Jmat.Real.lambertw, newton loop step

Average Accuracy: 77.8% → 96.3%

Time: 10.3s

Precision: binary64

Cost: 15552

Math

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

↓

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

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)
     (- (- (- -1.0 (* -2.0 t_0)) (* x -3.0)) (* 0.6666666666666666 x)))
    (+ (* (- 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) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (0.6666666666666666 * x))) + (((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) * ((((-1.0d0) - ((-2.0d0) * t_0)) - (x * (-3.0d0))) - (0.6666666666666666d0 * x))) + (((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) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (0.6666666666666666 * x))) + (((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) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (0.6666666666666666 * x))) + (((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(Float64(-1.0 - Float64(-2.0 * t_0)) - Float64(x * -3.0)) - Float64(0.6666666666666666 * x))) + 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) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (0.6666666666666666 * x))) + (((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[(N[(-1.0 - N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x * -3.0), $MachinePrecision]), $MachinePrecision] - N[(0.6666666666666666 * x), $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	77.8%
Target	78.7%
Herbie	96.3%

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

Derivation

1. Initial program 83.3%

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

2. Simplified 83.7%

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

3. Taylor expanded in wj around 0 97.5%

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

4. Final simplification 97.5%

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

Alternatives

Alternative 1
Accuracy	95.8%
Cost	7680

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

Alternative 2
Accuracy	95.8%
Cost	7424

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

Alternative 3
Accuracy	96.2%
Cost	7296

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

Alternative 4
Accuracy	85.6%
Cost	6848

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

Alternative 5
Accuracy	84.9%
Cost	1472

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

Alternative 6
Accuracy	84.8%
Cost	832

\[x + wj \cdot \left(x \cdot -2 - wj \cdot \left(x \cdot -2.5\right)\right) \]

Alternative 7
Accuracy	84.7%
Cost	576

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

Alternative 8
Accuracy	84.7%
Cost	576

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

Alternative 9
Accuracy	84.6%
Cost	448

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

Alternative 10
Accuracy	84.7%
Cost	448

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

Alternative 11
Accuracy	4.4%
Cost	64

\[wj \]

Alternative 12
Accuracy	84.1%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023157 -o generate:proofs
(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))))))