Jmat.Real.lambertw, newton loop step

Average Error: 14.2 → 0.6

Time: 15.6s

Precision: binary64

Cost: 13828

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;\left(x + x \cdot \left(-2 \cdot wj\right)\right) + \left(\left(-{wj}^{3}\right) + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(wj + \frac{-x}{e^{wj}}\right) \cdot \frac{1}{wj + 1}\\ \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (if (<= wj 3.7e-7)
   (+ (+ x (* x (* -2.0 wj))) (+ (- (pow wj 3.0)) (pow wj 2.0)))
   (- wj (* (+ wj (/ (- x) (exp wj))) (/ 1.0 (+ wj 1.0))))))

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 <= 3.7e-7) {
		tmp = (x + (x * (-2.0 * wj))) + (-pow(wj, 3.0) + pow(wj, 2.0));
	} else {
		tmp = wj - ((wj + (-x / exp(wj))) * (1.0 / (wj + 1.0)));
	}
	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 <= 3.7d-7) then
        tmp = (x + (x * ((-2.0d0) * wj))) + (-(wj ** 3.0d0) + (wj ** 2.0d0))
    else
        tmp = wj - ((wj + (-x / exp(wj))) * (1.0d0 / (wj + 1.0d0)))
    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 <= 3.7e-7) {
		tmp = (x + (x * (-2.0 * wj))) + (-Math.pow(wj, 3.0) + Math.pow(wj, 2.0));
	} else {
		tmp = wj - ((wj + (-x / Math.exp(wj))) * (1.0 / (wj + 1.0)));
	}
	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 <= 3.7e-7:
		tmp = (x + (x * (-2.0 * wj))) + (-math.pow(wj, 3.0) + math.pow(wj, 2.0))
	else:
		tmp = wj - ((wj + (-x / math.exp(wj))) * (1.0 / (wj + 1.0)))
	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 <= 3.7e-7)
		tmp = Float64(Float64(x + Float64(x * Float64(-2.0 * wj))) + Float64(Float64(-(wj ^ 3.0)) + (wj ^ 2.0)));
	else
		tmp = Float64(wj - Float64(Float64(wj + Float64(Float64(-x) / exp(wj))) * Float64(1.0 / Float64(wj + 1.0))));
	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 <= 3.7e-7)
		tmp = (x + (x * (-2.0 * wj))) + (-(wj ^ 3.0) + (wj ^ 2.0));
	else
		tmp = wj - ((wj + (-x / exp(wj))) * (1.0 / (wj + 1.0)));
	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, 3.7e-7], N[(N[(x + N[(x * N[(-2.0 * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-N[Power[wj, 3.0], $MachinePrecision]) + N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj + N[((-x) / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	14.2
Target	13.5
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if wj < 3.70000000000000004e-7

   1. Initial program 13.8

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

   2. Taylor expanded in wj around 0 0.6

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

   3. Simplified 0.6

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

      [Start] 0.6	\[ -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]
      rational.json-simplify-41 [=>] 0.6	\[ -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \color{blue}{\left(\left(-2 \cdot \left(wj \cdot x\right) + x\right) + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right)} \]
      rational.json-simplify-11 [=>] 0.6	\[ \color{blue}{\left(-2 \cdot \left(wj \cdot x\right) + x\right) + \left(-1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right)} \]
      rational.json-simplify-41 [=>] 0.6	\[ \color{blue}{\left(x + -2 \cdot \left(wj \cdot x\right)\right)} + \left(-1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right) \]
      rational.json-simplify-39 [=>] 0.6	\[ \left(x + -2 \cdot \color{blue}{\left(x \cdot wj\right)}\right) + \left(-1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right) \]
      rational.json-simplify-3 [=>] 0.6	\[ \left(x + \color{blue}{x \cdot \left(-2 \cdot wj\right)}\right) + \left(-1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right) \]

   4. Taylor expanded in x around 0 0.6

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

   5. Simplified 0.6

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

      [Start] 0.6	\[ \left(x + x \cdot \left(-2 \cdot wj\right)\right) + \left(-1 \cdot {wj}^{3} + \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2}\right) \]
      rational.json-simplify-39 [=>] 0.6	\[ \left(x + x \cdot \left(-2 \cdot wj\right)\right) + \left(\color{blue}{{wj}^{3} \cdot -1} + \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2}\right) \]
      rational.json-simplify-71 [<=] 0.6	\[ \left(x + x \cdot \left(-2 \cdot wj\right)\right) + \left(\color{blue}{\left(-{wj}^{3}\right)} + \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2}\right) \]

   6. Taylor expanded in x around 0 0.6

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

   if 3.70000000000000004e-7 < wj

   1. Initial program 32.9

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

   2. Applied egg-rr 33.0

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

   3. Taylor expanded in wj around inf 2.1

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

   4. Simplified 2.1

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

      [Start] 2.1	\[ wj - \left(-1 \cdot \frac{x}{e^{wj}} + wj\right) \cdot \frac{1}{wj + 1} \]
      rational.json-simplify-41 [=>] 2.1	\[ wj - \color{blue}{\left(wj + -1 \cdot \frac{x}{e^{wj}}\right)} \cdot \frac{1}{wj + 1} \]
      rational.json-simplify-20 [=>] 2.1	\[ wj - \left(wj + \color{blue}{\frac{x \cdot -1}{e^{wj}}}\right) \cdot \frac{1}{wj + 1} \]
      rational.json-simplify-71 [<=] 2.1	\[ wj - \left(wj + \frac{\color{blue}{-x}}{e^{wj}}\right) \cdot \frac{1}{wj + 1} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;\left(x + x \cdot \left(-2 \cdot wj\right)\right) + \left(\left(-{wj}^{3}\right) + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(wj + \frac{-x}{e^{wj}}\right) \cdot \frac{1}{wj + 1}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	13764

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+38}:\\ \;\;\;\;wj - \left(wj + \frac{-x}{e^{wj}}\right) \cdot \frac{1}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left({wj}^{2} + \left(x + wj \cdot \frac{x}{-0.5}\right)\right) - {wj}^{3}\\ \end{array} \]

Alternative 2
Error	0.9
Cost	7428

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.15 \cdot 10^{-8}:\\ \;\;\;\;x + \left({wj}^{2} + x \cdot \left(wj \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(wj + \frac{-x}{e^{wj}}\right) \cdot \frac{1}{wj + 1}\\ \end{array} \]

Alternative 3
Error	1.9
Cost	7040

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

Alternative 4
Error	9.5
Cost	6980

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.6 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{wj}^{2}\\ \end{array} \]

Alternative 5
Error	9.5
Cost	6980

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.25 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;{wj}^{2}\\ \end{array} \]

Alternative 6
Error	9.3
Cost	6660

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.25 \cdot 10^{-38}:\\ \;\;\;\;x \cdot \frac{1 - wj}{wj + 1} + \left(wj - \frac{wj}{wj + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;{wj}^{2}\\ \end{array} \]

Alternative 7
Error	7.9
Cost	1088

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

Alternative 8
Error	8.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;wj \leq 6.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{\frac{-1 - wj}{wj - 1}}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 9
Error	8.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;wj \leq 6.2 \cdot 10^{-15}:\\ \;\;\;\;-2 \cdot \left(wj \cdot x\right) + x\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 10
Error	9.3
Cost	448

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

Alternative 11
Error	61.2
Cost	64

\[wj \]

Alternative 12
Error	9.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023066 
(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))))))