Jmat.Real.lambertw, newton loop step

Average Error: 13.8 → 0.4

Time: 18.7s

Precision: binary64

Cost: 34948

Math

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

↓

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t_0 - x}{e^{wj} + t_0} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\left(x + x \cdot \left(-2 \cdot wj\right)\right) + \left(\left(1 - x \cdot -2.5\right) \cdot {wj}^{2} + {wj}^{3} \cdot \left(-\left(\left(1 + -2 \cdot \left(x \cdot -2.5\right)\right) + x \cdot -2.3333333333333335\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 5e-12)
     (+
      (+ x (* x (* -2.0 wj)))
      (+
       (* (- 1.0 (* x -2.5)) (pow wj 2.0))
       (*
        (pow wj 3.0)
        (- (+ (+ 1.0 (* -2.0 (* x -2.5))) (* x -2.3333333333333335))))))
     (- 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 t_0 = wj * exp(wj);
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 5e-12) {
		tmp = (x + (x * (-2.0 * wj))) + (((1.0 - (x * -2.5)) * pow(wj, 2.0)) + (pow(wj, 3.0) * -((1.0 + (-2.0 * (x * -2.5))) + (x * -2.3333333333333335))));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = wj * exp(wj)
    if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 5d-12) then
        tmp = (x + (x * ((-2.0d0) * wj))) + (((1.0d0 - (x * (-2.5d0))) * (wj ** 2.0d0)) + ((wj ** 3.0d0) * -((1.0d0 + ((-2.0d0) * (x * (-2.5d0)))) + (x * (-2.3333333333333335d0)))))
    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 t_0 = wj * Math.exp(wj);
	double tmp;
	if ((wj - ((t_0 - x) / (Math.exp(wj) + t_0))) <= 5e-12) {
		tmp = (x + (x * (-2.0 * wj))) + (((1.0 - (x * -2.5)) * Math.pow(wj, 2.0)) + (Math.pow(wj, 3.0) * -((1.0 + (-2.0 * (x * -2.5))) + (x * -2.3333333333333335))));
	} 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):
	t_0 = wj * math.exp(wj)
	tmp = 0
	if (wj - ((t_0 - x) / (math.exp(wj) + t_0))) <= 5e-12:
		tmp = (x + (x * (-2.0 * wj))) + (((1.0 - (x * -2.5)) * math.pow(wj, 2.0)) + (math.pow(wj, 3.0) * -((1.0 + (-2.0 * (x * -2.5))) + (x * -2.3333333333333335))))
	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)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 5e-12)
		tmp = Float64(Float64(x + Float64(x * Float64(-2.0 * wj))) + Float64(Float64(Float64(1.0 - Float64(x * -2.5)) * (wj ^ 2.0)) + Float64((wj ^ 3.0) * Float64(-Float64(Float64(1.0 + Float64(-2.0 * Float64(x * -2.5))) + Float64(x * -2.3333333333333335))))));
	else
		tmp = Float64(wj - Float64(Float64(wj - Float64(x / exp(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)
	t_0 = wj * exp(wj);
	tmp = 0.0;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 5e-12)
		tmp = (x + (x * (-2.0 * wj))) + (((1.0 - (x * -2.5)) * (wj ^ 2.0)) + ((wj ^ 3.0) * -((1.0 + (-2.0 * (x * -2.5))) + (x * -2.3333333333333335))));
	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_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-12], N[(N[(x + N[(x * N[(-2.0 * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 - N[(x * -2.5), $MachinePrecision]), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 3.0], $MachinePrecision] * (-N[(N[(1.0 + N[(-2.0 * N[(x * -2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * -2.3333333333333335), $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	13.8
Target	13.2
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 4.9999999999999997e-12

   1. Initial program 18.0

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

   2. Simplified 18.0

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

      [Start] 18.0	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      rational_best-simplify-59 [=>] 18.0	\[ wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{wj \cdot e^{wj} - \left(-e^{wj}\right)}} \]
      rational_best-simplify-11 [=>] 18.0	\[ wj - \frac{wj \cdot e^{wj} - x}{wj \cdot e^{wj} - \color{blue}{e^{wj} \cdot -1}} \]
      rational_best-simplify-62 [=>] 18.0	\[ wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj - -1\right)}} \]

   3. Taylor expanded in wj around 0 0.4

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

   4. Simplified 0.4

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

      [Start] 0.4	\[ -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]
      rational_best-simplify-47 [=>] 0.4	\[ \color{blue}{\left(-2 \cdot \left(wj \cdot x\right) + x\right) + \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)\right)} \]
      rational_best-simplify-3 [=>] 0.4	\[ \color{blue}{\left(x + -2 \cdot \left(wj \cdot x\right)\right)} + \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)\right) \]
      rational_best-simplify-50 [=>] 0.4	\[ \left(x + \color{blue}{x \cdot \left(wj \cdot -2\right)}\right) + \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)\right) \]
      rational_best-simplify-1 [=>] 0.4	\[ \left(x + x \cdot \color{blue}{\left(-2 \cdot wj\right)}\right) + \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)\right) \]
      rational_best-simplify-63 [=>] 0.4	\[ \left(x + x \cdot \left(-2 \cdot wj\right)\right) + \left(\left(1 - \color{blue}{x \cdot \left(-4 + 1.5\right)}\right) \cdot {wj}^{2} + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)\right) \]
      metadata-eval [=>] 0.4	\[ \left(x + x \cdot \left(-2 \cdot wj\right)\right) + \left(\left(1 - x \cdot \color{blue}{-2.5}\right) \cdot {wj}^{2} + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)\right) \]
      rational_best-simplify-50 [=>] 0.4	\[ \left(x + x \cdot \left(-2 \cdot wj\right)\right) + \left(\left(1 - x \cdot -2.5\right) \cdot {wj}^{2} + \color{blue}{{wj}^{3} \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot -1\right)}\right) \]
      rational_best-simplify-10 [=>] 0.4	\[ \left(x + x \cdot \left(-2 \cdot wj\right)\right) + \left(\left(1 - x \cdot -2.5\right) \cdot {wj}^{2} + {wj}^{3} \cdot \color{blue}{\left(-\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right)\right)}\right) \]
      rational_best-simplify-47 [=>] 0.4	\[ \left(x + x \cdot \left(-2 \cdot wj\right)\right) + \left(\left(1 - x \cdot -2.5\right) \cdot {wj}^{2} + {wj}^{3} \cdot \left(-\color{blue}{\left(\left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right) + \left(-3 \cdot x + 0.6666666666666666 \cdot x\right)\right)}\right)\right) \]

   if 4.9999999999999997e-12 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 2.6

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

   2. Simplified 2.7

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

      [Start] 2.6	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      rational_best-simplify-59 [=>] 2.6	\[ wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{wj \cdot e^{wj} - \left(-e^{wj}\right)}} \]
      rational_best-simplify-11 [=>] 2.6	\[ wj - \frac{wj \cdot e^{wj} - x}{wj \cdot e^{wj} - \color{blue}{e^{wj} \cdot -1}} \]
      rational_best-simplify-62 [=>] 2.7	\[ wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(wj - -1\right)}} \]

   3. Applied egg-rr 0.3

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

   4. Simplified 0.3

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

      [Start] 0.3	\[ wj - \left(\frac{wj}{wj + 1} - \frac{\frac{x}{wj + 1}}{e^{wj}}\right) \]
      rational_best-simplify-49 [=>] 0.3	\[ wj - \left(\frac{wj}{wj + 1} - \color{blue}{\frac{\frac{x}{e^{wj}}}{wj + 1}}\right) \]
      rational_best-simplify-66 [=>] 0.3	\[ wj - \color{blue}{\frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\left(x + x \cdot \left(-2 \cdot wj\right)\right) + \left(\left(1 - x \cdot -2.5\right) \cdot {wj}^{2} + {wj}^{3} \cdot \left(-\left(\left(1 + -2 \cdot \left(x \cdot -2.5\right)\right) + x \cdot -2.3333333333333335\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	7556

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.55 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(-2 \cdot wj\right) + \left(x + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	7368

\[\begin{array}{l} t_0 := wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{if}\;wj \leq -8.2 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;wj \leq 5.2 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \left(-2 \cdot wj\right) + \left(x + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.3
Cost	7172

\[\begin{array}{l} \mathbf{if}\;wj \leq 0.00068:\\ \;\;\;\;x \cdot \left(-2 \cdot wj\right) + \left(x + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{wj}{\frac{1}{wj + 1}} \cdot \frac{wj}{\left(wj + 1\right) \cdot \left(wj + 1\right)}\\ \end{array} \]

Alternative 4
Error	8.4
Cost	7112

\[\begin{array}{l} \mathbf{if}\;wj \leq -2.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{wj}{\frac{1}{wj + 1}} \cdot \frac{wj}{\left(wj + 1\right) \cdot \left(wj + 1\right)}\\ \mathbf{elif}\;wj \leq 1.95:\\ \;\;\;\;\frac{x}{\left(1 + wj\right) \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 5
Error	8.4
Cost	1352

\[\begin{array}{l} t_0 := \frac{wj}{\frac{1}{wj + 1}} \cdot \frac{wj}{\left(wj + 1\right) \cdot \left(wj + 1\right)}\\ \mathbf{if}\;wj \leq -3 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;wj \leq 0.000235:\\ \;\;\;\;-2 \cdot \left(wj \cdot x\right) + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	8.4
Cost	1092

\[\begin{array}{l} \mathbf{if}\;wj \leq -3.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{wj}{\left(1 + wj\right) \cdot \left(-1 - wj\right)} \cdot \left(wj \cdot \left(-1 - wj\right)\right)\\ \mathbf{elif}\;wj \leq 0.000235:\\ \;\;\;\;-2 \cdot \left(wj \cdot x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{wj}{wj + 1} \cdot wj\\ \end{array} \]

Alternative 7
Error	8.5
Cost	712

\[\begin{array}{l} t_0 := \frac{wj}{wj + 1} \cdot wj\\ \mathbf{if}\;wj \leq -3 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;wj \leq 0.00024:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	8.4
Cost	712

\[\begin{array}{l} t_0 := \frac{wj}{wj + 1} \cdot wj\\ \mathbf{if}\;wj \leq -3.1 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;wj \leq 0.00038:\\ \;\;\;\;-2 \cdot \left(wj \cdot x\right) + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	61.2
Cost	64

\[wj \]

Alternative 10
Error	9.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023100 
(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))))))