Jmat.Real.lambertw, newton loop step

Average Error: 79.1% → 99.2%

Time: 16.8s

Precision: binary64

Cost: 7560.00

Math

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

↓

\[\begin{array}{l} t_0 := 1 - wj \cdot wj\\ \mathbf{if}\;wj \leq -6 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.00027:\\ \;\;\;\;\left(wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{wj \cdot wj}{t_0} + \left(wj - \frac{wj}{t_0}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (- 1.0 (* wj wj))))
   (if (<= wj -6e-7)
     (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
     (if (<= wj 0.00027)
       (- (+ (* wj wj) (+ x (* -2.0 (* wj x)))) (pow wj 3.0))
       (+ (/ (* wj wj) t_0) (- wj (/ wj t_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 t_0 = 1.0 - (wj * wj);
	double tmp;
	if (wj <= -6e-7) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else if (wj <= 0.00027) {
		tmp = ((wj * wj) + (x + (-2.0 * (wj * x)))) - pow(wj, 3.0);
	} else {
		tmp = ((wj * wj) / t_0) + (wj - (wj / t_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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (wj * wj)
    if (wj <= (-6d-7)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else if (wj <= 0.00027d0) then
        tmp = ((wj * wj) + (x + ((-2.0d0) * (wj * x)))) - (wj ** 3.0d0)
    else
        tmp = ((wj * wj) / t_0) + (wj - (wj / t_0))
    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 t_0 = 1.0 - (wj * wj);
	double tmp;
	if (wj <= -6e-7) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else if (wj <= 0.00027) {
		tmp = ((wj * wj) + (x + (-2.0 * (wj * x)))) - Math.pow(wj, 3.0);
	} else {
		tmp = ((wj * wj) / t_0) + (wj - (wj / t_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):
	t_0 = 1.0 - (wj * wj)
	tmp = 0
	if wj <= -6e-7:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	elif wj <= 0.00027:
		tmp = ((wj * wj) + (x + (-2.0 * (wj * x)))) - math.pow(wj, 3.0)
	else:
		tmp = ((wj * wj) / t_0) + (wj - (wj / t_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)
	t_0 = Float64(1.0 - Float64(wj * wj))
	tmp = 0.0
	if (wj <= -6e-7)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	elseif (wj <= 0.00027)
		tmp = Float64(Float64(Float64(wj * wj) + Float64(x + Float64(-2.0 * Float64(wj * x)))) - (wj ^ 3.0));
	else
		tmp = Float64(Float64(Float64(wj * wj) / t_0) + Float64(wj - Float64(wj / t_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)
	t_0 = 1.0 - (wj * wj);
	tmp = 0.0;
	if (wj <= -6e-7)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	elseif (wj <= 0.00027)
		tmp = ((wj * wj) + (x + (-2.0 * (wj * x)))) - (wj ^ 3.0);
	else
		tmp = ((wj * wj) / t_0) + (wj - (wj / t_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_] := Block[{t$95$0 = N[(1.0 - N[(wj * wj), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -6e-7], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.00027], N[(N[(N[(wj * wj), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(wj * wj), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(wj - N[(wj / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	79.1%
Target	80.1%
Herbie	99.2%

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

Derivation

1. Split input into 3 regimes

2. if wj < -5.9999999999999997e-7

   1. Initial program 93.7

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

   2. Simplified 93.9

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

      [Start] 93.7	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 93.7	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-mul-1 [=>] 93.7	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      *-commutative [=>] 93.7	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
      *-commutative [<=] 93.7	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      neg-mul-1 [<=] 93.7	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-sub0 [=>] 93.7	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 93.7	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      associate--r- [=>] 93.7	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      +-commutative [=>] 93.7	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      sub0-neg [=>] 93.7	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [<=] 93.7	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   if -5.9999999999999997e-7 < wj < 2.70000000000000003e-4

   1. Initial program 79.5

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

   2. Simplified 79.5

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

      [Start] 79.5	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 79.5	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-mul-1 [=>] 79.5	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      *-commutative [=>] 79.5	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
      *-commutative [<=] 79.5	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      neg-mul-1 [<=] 79.5	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-sub0 [=>] 79.5	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 79.5	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      associate--r- [=>] 79.5	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      +-commutative [=>] 79.5	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      sub0-neg [=>] 79.5	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [<=] 79.5	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   3. Taylor expanded in wj around 0 99.8

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

   4. Taylor expanded in x around 0 99.8

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

   5. Simplified 99.8

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

      [Start] 99.8	\[ -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left({wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]
      unpow2 [=>] 99.8	\[ -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(\color{blue}{wj \cdot wj} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]

   6. Taylor expanded in x around 0 99.8

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

   if 2.70000000000000003e-4 < wj

   1. Initial program 50.3

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

   2. Simplified 98.8

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

      [Start] 50.3	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 50.3	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-mul-1 [=>] 50.3	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      *-commutative [=>] 50.3	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
      *-commutative [<=] 50.3	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      neg-mul-1 [<=] 50.3	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-sub0 [=>] 50.3	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 50.3	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      associate--r- [=>] 50.3	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      +-commutative [=>] 50.3	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      sub0-neg [=>] 50.3	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [<=] 50.3	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   3. Taylor expanded in wj around 0 50.0

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

   4. Simplified 50.0

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

      [Start] 50.0	\[ wj + \frac{\left(-1 \cdot \left(wj \cdot x\right) + x\right) - wj}{wj + 1} \]
      associate-*r* [=>] 50.0	\[ wj + \frac{\left(\color{blue}{\left(-1 \cdot wj\right) \cdot x} + x\right) - wj}{wj + 1} \]
      neg-mul-1 [<=] 50.0	\[ wj + \frac{\left(\color{blue}{\left(-wj\right)} \cdot x + x\right) - wj}{wj + 1} \]
      distribute-lft1-in [=>] 50.0	\[ wj + \frac{\color{blue}{\left(\left(-wj\right) + 1\right) \cdot x} - wj}{wj + 1} \]
      +-commutative [<=] 50.0	\[ wj + \frac{\color{blue}{\left(1 + \left(-wj\right)\right)} \cdot x - wj}{wj + 1} \]
      sub-neg [<=] 50.0	\[ wj + \frac{\color{blue}{\left(1 - wj\right)} \cdot x - wj}{wj + 1} \]

   5. Applied egg-rr 50.0

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

   6. Simplified 50.0

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

      [Start] 50.0	\[ wj + \frac{\left(1 - wj\right) \cdot x - wj}{1 - wj \cdot wj} \cdot \left(1 - wj\right) \]
      associate-*l/ [=>] 50.0	\[ wj + \color{blue}{\frac{\left(\left(1 - wj\right) \cdot x - wj\right) \cdot \left(1 - wj\right)}{1 - wj \cdot wj}} \]
      *-commutative [=>] 50.0	\[ wj + \frac{\left(\color{blue}{x \cdot \left(1 - wj\right)} - wj\right) \cdot \left(1 - wj\right)}{1 - wj \cdot wj} \]

   7. Taylor expanded in x around 0 75.6

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

   8. Simplified 75.6

      \[\leadsto \color{blue}{wj + \frac{wj \cdot wj - wj}{1 - wj \cdot wj}} \]
      Proof

      [Start] 75.6	\[ -1 \cdot \frac{\left(1 - wj\right) \cdot wj}{1 - {wj}^{2}} + wj \]
      +-commutative [=>] 75.6	\[ \color{blue}{wj + -1 \cdot \frac{\left(1 - wj\right) \cdot wj}{1 - {wj}^{2}}} \]
      associate-*r/ [=>] 75.6	\[ wj + \color{blue}{\frac{-1 \cdot \left(\left(1 - wj\right) \cdot wj\right)}{1 - {wj}^{2}}} \]
      mul-1-neg [=>] 75.6	\[ wj + \frac{\color{blue}{-\left(1 - wj\right) \cdot wj}}{1 - {wj}^{2}} \]
      *-commutative [=>] 75.6	\[ wj + \frac{-\color{blue}{wj \cdot \left(1 - wj\right)}}{1 - {wj}^{2}} \]
      distribute-lft-neg-out [<=] 75.6	\[ wj + \frac{\color{blue}{\left(-wj\right) \cdot \left(1 - wj\right)}}{1 - {wj}^{2}} \]
      sub-neg [=>] 75.6	\[ wj + \frac{\left(-wj\right) \cdot \color{blue}{\left(1 + \left(-wj\right)\right)}}{1 - {wj}^{2}} \]
      distribute-lft-in [=>] 75.6	\[ wj + \frac{\color{blue}{\left(-wj\right) \cdot 1 + \left(-wj\right) \cdot \left(-wj\right)}}{1 - {wj}^{2}} \]
      *-rgt-identity [=>] 75.6	\[ wj + \frac{\color{blue}{\left(-wj\right)} + \left(-wj\right) \cdot \left(-wj\right)}{1 - {wj}^{2}} \]
      distribute-lft-neg-in [<=] 75.6	\[ wj + \frac{\left(-wj\right) + \color{blue}{\left(-wj \cdot \left(-wj\right)\right)}}{1 - {wj}^{2}} \]
      distribute-rgt-neg-out [=>] 75.6	\[ wj + \frac{\left(-wj\right) + \left(-\color{blue}{\left(-wj \cdot wj\right)}\right)}{1 - {wj}^{2}} \]
      remove-double-neg [=>] 75.6	\[ wj + \frac{\left(-wj\right) + \color{blue}{wj \cdot wj}}{1 - {wj}^{2}} \]
      +-commutative [<=] 75.6	\[ wj + \frac{\color{blue}{wj \cdot wj + \left(-wj\right)}}{1 - {wj}^{2}} \]
      sub-neg [<=] 75.6	\[ wj + \frac{\color{blue}{wj \cdot wj - wj}}{1 - {wj}^{2}} \]
      unpow2 [=>] 75.6	\[ wj + \frac{wj \cdot wj - wj}{1 - \color{blue}{wj \cdot wj}} \]

   9. Applied egg-rr 75.5

      \[\leadsto \color{blue}{\frac{wj \cdot wj}{1 - wj \cdot wj} - \left(\frac{wj}{1 - wj \cdot wj} - wj\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -6 \cdot 10^{-7}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.00027:\\ \;\;\;\;\left(wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{wj \cdot wj}{1 - wj \cdot wj} + \left(wj - \frac{wj}{1 - wj \cdot wj}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	99.0%
Cost	33604.00

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t_0}{e^{wj} + t_0} \leq 10^{-13}:\\ \;\;\;\;\left(wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \]

Alternative 2
Error	98.9%
Cost	7304.00

\[\begin{array}{l} t_0 := 1 - wj \cdot wj\\ \mathbf{if}\;wj \leq -5.8 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 0.00027:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + {wj}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{wj \cdot wj}{t_0} + \left(wj - \frac{wj}{t_0}\right)\\ \end{array} \]

Alternative 3
Error	97.9%
Cost	7172.00

\[\begin{array}{l} t_0 := 1 - wj \cdot wj\\ \mathbf{if}\;wj \leq 0.00031:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + {wj}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{wj \cdot wj}{t_0} + \left(wj - \frac{wj}{t_0}\right)\\ \end{array} \]

Alternative 4
Error	87.5%
Cost	6980.00

\[\begin{array}{l} t_0 := 1 - wj \cdot wj\\ \mathbf{if}\;wj \leq 0.00031:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{wj \cdot wj}{t_0} + \left(wj - \frac{wj}{t_0}\right)\\ \end{array} \]

Alternative 5
Error	86.8%
Cost	1481.00

\[\begin{array}{l} t_0 := 1 - wj \cdot wj\\ \mathbf{if}\;wj \leq -3 \cdot 10^{-23} \lor \neg \left(wj \leq 7.8 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{wj \cdot wj}{t_0} + \left(wj - \frac{wj}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \end{array} \]

Alternative 6
Error	88.0%
Cost	1480.00

\[\begin{array}{l} t_0 := 1 - wj \cdot wj\\ \mathbf{if}\;wj \leq -5.6 \cdot 10^{-17}:\\ \;\;\;\;wj + \frac{\left(x + \left(\left(wj \cdot wj\right) \cdot \left(x \cdot 0.5\right) - wj \cdot x\right)\right) - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{wj \cdot wj}{t_0} + \left(wj - \frac{wj}{t_0}\right)\\ \end{array} \]

Alternative 7
Error	87.9%
Cost	964.00

\[\begin{array}{l} \mathbf{if}\;wj \leq -5.6 \cdot 10^{-17}:\\ \;\;\;\;wj + \frac{x \cdot \left(1 - wj\right) - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 3.3 \cdot 10^{-5}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 8
Error	86.9%
Cost	708.00

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;\left(1 - wj\right) \cdot \frac{x}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 9
Error	86.9%
Cost	580.00

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

Alternative 10
Error	87.0%
Cost	580.00

\[\begin{array}{l} \mathbf{if}\;wj \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{1 - wj \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 11
Error	85.5%
Cost	448.00

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

Alternative 12
Error	4.4%
Cost	64.00

\[wj \]

Alternative 13
Error	85.0%
Cost	64.00

\[x \]

Reproduce

herbie shell --seed 2023098 
(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))))))