	tmp_2 = 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_] := If[LessEqual[wj, -3.3e-7], N[(wj - N[(N[(N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(N[(wj + 1.0), $MachinePrecision] * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 5.8e-7], N[(N[(x + N[(wj * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 2.0], $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(1.0 + (-N[(N[(1.0 + N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / wj), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Target

Original	14.1
Target	13.4
Herbie	0.9

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

Derivation

1. Split input into 3 regimes

2. if wj < -3.3000000000000002e-7

   1. Initial program 3.6

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

   2. Applied egg-rr 3.7

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

   if -3.3000000000000002e-7 < wj < 5.7999999999999995e-7

   1. Initial program 13.9

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

   2. Taylor expanded in wj around 0 0.1

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

   3. Simplified 0.1

      \[\leadsto \color{blue}{\left(x + wj \cdot \left(x \cdot -2\right)\right) + \left(\left(1 - x \cdot -2.5\right) \cdot {wj}^{2} + \left(-\left(\left(1 + -2 \cdot \left(x \cdot -2.5\right)\right) + x \cdot -2.3333333333333335\right) \cdot {wj}^{3}\right)\right)} \]
      Proof

      [Start] 0.1	\[ -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]
      rational_best.json-simplify-43 [=>] 0.1	\[ \color{blue}{\left(-2 \cdot \left(wj \cdot x\right) + x\right) + \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)\right)} \]
      rational_best.json-simplify-1 [=>] 0.1	\[ \color{blue}{\left(x + -2 \cdot \left(wj \cdot x\right)\right)} + \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)\right) \]
      rational_best.json-simplify-44 [=>] 0.1	\[ \left(x + \color{blue}{wj \cdot \left(-2 \cdot x\right)}\right) + \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)\right) \]
      rational_best.json-simplify-2 [=>] 0.1	\[ \left(x + wj \cdot \color{blue}{\left(x \cdot -2\right)}\right) + \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)\right) \]
      rational_best.json-simplify-2 [=>] 0.1	\[ \left(x + wj \cdot \left(x \cdot -2\right)\right) + \left(\left(1 - \left(\color{blue}{x \cdot -4} + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)\right) \]
      rational_best.json-simplify-2 [=>] 0.1	\[ \left(x + wj \cdot \left(x \cdot -2\right)\right) + \left(\left(1 - \left(x \cdot -4 + \color{blue}{x \cdot 1.5}\right)\right) \cdot {wj}^{2} + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)\right) \]
      rational_best.json-simplify-47 [=>] 0.1	\[ \left(x + wj \cdot \left(x \cdot -2\right)\right) + \left(\left(1 - \color{blue}{x \cdot \left(1.5 + -4\right)}\right) \cdot {wj}^{2} + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)\right) \]
      metadata-eval [=>] 0.1	\[ \left(x + wj \cdot \left(x \cdot -2\right)\right) + \left(\left(1 - x \cdot \color{blue}{-2.5}\right) \cdot {wj}^{2} + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)\right) \]
      rational_best.json-simplify-2 [=>] 0.1	\[ \left(x + wj \cdot \left(x \cdot -2\right)\right) + \left(\left(1 - x \cdot -2.5\right) \cdot {wj}^{2} + \color{blue}{\left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \cdot -1}\right) \]
      rational_best.json-simplify-12 [=>] 0.1	\[ \left(x + wj \cdot \left(x \cdot -2\right)\right) + \left(\left(1 - x \cdot -2.5\right) \cdot {wj}^{2} + \color{blue}{\left(-\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)}\right) \]

   4. Taylor expanded in x around 0 0.0

      \[\leadsto \left(x + wj \cdot \left(x \cdot -2\right)\right) + \left(\left(1 - x \cdot -2.5\right) \cdot {wj}^{2} + \left(-\color{blue}{{wj}^{3}}\right)\right) \]

   5. Taylor expanded in x around 0 0.1

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

   if 5.7999999999999995e-7 < wj

   1. Initial program 29.9

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

   2. Taylor expanded in wj around -inf 31.2

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

   3. Simplified 31.2

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

      [Start] 31.2	\[ wj - \left(1 + -1 \cdot \frac{1 + \frac{x}{e^{wj}}}{wj}\right) \]
      rational_best.json-simplify-2 [=>] 31.2	\[ wj - \left(1 + \color{blue}{\frac{1 + \frac{x}{e^{wj}}}{wj} \cdot -1}\right) \]
      rational_best.json-simplify-12 [=>] 31.2	\[ wj - \left(1 + \color{blue}{\left(-\frac{1 + \frac{x}{e^{wj}}}{wj}\right)}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.3 \cdot 10^{-7}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\\ \mathbf{elif}\;wj \leq 5.8 \cdot 10^{-7}:\\ \;\;\;\;\left(x + wj \cdot \left(x \cdot -2\right)\right) + \left({wj}^{2} - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(1 + \left(-\frac{1 + \frac{x}{e^{wj}}}{wj}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.9
Cost	14848

\[\left(x + wj \cdot \left(x \cdot -2\right)\right) + \left(\left(1 - x \cdot -2.5\right) \cdot {wj}^{2} + \left(-\left(\left(1 + -2 \cdot \left(x \cdot -2.5\right)\right) + x \cdot -2.3333333333333335\right) \cdot {wj}^{3}\right)\right) \]

Alternative 2
Error	1.9
Cost	14080

\[\left(x + wj \cdot \left(x \cdot -2\right)\right) + \left(\left(1 - x \cdot -2.5\right) \cdot {wj}^{2} + \left(-{wj}^{3}\right)\right) \]

Alternative 3
Error	2.1
Cost	7424

\[x + \left(x \cdot \left(-2 \cdot wj\right) + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\right) \]

Alternative 4
Error	2.2
Cost	7040

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

Alternative 5
Error	10.0
Cost	6792

\[\begin{array}{l} t_0 := -2 \cdot wj + 1\\ t_1 := x + x \cdot \left(-2 \cdot wj\right)\\ \mathbf{if}\;wj \leq -2 \cdot 10^{-65}:\\ \;\;\;\;x \cdot \left(\left(t_0 \cdot t_1\right) \cdot \frac{\frac{x \cdot t_0}{t_1}}{t_1}\right)\\ \mathbf{elif}\;wj \leq -5.2 \cdot 10^{-70}:\\ \;\;\;\;{wj}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot x\\ \end{array} \]

Alternative 6
Error	9.8
Cost	448

\[\left(-2 \cdot wj + 1\right) \cdot x \]

Alternative 7
Error	61.2
Cost	64

\[wj \]

Alternative 8
Error	10.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(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))))))