Jmat.Real.lambertw, newton loop step

Average Accuracy: 79.5% → 98.9%

Time: 13.7s

Precision: binary64

Cost: 15684

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq 1.85 \cdot 10^{-6}:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot t_0\right) - x \cdot -3\right) - 0.6666666666666666 \cdot x\right) + \left(\left(1 - t_0\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + {\left(\frac{wj + 1}{\frac{x}{e^{wj}} - wj}\right)}^{-1}\\ \end{array} \]

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 (<= wj 1.85e-6)
     (+
      (*
       (pow wj 3.0)
       (- (- (- -1.0 (* -2.0 t_0)) (* x -3.0)) (* 0.6666666666666666 x)))
      (+ (* (- 1.0 t_0) (pow wj 2.0)) (+ x (* -2.0 (* wj x)))))
     (+ wj (pow (/ (+ wj 1.0) (- (/ x (exp wj)) 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 t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= 1.85e-6) {
		tmp = (pow(wj, 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (0.6666666666666666 * x))) + (((1.0 - t_0) * pow(wj, 2.0)) + (x + (-2.0 * (wj * x))));
	} else {
		tmp = wj + pow(((wj + 1.0) / ((x / exp(wj)) - 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) :: t_0
    real(8) :: tmp
    t_0 = (x * (-4.0d0)) + (x * 1.5d0)
    if (wj <= 1.85d-6) then
        tmp = ((wj ** 3.0d0) * ((((-1.0d0) - ((-2.0d0) * t_0)) - (x * (-3.0d0))) - (0.6666666666666666d0 * x))) + (((1.0d0 - t_0) * (wj ** 2.0d0)) + (x + ((-2.0d0) * (wj * x))))
    else
        tmp = wj + (((wj + 1.0d0) / ((x / exp(wj)) - 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 t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= 1.85e-6) {
		tmp = (Math.pow(wj, 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (0.6666666666666666 * x))) + (((1.0 - t_0) * Math.pow(wj, 2.0)) + (x + (-2.0 * (wj * x))));
	} else {
		tmp = wj + Math.pow(((wj + 1.0) / ((x / Math.exp(wj)) - 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):
	t_0 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if wj <= 1.85e-6:
		tmp = (math.pow(wj, 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (0.6666666666666666 * x))) + (((1.0 - t_0) * math.pow(wj, 2.0)) + (x + (-2.0 * (wj * x))))
	else:
		tmp = wj + math.pow(((wj + 1.0) / ((x / math.exp(wj)) - 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)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (wj <= 1.85e-6)
		tmp = Float64(Float64((wj ^ 3.0) * Float64(Float64(Float64(-1.0 - Float64(-2.0 * t_0)) - Float64(x * -3.0)) - Float64(0.6666666666666666 * x))) + Float64(Float64(Float64(1.0 - t_0) * (wj ^ 2.0)) + Float64(x + Float64(-2.0 * Float64(wj * x)))));
	else
		tmp = Float64(wj + (Float64(Float64(wj + 1.0) / Float64(Float64(x / exp(wj)) - 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)
	t_0 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if (wj <= 1.85e-6)
		tmp = ((wj ^ 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (0.6666666666666666 * x))) + (((1.0 - t_0) * (wj ^ 2.0)) + (x + (-2.0 * (wj * x))));
	else
		tmp = wj + (((wj + 1.0) / ((x / exp(wj)) - 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_] := Block[{t$95$0 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, 1.85e-6], 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[(0.6666666666666666 * x), $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], N[(wj + N[Power[N[(N[(wj + 1.0), $MachinePrecision] / N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Target

Original	79.5%
Target	80.4%
Herbie	98.9%

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

Derivation

1. Split input into 2 regimes

2. if wj < 1.8500000000000001e-6

   1. Initial program 80.0%

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

   2. Simplified 80.0%

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

      [Start] 80.0	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 80.0	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 80.0	\[ wj + \left(-\color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [=>] 80.0	\[ wj + \left(-\color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} + \left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]
      +-commutative [=>] 80.0	\[ wj + \left(-\color{blue}{\left(\left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      distribute-neg-in [=>] 80.0	\[ wj + \color{blue}{\left(\left(-\left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\right) + \left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      remove-double-neg [=>] 80.0	\[ wj + \left(\color{blue}{\frac{x}{e^{wj} + wj \cdot e^{wj}}} + \left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right) \]
      sub-neg [<=] 80.0	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [<=] 80.0	\[ wj + \color{blue}{\frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} \]
      distribute-rgt1-in [=>] 80.0	\[ wj + \frac{x - wj \cdot e^{wj}}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]
      associate-/l/ [<=] 80.0	\[ wj + \color{blue}{\frac{\frac{x - wj \cdot e^{wj}}{e^{wj}}}{wj + 1}} \]

   3. Taylor expanded in wj around 0 99.0%

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

   if 1.8500000000000001e-6 < wj

   1. Initial program 59.0%

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

   2. Simplified 97.3%

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

      [Start] 59.0	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 59.0	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 59.1	\[ wj + \left(-\color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [=>] 59.1	\[ wj + \left(-\color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} + \left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]
      +-commutative [=>] 59.1	\[ wj + \left(-\color{blue}{\left(\left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      distribute-neg-in [=>] 59.1	\[ wj + \color{blue}{\left(\left(-\left(-\frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\right) + \left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      remove-double-neg [=>] 59.1	\[ wj + \left(\color{blue}{\frac{x}{e^{wj} + wj \cdot e^{wj}}} + \left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right) \]
      sub-neg [<=] 59.1	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [<=] 59.0	\[ wj + \color{blue}{\frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} \]
      distribute-rgt1-in [=>] 59.0	\[ wj + \frac{x - wj \cdot e^{wj}}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}} \]
      associate-/l/ [<=] 59.0	\[ wj + \color{blue}{\frac{\frac{x - wj \cdot e^{wj}}{e^{wj}}}{wj + 1}} \]

   3. Applied egg-rr 97.3%

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

      [Start] 97.3	\[ wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \]
      clear-num [=>] 97.3	\[ wj + \color{blue}{\frac{1}{\frac{wj + 1}{\frac{x}{e^{wj}} - wj}}} \]
      inv-pow [=>] 97.3	\[ wj + \color{blue}{{\left(\frac{wj + 1}{\frac{x}{e^{wj}} - wj}\right)}^{-1}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.85 \cdot 10^{-6}:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot -3\right) - 0.6666666666666666 \cdot x\right) + \left(\left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + {\left(\frac{wj + 1}{\frac{x}{e^{wj}} - wj}\right)}^{-1}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.8%
Cost	13700

\[\begin{array}{l} \mathbf{if}\;wj \leq 7.6 \cdot 10^{-7}:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot -3\right) - 0.6666666666666666 \cdot x\right) + \left(\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj + {\left(\frac{wj + 1}{\frac{x}{e^{wj}} - wj}\right)}^{-1}\\ \end{array} \]

Alternative 2
Accuracy	98.8%
Cost	8708

\[\begin{array}{l} \mathbf{if}\;wj \leq 7.6 \cdot 10^{-7}:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot -3\right) - 0.6666666666666666 \cdot x\right) + \left(\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]

Alternative 3
Accuracy	98.6%
Cost	7812

\[\begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-10}:\\ \;\;\;\;\left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]

Alternative 4
Accuracy	98.7%
Cost	7236

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

Alternative 5
Accuracy	97.7%
Cost	964

\[\begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-10}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\left(x - wj \cdot x\right) - wj}{wj + 1}\\ \end{array} \]

Alternative 6
Accuracy	98.0%
Cost	836

\[\begin{array}{l} \mathbf{if}\;wj \leq 0.00017:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 7
Accuracy	97.4%
Cost	580

\[\begin{array}{l} \mathbf{if}\;wj \leq 0.000115:\\ \;\;\;\;x + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 8
Accuracy	85.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-274}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-248}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Accuracy	96.3%
Cost	320

\[x + wj \cdot wj \]

Alternative 10
Accuracy	4.3%
Cost	64

\[wj \]

Alternative 11
Accuracy	85.8%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023147 
(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))))))