Jmat.Real.lambertw, newton loop step

Average Error: 13.9 → 1.6

Time: 15.0s

Precision: binary64

Cost: 15552

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot 4 + x \cdot -1.5\\ {wj}^{3} \cdot \left(x \cdot -0.6666666666666666 + \left(x \cdot 3 - \left(1 + 2 \cdot t_0\right)\right)\right) + \left(\left(1 + t_0\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(x \cdot wj\right)\right)\right) \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x 4.0) (* x -1.5))))
   (+
    (*
     (pow wj 3.0)
     (+ (* x -0.6666666666666666) (- (* x 3.0) (+ 1.0 (* 2.0 t_0)))))
    (+ (* (+ 1.0 t_0) (pow wj 2.0)) (+ x (* -2.0 (* x 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 t_0 = (x * 4.0) + (x * -1.5);
	return (pow(wj, 3.0) * ((x * -0.6666666666666666) + ((x * 3.0) - (1.0 + (2.0 * t_0))))) + (((1.0 + t_0) * pow(wj, 2.0)) + (x + (-2.0 * (x * wj))));
}

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
    t_0 = (x * 4.0d0) + (x * (-1.5d0))
    code = ((wj ** 3.0d0) * ((x * (-0.6666666666666666d0)) + ((x * 3.0d0) - (1.0d0 + (2.0d0 * t_0))))) + (((1.0d0 + t_0) * (wj ** 2.0d0)) + (x + ((-2.0d0) * (x * wj))))
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);
	return (Math.pow(wj, 3.0) * ((x * -0.6666666666666666) + ((x * 3.0) - (1.0 + (2.0 * t_0))))) + (((1.0 + t_0) * Math.pow(wj, 2.0)) + (x + (-2.0 * (x * wj))));
}

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)
	return (math.pow(wj, 3.0) * ((x * -0.6666666666666666) + ((x * 3.0) - (1.0 + (2.0 * t_0))))) + (((1.0 + t_0) * math.pow(wj, 2.0)) + (x + (-2.0 * (x * wj))))

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))
	return Float64(Float64((wj ^ 3.0) * Float64(Float64(x * -0.6666666666666666) + Float64(Float64(x * 3.0) - Float64(1.0 + Float64(2.0 * t_0))))) + Float64(Float64(Float64(1.0 + t_0) * (wj ^ 2.0)) + Float64(x + Float64(-2.0 * Float64(x * wj)))))
end

MATLAB

function tmp = code(wj, x)
	tmp = wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))));
end

↓

function tmp = code(wj, x)
	t_0 = (x * 4.0) + (x * -1.5);
	tmp = ((wj ^ 3.0) * ((x * -0.6666666666666666) + ((x * 3.0) - (1.0 + (2.0 * t_0))))) + (((1.0 + t_0) * (wj ^ 2.0)) + (x + (-2.0 * (x * wj))));
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]}, 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 * t$95$0), $MachinePrecision]), $MachinePrecision]), $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[(x * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	13.9
Target	13.3
Herbie	1.6

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

Derivation

1. Initial program 13.9

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

2. Simplified 13.3

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

   [Start] 13.9	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   sub-neg [=>] 13.9	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
   neg-mul-1 [=>] 13.9	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
   *-commutative [=>] 13.9	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
   *-commutative [<=] 13.9	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
   neg-mul-1 [<=] 13.9	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
   neg-sub0 [=>] 13.9	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
   div-sub [=>] 13.9	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
   associate--r- [=>] 13.9	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
   +-commutative [=>] 13.9	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
   sub0-neg [=>] 13.9	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
   sub-neg [<=] 13.9	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

3. Taylor expanded in wj around 0 1.6

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

4. Final simplification 1.6

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

Alternatives

Alternative 1
Error	1.7
Cost	14400

\[\begin{array}{l} t_0 := x + \left(x + 1\right)\\ {wj}^{2} \cdot t_0 - \left({wj}^{3} \cdot t_0 + \left(2 \cdot \left(x \cdot wj\right) - x\right)\right) \end{array} \]

Alternative 2
Error	1.7
Cost	7296

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

Alternative 3
Error	8.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.45 \cdot 10^{-13}:\\ \;\;\;\;\frac{x \cdot \left(1 - wj\right)}{1 + wj}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \]

Alternative 4
Error	1.9
Cost	704

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

Alternative 5
Error	8.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.45 \cdot 10^{-13}:\\ \;\;\;\;x + -2 \cdot \left(x \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \]

Alternative 6
Error	8.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.45 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \]

Alternative 7
Error	9.1
Cost	448

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

Alternative 8
Error	61.2
Cost	64

\[wj \]

Alternative 9
Error	9.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023059 
(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))))))