Jmat.Real.lambertw, newton loop step

Average Error: 13.4 → 1.2

Time: 1.2min

Precision: binary64

Cost: 13764

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj \leq 0.0036:\\ \;\;\;\;{wj}^{2} - \left({wj}^{3} - \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{wj}{wj + 1} \cdot wj\\ \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0036)
   (- (pow wj 2.0) (- (pow wj 3.0) (+ x (* -2.0 (* wj x)))))
   (* (/ wj (+ wj 1.0)) 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 <= 0.0036) {
		tmp = pow(wj, 2.0) - (pow(wj, 3.0) - (x + (-2.0 * (wj * x))));
	} else {
		tmp = (wj / (wj + 1.0)) * 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 <= 0.0036d0) then
        tmp = (wj ** 2.0d0) - ((wj ** 3.0d0) - (x + ((-2.0d0) * (wj * x))))
    else
        tmp = (wj / (wj + 1.0d0)) * 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 <= 0.0036) {
		tmp = Math.pow(wj, 2.0) - (Math.pow(wj, 3.0) - (x + (-2.0 * (wj * x))));
	} else {
		tmp = (wj / (wj + 1.0)) * 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 <= 0.0036:
		tmp = math.pow(wj, 2.0) - (math.pow(wj, 3.0) - (x + (-2.0 * (wj * x))))
	else:
		tmp = (wj / (wj + 1.0)) * 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 <= 0.0036)
		tmp = Float64((wj ^ 2.0) - Float64((wj ^ 3.0) - Float64(x + Float64(-2.0 * Float64(wj * x)))));
	else
		tmp = Float64(Float64(wj / Float64(wj + 1.0)) * 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 <= 0.0036)
		tmp = (wj ^ 2.0) - ((wj ^ 3.0) - (x + (-2.0 * (wj * x))));
	else
		tmp = (wj / (wj + 1.0)) * 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, 0.0036], N[(N[Power[wj, 2.0], $MachinePrecision] - N[(N[Power[wj, 3.0], $MachinePrecision] - N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision]]

Target

Original	13.4
Target	12.8
Herbie	1.2

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

Derivation

1. Split input into 2 regimes

2. if wj < 0.0035999999999999999

   1. Initial program 13.1

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

   2. Taylor expanded in wj around 0 0.8

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

   3. Simplified 0.8

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

      [Start] 0.8	\[ -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]
      rational_best-simplify-3 [=>] 0.8	\[ \color{blue}{\left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)} \]
      rational_best-simplify-3 [=>] 0.8	\[ \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + \color{blue}{\left(x + -2 \cdot \left(wj \cdot x\right)\right)}\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \]
      rational_best-simplify-47 [=>] 0.8	\[ \color{blue}{\left(-2 \cdot \left(wj \cdot x\right) + \left(x + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right)\right)} + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \]
      rational_best-simplify-50 [=>] 0.8	\[ \left(\color{blue}{x \cdot \left(wj \cdot -2\right)} + \left(x + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right)\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \]
      rational_best-simplify-1 [=>] 0.8	\[ \left(x \cdot \color{blue}{\left(-2 \cdot wj\right)} + \left(x + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right)\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \]
      rational_best-simplify-63 [=>] 0.8	\[ \left(x \cdot \left(-2 \cdot wj\right) + \left(x + \left(1 - \color{blue}{x \cdot \left(-4 + 1.5\right)}\right) \cdot {wj}^{2}\right)\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \]
      metadata-eval [=>] 0.8	\[ \left(x \cdot \left(-2 \cdot wj\right) + \left(x + \left(1 - x \cdot \color{blue}{-2.5}\right) \cdot {wj}^{2}\right)\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \]
      rational_best-simplify-50 [=>] 0.8	\[ \left(x \cdot \left(-2 \cdot wj\right) + \left(x + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\right)\right) + \color{blue}{{wj}^{3} \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot -1\right)} \]
      rational_best-simplify-10 [=>] 0.8	\[ \left(x \cdot \left(-2 \cdot wj\right) + \left(x + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\right)\right) + {wj}^{3} \cdot \color{blue}{\left(-\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right)\right)} \]

   4. Taylor expanded in x around 0 0.8

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

   5. Simplified 0.8

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

      [Start] 0.8	\[ \left(x \cdot \left(-2 \cdot wj\right) + \left(x + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\right)\right) + -1 \cdot {wj}^{3} \]
      rational_best-simplify-1 [=>] 0.8	\[ \left(x \cdot \left(-2 \cdot wj\right) + \left(x + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\right)\right) + \color{blue}{{wj}^{3} \cdot -1} \]
      rational_best-simplify-10 [=>] 0.8	\[ \left(x \cdot \left(-2 \cdot wj\right) + \left(x + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\right)\right) + \color{blue}{\left(-{wj}^{3}\right)} \]

   6. Taylor expanded in x around 0 0.9

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

   7. Taylor expanded in x around 0 0.9

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

   8. Simplified 0.9

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

      [Start] 0.9	\[ \left(\left(1 + -2 \cdot wj\right) \cdot x + {wj}^{2}\right) - {wj}^{3} \]
      rational_best-simplify-3 [=>] 0.9	\[ \color{blue}{\left({wj}^{2} + \left(1 + -2 \cdot wj\right) \cdot x\right)} - {wj}^{3} \]
      rational_best-simplify-59 [=>] 0.9	\[ \color{blue}{\left(\left(1 + -2 \cdot wj\right) \cdot x - \left(-{wj}^{2}\right)\right)} - {wj}^{3} \]
      rational_best-simplify-52 [=>] 0.9	\[ \color{blue}{\left(1 + -2 \cdot wj\right) \cdot x - \left({wj}^{3} + \left(-{wj}^{2}\right)\right)} \]
      rational_best-simplify-1 [=>] 0.9	\[ \color{blue}{x \cdot \left(1 + -2 \cdot wj\right)} - \left({wj}^{3} + \left(-{wj}^{2}\right)\right) \]
      rational_best-simplify-59 [=>] 0.9	\[ x \cdot \color{blue}{\left(-2 \cdot wj - \left(-1\right)\right)} - \left({wj}^{3} + \left(-{wj}^{2}\right)\right) \]
      metadata-eval [=>] 0.9	\[ x \cdot \left(-2 \cdot wj - \color{blue}{-1}\right) - \left({wj}^{3} + \left(-{wj}^{2}\right)\right) \]
      rational_best-simplify-62 [<=] 0.9	\[ \color{blue}{\left(\left(-2 \cdot wj\right) \cdot x - x \cdot -1\right)} - \left({wj}^{3} + \left(-{wj}^{2}\right)\right) \]
      rational_best-simplify-1 [<=] 0.9	\[ \left(\color{blue}{x \cdot \left(-2 \cdot wj\right)} - x \cdot -1\right) - \left({wj}^{3} + \left(-{wj}^{2}\right)\right) \]
      rational_best-simplify-11 [<=] 0.9	\[ \left(x \cdot \left(-2 \cdot wj\right) - \color{blue}{\left(-x\right)}\right) - \left({wj}^{3} + \left(-{wj}^{2}\right)\right) \]
      rational_best-simplify-59 [<=] 0.9	\[ \color{blue}{\left(x + x \cdot \left(-2 \cdot wj\right)\right)} - \left({wj}^{3} + \left(-{wj}^{2}\right)\right) \]
      rational_best-simplify-57 [=>] 0.9	\[ \color{blue}{\left(\left(x + x \cdot \left(-2 \cdot wj\right)\right) - {wj}^{3}\right) + \left(-\left(-{wj}^{2}\right)\right)} \]
      rational_best-simplify-59 [=>] 0.9	\[ \left(\color{blue}{\left(x \cdot \left(-2 \cdot wj\right) - \left(-x\right)\right)} - {wj}^{3}\right) + \left(-\left(-{wj}^{2}\right)\right) \]
      rational_best-simplify-52 [=>] 0.9	\[ \color{blue}{\left(x \cdot \left(-2 \cdot wj\right) - \left({wj}^{3} + \left(-x\right)\right)\right)} + \left(-\left(-{wj}^{2}\right)\right) \]
      rational_best-simplify-56 [=>] 0.9	\[ \color{blue}{x \cdot \left(-2 \cdot wj\right) - \left(\left({wj}^{3} + \left(-x\right)\right) + \left(-{wj}^{2}\right)\right)} \]
      trig-simplify-13 [<=] 0.9	\[ x \cdot \left(-2 \cdot wj\right) - \color{blue}{\left(\left(-{wj}^{2}\right) + \left({wj}^{3} + \left(-x\right)\right)\right)} \]
      rational_best-simplify-56 [<=] 0.9	\[ \color{blue}{\left(x \cdot \left(-2 \cdot wj\right) - \left(-{wj}^{2}\right)\right) + \left(-\left({wj}^{3} + \left(-x\right)\right)\right)} \]

   if 0.0035999999999999999 < wj

   1. Initial program 33.1

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

   2. Simplified 33.1

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

      [Start] 33.1	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      rational_best-simplify-59 [=>] 33.1	\[ wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{wj \cdot e^{wj} - \left(-e^{wj}\right)}} \]
      rational_best-simplify-11 [=>] 33.1	\[ wj - \frac{wj \cdot e^{wj} - x}{wj \cdot e^{wj} - \color{blue}{e^{wj} \cdot -1}} \]
      rational_best-simplify-62 [=>] 33.1	\[ wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj - -1\right)}} \]

   3. Taylor expanded in x around 0 16.5

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

   4. Simplified 16.5

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

      [Start] 16.5	\[ wj - \frac{wj}{1 + wj} \]
      rational_best-simplify-3 [<=] 16.5	\[ wj - \frac{wj}{\color{blue}{wj + 1}} \]

   5. Applied egg-rr 16.4

      \[\leadsto \color{blue}{\frac{wj}{wj + 1} \cdot wj} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.0036:\\ \;\;\;\;{wj}^{2} - \left({wj}^{3} - \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{wj}{wj + 1} \cdot wj\\ \end{array} \]

Alternatives

Alternative 1
Error	1.5
Cost	7172

\[\begin{array}{l} \mathbf{if}\;wj \leq 0.0036:\\ \;\;\;\;x \cdot \left(-2 \cdot wj\right) + \left(x + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{wj}{wj + 1} \cdot wj\\ \end{array} \]

Alternative 2
Error	9.0
Cost	7112

\[\begin{array}{l} t_0 := \frac{\frac{x}{wj + 1}}{e^{wj}}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-230}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-303}:\\ \;\;\;\;\frac{wj}{wj + 1} \cdot wj\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	9.0
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-230}:\\ \;\;\;\;\frac{e^{-wj} \cdot x}{1 + wj}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-303}:\\ \;\;\;\;\frac{wj}{wj + 1} \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}}\\ \end{array} \]

Alternative 4
Error	8.4
Cost	840

\[\begin{array}{l} t_0 := \frac{wj}{wj + 1} \cdot wj\\ \mathbf{if}\;wj \leq -7.5 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;wj \leq 0.0036:\\ \;\;\;\;\frac{\left(1 - wj\right) \cdot x}{1 + wj}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	8.4
Cost	840

\[\begin{array}{l} t_0 := \frac{wj}{wj + 1} \cdot wj\\ \mathbf{if}\;wj \leq -7.5 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;wj \leq 0.0036:\\ \;\;\;\;\frac{wj \cdot x - x}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	8.6
Cost	712

\[\begin{array}{l} t_0 := \frac{wj}{wj + 1} \cdot wj\\ \mathbf{if}\;wj \leq -2.4 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;wj \leq 0.0036:\\ \;\;\;\;\frac{x}{1 + wj}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	8.4
Cost	712

\[\begin{array}{l} t_0 := \frac{wj}{wj + 1} \cdot wj\\ \mathbf{if}\;wj \leq -7.5 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;wj \leq 0.0036:\\ \;\;\;\;-2 \cdot \left(wj \cdot x\right) + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	10.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-230}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-302}:\\ \;\;\;\;1 \cdot \left(wj \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	10.1
Cost	584

\[\begin{array}{l} t_0 := \frac{x}{1 + wj}\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-231}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-302}:\\ \;\;\;\;1 \cdot \left(wj \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	61.2
Cost	64

\[wj \]

Alternative 11
Error	9.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023099 
(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))))))