Jmat.Real.lambertw, newton loop step

Average Error: 14.1 → 1.9

Time: 11.1s

Precision: binary64

Cost: 15104

Math

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

↓

\[\left(x + wj \cdot \left(x \cdot -2\right)\right) + \left(\left(-\left(x \cdot -3 + \left(1 + \left(x \cdot 0.6666666666666666 + -2 \cdot \left(x \cdot -2.5\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\right) \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (+
  (+ x (* wj (* x -2.0)))
  (+
   (-
    (*
     (+ (* x -3.0) (+ 1.0 (+ (* x 0.6666666666666666) (* -2.0 (* x -2.5)))))
     (pow wj 3.0)))
   (* (- 1.0 (* x -2.5)) (pow wj 2.0)))))

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 + (wj * (x * -2.0))) + (-(((x * -3.0) + (1.0 + ((x * 0.6666666666666666) + (-2.0 * (x * -2.5))))) * pow(wj, 3.0)) + ((1.0 - (x * -2.5)) * pow(wj, 2.0)));
}

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 + (wj * (x * (-2.0d0)))) + (-(((x * (-3.0d0)) + (1.0d0 + ((x * 0.6666666666666666d0) + ((-2.0d0) * (x * (-2.5d0)))))) * (wj ** 3.0d0)) + ((1.0d0 - (x * (-2.5d0))) * (wj ** 2.0d0)))
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 + (wj * (x * -2.0))) + (-(((x * -3.0) + (1.0 + ((x * 0.6666666666666666) + (-2.0 * (x * -2.5))))) * Math.pow(wj, 3.0)) + ((1.0 - (x * -2.5)) * Math.pow(wj, 2.0)));
}

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 + (wj * (x * -2.0))) + (-(((x * -3.0) + (1.0 + ((x * 0.6666666666666666) + (-2.0 * (x * -2.5))))) * math.pow(wj, 3.0)) + ((1.0 - (x * -2.5)) * math.pow(wj, 2.0)))

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(x + Float64(wj * Float64(x * -2.0))) + Float64(Float64(-Float64(Float64(Float64(x * -3.0) + Float64(1.0 + Float64(Float64(x * 0.6666666666666666) + Float64(-2.0 * Float64(x * -2.5))))) * (wj ^ 3.0))) + Float64(Float64(1.0 - Float64(x * -2.5)) * (wj ^ 2.0))))
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 + (wj * (x * -2.0))) + (-(((x * -3.0) + (1.0 + ((x * 0.6666666666666666) + (-2.0 * (x * -2.5))))) * (wj ^ 3.0)) + ((1.0 - (x * -2.5)) * (wj ^ 2.0)));
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[(x + N[(wj * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-N[(N[(N[(x * -3.0), $MachinePrecision] + N[(1.0 + N[(N[(x * 0.6666666666666666), $MachinePrecision] + N[(-2.0 * N[(x * -2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]) + N[(N[(1.0 - N[(x * -2.5), $MachinePrecision]), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	14.1
Target	13.4
Herbie	1.9

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

Derivation

1. Initial program 14.1

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

2. Taylor expanded in wj around 0 1.9

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

3. Simplified 1.9

   \[\leadsto \color{blue}{\left(x + wj \cdot \left(x \cdot -2\right)\right) + \left(\left(-\left(x \cdot -3 + \left(1 + \left(x \cdot 0.6666666666666666 + -2 \cdot \left(x \cdot -2.5\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\right)} \]
   Proof

   [Start] 1.9	\[ -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 1.9	\[ -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \color{blue}{\left(\left(-2 \cdot \left(wj \cdot x\right) + x\right) + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-82 [=>] 1.9	\[ \color{blue}{\left(-2 \cdot \left(wj \cdot x\right) + x\right) + \left(-1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 1.9	\[ \color{blue}{\left(x + -2 \cdot \left(wj \cdot x\right)\right)} + \left(-1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 1.9	\[ \left(x + \color{blue}{wj \cdot \left(-2 \cdot x\right)}\right) + \left(-1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 1.9	\[ \left(x + wj \cdot \color{blue}{\left(x \cdot -2\right)}\right) + \left(-1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right) \]

4. Final simplification 1.9

   \[\leadsto \left(x + wj \cdot \left(x \cdot -2\right)\right) + \left(\left(-\left(x \cdot -3 + \left(1 + \left(x \cdot 0.6666666666666666 + -2 \cdot \left(x \cdot -2.5\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\right) \]

Alternatives

Alternative 1
Error	1.9
Cost	14080

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

Alternative 2
Error	2.1
Cost	7424

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

Alternative 3
Error	9.8
Cost	7376

\[\begin{array}{l} t_0 := \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{if}\;wj \leq -2.1 \cdot 10^{-20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;wj \leq -2.6 \cdot 10^{-45}:\\ \;\;\;\;{wj}^{2}\\ \mathbf{elif}\;wj \leq -2 \cdot 10^{-65}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq -5.2 \cdot 10^{-70}:\\ \;\;\;\;{wj}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	2.2
Cost	7040

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

Alternative 5
Error	10.0
Cost	6792

\[\begin{array}{l} \mathbf{if}\;wj \leq -2 \cdot 10^{-65}:\\ \;\;\;\;\left(-2 \cdot wj + 1\right) \cdot x\\ \mathbf{elif}\;wj \leq -5.2 \cdot 10^{-70}:\\ \;\;\;\;{wj}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \end{array} \]

Alternative 6
Error	9.8
Cost	448

\[\left(-2 \cdot wj + 1\right) \cdot x \]

Alternative 7
Error	9.7
Cost	448

\[\frac{x}{1 + wj \cdot 2} \]

Alternative 8
Error	61.2
Cost	64

\[wj \]

Alternative 9
Error	10.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(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))))))