Jmat.Real.lambertw, newton loop step

Average Error: 14.2 → 1.7

Time: 36.3s

Precision: binary64

Cost: 14848

Math

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

↓

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

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (+
  (+ (+ x (* x (* -2.0 wj))) (* (- 1.0 (* x -2.5)) (pow wj 2.0)))
  (*
   (+ (+ 1.0 (* -2.0 (* x -2.5))) (* x -2.3333333333333335))
   (- (pow wj 3.0)))))

C

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

↓

double code(double wj, double x) {
	return ((x + (x * (-2.0 * wj))) + ((1.0 - (x * -2.5)) * pow(wj, 2.0))) + (((1.0 + (-2.0 * (x * -2.5))) + (x * -2.3333333333333335)) * -pow(wj, 3.0));
}

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 = ((x + (x * ((-2.0d0) * wj))) + ((1.0d0 - (x * (-2.5d0))) * (wj ** 2.0d0))) + (((1.0d0 + ((-2.0d0) * (x * (-2.5d0)))) + (x * (-2.3333333333333335d0))) * -(wj ** 3.0d0))
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 ((x + (x * (-2.0 * wj))) + ((1.0 - (x * -2.5)) * Math.pow(wj, 2.0))) + (((1.0 + (-2.0 * (x * -2.5))) + (x * -2.3333333333333335)) * -Math.pow(wj, 3.0));
}

Python

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

↓

def code(wj, x):
	return ((x + (x * (-2.0 * wj))) + ((1.0 - (x * -2.5)) * math.pow(wj, 2.0))) + (((1.0 + (-2.0 * (x * -2.5))) + (x * -2.3333333333333335)) * -math.pow(wj, 3.0))

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(x + Float64(x * Float64(-2.0 * wj))) + Float64(Float64(1.0 - Float64(x * -2.5)) * (wj ^ 2.0))) + Float64(Float64(Float64(1.0 + Float64(-2.0 * Float64(x * -2.5))) + Float64(x * -2.3333333333333335)) * Float64(-(wj ^ 3.0))))
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 = ((x + (x * (-2.0 * wj))) + ((1.0 - (x * -2.5)) * (wj ^ 2.0))) + (((1.0 + (-2.0 * (x * -2.5))) + (x * -2.3333333333333335)) * -(wj ^ 3.0));
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[(x + N[(x * N[(-2.0 * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[(x * -2.5), $MachinePrecision]), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(-2.0 * N[(x * -2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * -2.3333333333333335), $MachinePrecision]), $MachinePrecision] * (-N[Power[wj, 3.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Target

Original	14.2
Target	13.6
Herbie	1.7

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

Derivation

1. Initial program 14.2

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

2. Taylor expanded in wj around 0 1.7

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

3. Simplified 1.7

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

   [Start] 1.7	\[ -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]
   rational.json-simplify-1 [=>] 1.7	\[ \color{blue}{\left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)} \]
   rational.json-simplify-1 [=>] 1.7	\[ \color{blue}{\left(\left(-2 \cdot \left(wj \cdot x\right) + x\right) + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right)} + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \]
   rational.json-simplify-1 [=>] 1.7	\[ \left(\color{blue}{\left(x + -2 \cdot \left(wj \cdot x\right)\right)} + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \]
   rational.json-simplify-43 [=>] 1.7	\[ \left(\left(x + \color{blue}{wj \cdot \left(x \cdot -2\right)}\right) + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \]
   rational.json-simplify-43 [=>] 1.7	\[ \left(\left(x + \color{blue}{x \cdot \left(-2 \cdot wj\right)}\right) + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \]
   rational.json-simplify-2 [=>] 1.7	\[ \left(\left(x + x \cdot \left(-2 \cdot wj\right)\right) + \left(1 - \left(\color{blue}{x \cdot -4} + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \]
   rational.json-simplify-51 [=>] 1.7	\[ \left(\left(x + x \cdot \left(-2 \cdot wj\right)\right) + \left(1 - \color{blue}{x \cdot \left(1.5 + -4\right)}\right) \cdot {wj}^{2}\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \]
   metadata-eval [=>] 1.7	\[ \left(\left(x + x \cdot \left(-2 \cdot wj\right)\right) + \left(1 - x \cdot \color{blue}{-2.5}\right) \cdot {wj}^{2}\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \]
   rational.json-simplify-43 [=>] 1.7	\[ \left(\left(x + x \cdot \left(-2 \cdot wj\right)\right) + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\right) + \color{blue}{\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot \left({wj}^{3} \cdot -1\right)} \]

4. Final simplification 1.7

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

Alternatives

Alternative 1
Error	1.9
Cost	7424

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

Alternative 2
Error	9.2
Cost	7112

\[\begin{array}{l} t_0 := \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{-258}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-265}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	9.2
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-261}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-272}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \]

Alternative 4
Error	9.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-258}:\\ \;\;\;\;\frac{e^{-wj}}{wj + 1} \cdot x\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-272}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \]

Alternative 5
Error	2.1
Cost	7040

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

Alternative 6
Error	9.7
Cost	904

\[\begin{array}{l} t_0 := \frac{x + x \cdot \left(-wj\right)}{wj + 1}\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{-260}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-263}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	9.7
Cost	712

\[\begin{array}{l} t_0 := -2 \cdot \left(wj \cdot x\right) + x\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{-260}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-271}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	9.8
Cost	712

\[\begin{array}{l} t_0 := \frac{x}{1 + wj \cdot 2}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-260}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-266}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	10.1
Cost	584

\[\begin{array}{l} t_0 := \frac{x}{wj + 1}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-260}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-267}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	10.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-261}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-269}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	61.2
Cost	64

\[wj \]

Alternative 12
Error	10.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023064 
(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))))))