Jmat.Real.lambertw, newton loop step

Average Error: 13.6 → 2.1

Time: 9.9s

Precision: binary64

Cost: 7040

Math

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

↓

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

FPCore

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

↓

(FPCore (wj x) :precision binary64 (+ (pow wj 2.0) (+ x (* -2.0 (* wj x)))))

C

double code(double wj, double x) {
	return wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))));
}

↓

double code(double wj, double x) {
	return pow(wj, 2.0) + (x + (-2.0 * (wj * x)));
}

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
    code = (wj ** 2.0d0) + (x + ((-2.0d0) * (wj * x)))
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) {
	return Math.pow(wj, 2.0) + (x + (-2.0 * (wj * x)));
}

Python

def code(wj, x):
	return wj - (((wj * math.exp(wj)) - x) / (math.exp(wj) + (wj * math.exp(wj))))

↓

def code(wj, x):
	return math.pow(wj, 2.0) + (x + (-2.0 * (wj * x)))

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)
	return Float64((wj ^ 2.0) + Float64(x + Float64(-2.0 * Float64(wj * x))))
end

MATLAB

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

↓

function tmp = code(wj, x)
	tmp = (wj ^ 2.0) + (x + (-2.0 * (wj * x)));
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_] := N[(N[Power[wj, 2.0], $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	13.6
Target	12.9
Herbie	2.1

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

Derivation

1. Initial program 13.6

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

2. Simplified 12.9

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

3. Taylor expanded in wj around 0 1.8

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

4. Taylor expanded in x around 0 2.0

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

5. Simplified 2.0

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

6. Taylor expanded in x around 0 1.9

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

7. Taylor expanded in wj around 0 2.1

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

8. Final simplification 2.1

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

Alternatives

Alternative 1
Error	9.0
Cost	1228

\[\begin{array}{l} \mathbf{if}\;wj \leq -6.134586665876447 \cdot 10^{-28}:\\ \;\;\;\;wj \cdot \left(wj \cdot \left(wj \cdot wj + \left(1 - wj\right)\right)\right)\\ \mathbf{elif}\;wj \leq 6.100469428151673 \cdot 10^{-49}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 5.972537427254605 \cdot 10^{-18}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\left(x - wj \cdot x\right) - wj}{wj + 1}\\ \end{array} \]

Alternative 2
Error	10.0
Cost	844

\[\begin{array}{l} \mathbf{if}\;wj \leq -6.134586665876447 \cdot 10^{-28}:\\ \;\;\;\;wj \cdot \left(wj - wj \cdot wj\right)\\ \mathbf{elif}\;wj \leq 6.100469428151673 \cdot 10^{-49}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 5.972537427254605 \cdot 10^{-18}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \end{array} \]

Alternative 3
Error	9.9
Cost	844

\[\begin{array}{l} \mathbf{if}\;wj \leq -6.134586665876447 \cdot 10^{-28}:\\ \;\;\;\;wj \cdot \left(wj \cdot \left(wj \cdot wj + \left(1 - wj\right)\right)\right)\\ \mathbf{elif}\;wj \leq 6.100469428151673 \cdot 10^{-49}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 5.972537427254605 \cdot 10^{-18}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \end{array} \]

Alternative 4
Error	10.1
Cost	588

\[\begin{array}{l} \mathbf{if}\;wj \leq -6.134586665876447 \cdot 10^{-28}:\\ \;\;\;\;wj \cdot \left(wj - wj \cdot wj\right)\\ \mathbf{elif}\;wj \leq 6.100469428151673 \cdot 10^{-49}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 5.972537427254605 \cdot 10^{-18}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	9.9
Cost	456

\[\begin{array}{l} \mathbf{if}\;wj \leq 6.100469428151673 \cdot 10^{-49}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 5.972537427254605 \cdot 10^{-18}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	9.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022228 
(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))))))