Jmat.Real.lambertw, newton loop step

Average Error: 13.7 → 0.7

Time: 1.9min

Precision: binary64

Cost: 7936

Math

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

↓

\[\frac{wj \cdot x + \left({wj}^{2} \cdot \left(\left(0.5 \cdot x + -1 \cdot x\right) - 1\right) + -1 \cdot x\right)}{-1 - wj} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (/
  (+ (* wj x) (+ (* (pow wj 2.0) (- (+ (* 0.5 x) (* -1.0 x)) 1.0)) (* -1.0 x)))
  (- -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) {
	return ((wj * x) + ((pow(wj, 2.0) * (((0.5 * x) + (-1.0 * x)) - 1.0)) + (-1.0 * x))) / (-1.0 - 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
    code = ((wj * x) + (((wj ** 2.0d0) * (((0.5d0 * x) + ((-1.0d0) * x)) - 1.0d0)) + ((-1.0d0) * x))) / ((-1.0d0) - 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) {
	return ((wj * x) + ((Math.pow(wj, 2.0) * (((0.5 * x) + (-1.0 * x)) - 1.0)) + (-1.0 * x))) / (-1.0 - wj);
}

Python

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

↓

def code(wj, x):
	return ((wj * x) + ((math.pow(wj, 2.0) * (((0.5 * x) + (-1.0 * x)) - 1.0)) + (-1.0 * x))) / (-1.0 - 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)
	return Float64(Float64(Float64(wj * x) + Float64(Float64((wj ^ 2.0) * Float64(Float64(Float64(0.5 * x) + Float64(-1.0 * x)) - 1.0)) + Float64(-1.0 * x))) / Float64(-1.0 - 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)
	tmp = ((wj * x) + (((wj ^ 2.0) * (((0.5 * x) + (-1.0 * x)) - 1.0)) + (-1.0 * x))) / (-1.0 - 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_] := N[(N[(N[(wj * x), $MachinePrecision] + N[(N[(N[Power[wj, 2.0], $MachinePrecision] * N[(N[(N[(0.5 * x), $MachinePrecision] + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]

Target

Original	13.7
Target	13.1
Herbie	0.7

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

Derivation

1. Initial program 13.7

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

2. Simplified 13.1

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

3. Applied egg-rr 13.1

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

4. Taylor expanded in wj around 0 0.7

   \[\leadsto \frac{\color{blue}{wj \cdot x + \left({wj}^{2} \cdot \left(\left(0.5 \cdot x + -1 \cdot x\right) - 1\right) + -1 \cdot x\right)}}{-1 - wj} \]

Alternatives

Alternative 1
Error	9.0
Cost	7112

\[\begin{array}{l} t_0 := \frac{x}{\left(1 + wj\right) \cdot e^{wj}}\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{-291}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-256}:\\ \;\;\;\;\frac{\left(-wj\right) \cdot wj}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	2.2
Cost	7040

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

Alternative 3
Error	9.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-292}:\\ \;\;\;\;\frac{x}{wj \cdot 2 - -1}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-257}:\\ \;\;\;\;\frac{\left(-wj\right) \cdot wj}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;\frac{wj \cdot x - x}{-1 - wj}\\ \end{array} \]

Alternative 4
Error	9.5
Cost	776

\[\begin{array}{l} t_0 := \frac{x}{wj \cdot 2 - -1}\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{-291}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-257}:\\ \;\;\;\;\frac{\left(-wj\right) \cdot wj}{-1 - wj}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	9.7
Cost	712

\[\begin{array}{l} t_0 := -2 \cdot \left(wj \cdot x\right) + x\\ \mathbf{if}\;x \leq -3.15 \cdot 10^{-291}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-257}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	9.6
Cost	712

\[\begin{array}{l} t_0 := \frac{x}{wj \cdot 2 - -1}\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{-291}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-256}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	10.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-291}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-255}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x - wj \cdot x\\ \end{array} \]

Alternative 8
Error	10.0
Cost	584

\[\begin{array}{l} t_0 := \frac{x}{wj - -1}\\ \mathbf{if}\;x \leq -9 \cdot 10^{-292}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-255}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	10.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-291}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-257}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	61.2
Cost	64

\[wj \]

Alternative 11
Error	9.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023033 
(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))))))