Jmat.Real.lambertw, newton loop step

Average Accuracy: 78.8% → 99.2%

Time: 16.9s

Precision: binary64

Cost: 15684

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj \leq 3.1 \cdot 10^{-6}:\\ \;\;\;\;{wj}^{3} \cdot \left(x \cdot -0.6666666666666666 + \left(x \cdot 3 - \left(1 + -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\right) + \left(\left(1 + \left(x \cdot 4 + x \cdot -1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - x \cdot e^{-wj}}{wj + 1}\\ \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (if (<= wj 3.1e-6)
   (+
    (*
     (pow wj 3.0)
     (+
      (* x -0.6666666666666666)
      (- (* x 3.0) (+ 1.0 (* -2.0 (+ (* x -4.0) (* x 1.5)))))))
    (+
     (* (+ 1.0 (+ (* x 4.0) (* x -1.5))) (pow wj 2.0))
     (+ x (* -2.0 (* wj x)))))
   (- wj (/ (- wj (* 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 tmp;
	if (wj <= 3.1e-6) {
		tmp = (pow(wj, 3.0) * ((x * -0.6666666666666666) + ((x * 3.0) - (1.0 + (-2.0 * ((x * -4.0) + (x * 1.5))))))) + (((1.0 + ((x * 4.0) + (x * -1.5))) * pow(wj, 2.0)) + (x + (-2.0 * (wj * x))));
	} else {
		tmp = wj - ((wj - (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) :: tmp
    if (wj <= 3.1d-6) then
        tmp = ((wj ** 3.0d0) * ((x * (-0.6666666666666666d0)) + ((x * 3.0d0) - (1.0d0 + ((-2.0d0) * ((x * (-4.0d0)) + (x * 1.5d0))))))) + (((1.0d0 + ((x * 4.0d0) + (x * (-1.5d0)))) * (wj ** 2.0d0)) + (x + ((-2.0d0) * (wj * x))))
    else
        tmp = wj - ((wj - (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 tmp;
	if (wj <= 3.1e-6) {
		tmp = (Math.pow(wj, 3.0) * ((x * -0.6666666666666666) + ((x * 3.0) - (1.0 + (-2.0 * ((x * -4.0) + (x * 1.5))))))) + (((1.0 + ((x * 4.0) + (x * -1.5))) * Math.pow(wj, 2.0)) + (x + (-2.0 * (wj * x))));
	} else {
		tmp = wj - ((wj - (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):
	tmp = 0
	if wj <= 3.1e-6:
		tmp = (math.pow(wj, 3.0) * ((x * -0.6666666666666666) + ((x * 3.0) - (1.0 + (-2.0 * ((x * -4.0) + (x * 1.5))))))) + (((1.0 + ((x * 4.0) + (x * -1.5))) * math.pow(wj, 2.0)) + (x + (-2.0 * (wj * x))))
	else:
		tmp = wj - ((wj - (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)
	tmp = 0.0
	if (wj <= 3.1e-6)
		tmp = Float64(Float64((wj ^ 3.0) * Float64(Float64(x * -0.6666666666666666) + Float64(Float64(x * 3.0) - Float64(1.0 + Float64(-2.0 * Float64(Float64(x * -4.0) + Float64(x * 1.5))))))) + Float64(Float64(Float64(1.0 + Float64(Float64(x * 4.0) + Float64(x * -1.5))) * (wj ^ 2.0)) + Float64(x + Float64(-2.0 * Float64(wj * x)))));
	else
		tmp = Float64(wj - Float64(Float64(wj - Float64(x * exp(Float64(-wj)))) / 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 <= 3.1e-6)
		tmp = ((wj ^ 3.0) * ((x * -0.6666666666666666) + ((x * 3.0) - (1.0 + (-2.0 * ((x * -4.0) + (x * 1.5))))))) + (((1.0 + ((x * 4.0) + (x * -1.5))) * (wj ^ 2.0)) + (x + (-2.0 * (wj * x))));
	else
		tmp = wj - ((wj - (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_] := If[LessEqual[wj, 3.1e-6], N[(N[(N[Power[wj, 3.0], $MachinePrecision] * N[(N[(x * -0.6666666666666666), $MachinePrecision] + N[(N[(x * 3.0), $MachinePrecision] - N[(1.0 + N[(-2.0 * N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(N[(x * 4.0), $MachinePrecision] + N[(x * -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj - N[(x * N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	78.8%
Target	79.7%
Herbie	99.2%

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

Derivation

1. Split input into 2 regimes

2. if wj < 3.1e-6

   1. Initial program 79.3%

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

   2. Simplified 79.4%

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

      [Start] 79.3	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 79.3	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-mul-1 [=>] 79.3	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      *-commutative [=>] 79.3	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
      *-commutative [<=] 79.3	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      neg-mul-1 [<=] 79.3	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-sub0 [=>] 79.3	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 79.3	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      associate--r- [=>] 79.3	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      +-commutative [=>] 79.3	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      sub0-neg [=>] 79.3	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [<=] 79.3	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   3. Taylor expanded in wj around 0 99.2%

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

   if 3.1e-6 < wj

   1. Initial program 52.3%

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

   2. Simplified 97.4%

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

      [Start] 52.3	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 52.3	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-mul-1 [=>] 52.3	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      *-commutative [=>] 52.3	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
      *-commutative [<=] 52.3	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      neg-mul-1 [<=] 52.3	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-sub0 [=>] 52.3	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 52.3	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      associate--r- [=>] 52.3	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      +-commutative [=>] 52.3	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      sub0-neg [=>] 52.3	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [<=] 52.3	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   3. Applied egg-rr 97.4%

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

      [Start] 97.4	\[ wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \]
      div-inv [=>] 97.4	\[ wj + \frac{\color{blue}{x \cdot \frac{1}{e^{wj}}} - wj}{wj + 1} \]
      *-commutative [=>] 97.4	\[ wj + \frac{\color{blue}{\frac{1}{e^{wj}} \cdot x} - wj}{wj + 1} \]
      rec-exp [=>] 97.4	\[ wj + \frac{\color{blue}{e^{-wj}} \cdot x - wj}{wj + 1} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 3.1 \cdot 10^{-6}:\\ \;\;\;\;{wj}^{3} \cdot \left(x \cdot -0.6666666666666666 + \left(x \cdot 3 - \left(1 + -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\right) + \left(\left(1 + \left(x \cdot 4 + x \cdot -1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - x \cdot e^{-wj}}{wj + 1}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.0%
Cost	8708

\[\begin{array}{l} \mathbf{if}\;wj \leq 8.4 \cdot 10^{-8}:\\ \;\;\;\;{wj}^{3} \cdot \left(x \cdot -0.6666666666666666 + \left(x \cdot 3 - \left(1 + -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\right) + \left(\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - x \cdot e^{-wj}}{wj + 1}\\ \end{array} \]

Alternative 2
Accuracy	99.1%
Cost	7428

\[\begin{array}{l} \mathbf{if}\;wj \leq 9.6 \cdot 10^{-8}:\\ \;\;\;\;\left(\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - x \cdot e^{-wj}}{wj + 1}\\ \end{array} \]

Alternative 3
Accuracy	98.6%
Cost	7300

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.75 \cdot 10^{-13}:\\ \;\;\;\;{wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - x \cdot e^{-wj}}{wj + 1}\\ \end{array} \]

Alternative 4
Accuracy	98.7%
Cost	7236

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

Alternative 5
Accuracy	97.0%
Cost	7040

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

Alternative 6
Accuracy	87.3%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-275}:\\ \;\;\;\;\frac{x}{\left(wj + 1\right) \cdot e^{wj}}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-297}:\\ \;\;\;\;\frac{1}{\frac{1}{wj} + \frac{x + 1}{wj \cdot wj}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array} \]

Alternative 7
Accuracy	87.8%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;x \leq -3.35 \cdot 10^{-274} \lor \neg \left(x \leq 1.02 \cdot 10^{-290}\right):\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}} + \left(wj - \frac{wj}{wj + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{wj} + \frac{x + 1}{wj \cdot wj}}\\ \end{array} \]

Alternative 8
Accuracy	86.4%
Cost	964

\[\begin{array}{l} \mathbf{if}\;wj \leq -3.35 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{\frac{1}{wj} + \frac{x + 1}{wj \cdot wj}}\\ \mathbf{elif}\;wj \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 9
Accuracy	86.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.15 \cdot 10^{-6}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 10
Accuracy	86.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;wj \leq 8.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 11
Accuracy	85.6%
Cost	448

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

Alternative 12
Accuracy	4.3%
Cost	64

\[wj \]

Alternative 13
Accuracy	85.2%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023136 
(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))))))