Jmat.Real.lambertw, newton loop step

Average Error: 13.8 → 0.6

Time: 6.4s

Precision: binary64

Cost: 35780

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ t_1 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t_1 - x}{e^{wj} + t_1} \leq 10^{-16}:\\ \;\;\;\;\left(\left(x + x \cdot \left(-2 \cdot wj\right)\right) + \left(1 - t_0\right) \cdot {wj}^{2}\right) + \left(x \cdot 0.6666666666666666 + \left(x \cdot -3 + \left(1 + -2 \cdot t_0\right)\right)\right) \cdot \left(-1 \cdot {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right) - \frac{wj}{wj + 1}\\ \end{array} \]

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))) (t_1 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_1 x) (+ (exp wj) t_1))) 1e-16)
     (+
      (+ (+ x (* x (* -2.0 wj))) (* (- 1.0 t_0) (pow wj 2.0)))
      (*
       (+ (* x 0.6666666666666666) (+ (* x -3.0) (+ 1.0 (* -2.0 t_0))))
       (* -1.0 (pow wj 3.0))))
     (- (+ wj (/ x (* (exp wj) (+ wj 1.0)))) (/ wj (+ wj 1.0))))))

C

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

↓

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double t_1 = wj * exp(wj);
	double tmp;
	if ((wj - ((t_1 - x) / (exp(wj) + t_1))) <= 1e-16) {
		tmp = ((x + (x * (-2.0 * wj))) + ((1.0 - t_0) * pow(wj, 2.0))) + (((x * 0.6666666666666666) + ((x * -3.0) + (1.0 + (-2.0 * t_0)))) * (-1.0 * pow(wj, 3.0)));
	} else {
		tmp = (wj + (x / (exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * (-4.0d0)) + (x * 1.5d0)
    t_1 = wj * exp(wj)
    if ((wj - ((t_1 - x) / (exp(wj) + t_1))) <= 1d-16) then
        tmp = ((x + (x * ((-2.0d0) * wj))) + ((1.0d0 - t_0) * (wj ** 2.0d0))) + (((x * 0.6666666666666666d0) + ((x * (-3.0d0)) + (1.0d0 + ((-2.0d0) * t_0)))) * ((-1.0d0) * (wj ** 3.0d0)))
    else
        tmp = (wj + (x / (exp(wj) * (wj + 1.0d0)))) - (wj / (wj + 1.0d0))
    end if
    code = tmp
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);
	double t_1 = wj * Math.exp(wj);
	double tmp;
	if ((wj - ((t_1 - x) / (Math.exp(wj) + t_1))) <= 1e-16) {
		tmp = ((x + (x * (-2.0 * wj))) + ((1.0 - t_0) * Math.pow(wj, 2.0))) + (((x * 0.6666666666666666) + ((x * -3.0) + (1.0 + (-2.0 * t_0)))) * (-1.0 * Math.pow(wj, 3.0)));
	} else {
		tmp = (wj + (x / (Math.exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0));
	}
	return tmp;
}

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)
	t_1 = wj * math.exp(wj)
	tmp = 0
	if (wj - ((t_1 - x) / (math.exp(wj) + t_1))) <= 1e-16:
		tmp = ((x + (x * (-2.0 * wj))) + ((1.0 - t_0) * math.pow(wj, 2.0))) + (((x * 0.6666666666666666) + ((x * -3.0) + (1.0 + (-2.0 * t_0)))) * (-1.0 * math.pow(wj, 3.0)))
	else:
		tmp = (wj + (x / (math.exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0))
	return tmp

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))
	t_1 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_1 - x) / Float64(exp(wj) + t_1))) <= 1e-16)
		tmp = Float64(Float64(Float64(x + Float64(x * Float64(-2.0 * wj))) + Float64(Float64(1.0 - t_0) * (wj ^ 2.0))) + Float64(Float64(Float64(x * 0.6666666666666666) + Float64(Float64(x * -3.0) + Float64(1.0 + Float64(-2.0 * t_0)))) * Float64(-1.0 * (wj ^ 3.0))));
	else
		tmp = Float64(Float64(wj + Float64(x / Float64(exp(wj) * Float64(wj + 1.0)))) - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(wj, x)
	t_0 = (x * -4.0) + (x * 1.5);
	t_1 = wj * exp(wj);
	tmp = 0.0;
	if ((wj - ((t_1 - x) / (exp(wj) + t_1))) <= 1e-16)
		tmp = ((x + (x * (-2.0 * wj))) + ((1.0 - t_0) * (wj ^ 2.0))) + (((x * 0.6666666666666666) + ((x * -3.0) + (1.0 + (-2.0 * t_0)))) * (-1.0 * (wj ^ 3.0)));
	else
		tmp = (wj + (x / (exp(wj) * (wj + 1.0)))) - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
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]}, Block[{t$95$1 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$1 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-16], N[(N[(N[(x + N[(x * N[(-2.0 * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - t$95$0), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x * 0.6666666666666666), $MachinePrecision] + N[(N[(x * -3.0), $MachinePrecision] + N[(1.0 + N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 * N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(wj + N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	13.8
Target	13.1
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 9.9999999999999998e-17

   1. Initial program 18.2

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

   2. Simplified 18.2

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

      [Start] 18.2	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      rational.json-simplify-2 [=>] 18.2	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + \color{blue}{e^{wj} \cdot wj}} \]
      rational.json-simplify-19 [=>] 18.2	\[ wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(1 + wj\right)}} \]

   3. Taylor expanded in wj around 0 0.5

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

   4. Simplified 0.5

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

      [Start] 0.5	\[ -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]
      rational.json-simplify-1 [=>] 0.5	\[ \color{blue}{\left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)} \]

   if 9.9999999999999998e-17 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 2.9

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

   2. Simplified 3.0

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

      [Start] 2.9	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      rational.json-simplify-2 [=>] 2.9	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + \color{blue}{e^{wj} \cdot wj}} \]
      rational.json-simplify-19 [=>] 3.0	\[ wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{e^{wj} \cdot \left(1 + wj\right)}} \]

   3. Taylor expanded in x around 0 0.6

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

   4. Simplified 0.6

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

      [Start] 0.6	\[ \left(wj + \frac{x}{e^{wj} \cdot \left(1 + wj\right)}\right) - \frac{wj}{1 + wj} \]
      rational.json-simplify-34 [<=] 0.6	\[ \color{blue}{\frac{x}{e^{wj} \cdot \left(1 + wj\right)} - \left(\frac{wj}{1 + wj} - wj\right)} \]
      rational.json-simplify-40 [=>] 0.6	\[ \frac{x}{e^{wj} \cdot \left(1 + wj\right)} - \color{blue}{\left(-\left(wj - \frac{wj}{1 + wj}\right)\right)} \]
      rational.json-simplify-41 [=>] 0.6	\[ \frac{x}{e^{wj} \cdot \left(1 + wj\right)} - \left(-\color{blue}{\left(\left(-\frac{wj}{1 + wj}\right) + wj\right)}\right) \]
      rational.json-simplify-27 [=>] 0.6	\[ \frac{x}{e^{wj} \cdot \left(1 + wj\right)} - \color{blue}{\left(\left(-\left(-\frac{wj}{1 + wj}\right)\right) - wj\right)} \]
      rational.json-simplify-34 [=>] 0.6	\[ \color{blue}{\left(wj + \frac{x}{e^{wj} \cdot \left(1 + wj\right)}\right) - \left(-\left(-\frac{wj}{1 + wj}\right)\right)} \]
      rational.json-simplify-1 [=>] 0.6	\[ \left(wj + \frac{x}{e^{wj} \cdot \color{blue}{\left(wj + 1\right)}}\right) - \left(-\left(-\frac{wj}{1 + wj}\right)\right) \]
      rational.json-simplify-28 [<=] 0.6	\[ \left(wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right) - \left(-\color{blue}{\frac{-wj}{1 + wj}}\right) \]
      rational.json-simplify-28 [<=] 0.6	\[ \left(wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right) - \color{blue}{\frac{-\left(-wj\right)}{1 + wj}} \]
      rational.json-simplify-7 [<=] 0.6	\[ \left(wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right) - \frac{-\color{blue}{\left(0 - wj\right)}}{1 + wj} \]
      rational.json-simplify-40 [<=] 0.6	\[ \left(wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right) - \frac{\color{blue}{wj - 0}}{1 + wj} \]
      rational.json-simplify-5 [=>] 0.6	\[ \left(wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right) - \frac{\color{blue}{wj}}{1 + wj} \]
      rational.json-simplify-1 [=>] 0.6	\[ \left(wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right) - \frac{wj}{\color{blue}{wj + 1}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 10^{-16}:\\ \;\;\;\;\left(\left(x + x \cdot \left(-2 \cdot wj\right)\right) + \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2}\right) + \left(x \cdot 0.6666666666666666 + \left(x \cdot -3 + \left(1 + -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\right) \cdot \left(-1 \cdot {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right) - \frac{wj}{wj + 1}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	7556

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.42 \cdot 10^{-8}:\\ \;\;\;\;x + \left(x \cdot \left(-2 \cdot wj\right) + {wj}^{2} \cdot \left(1 + 2.5 \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right) - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 2
Error	1.0
Cost	7492

\[\begin{array}{l} \mathbf{if}\;wj \leq 7 \cdot 10^{-9}:\\ \;\;\;\;x + \left(x \cdot \left(-2 \cdot wj\right) + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\right) - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 3
Error	1.4
Cost	7172

\[\begin{array}{l} \mathbf{if}\;wj \leq 0.39:\\ \;\;\;\;x + \left(x \cdot \left(-2 \cdot wj\right) + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 4
Error	1.8
Cost	6788

\[\begin{array}{l} \mathbf{if}\;wj \leq 6.4:\\ \;\;\;\;x + {wj}^{2}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 5
Error	8.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;wj \leq 0.52:\\ \;\;\;\;-2 \cdot \left(wj \cdot x\right) + x\\ \mathbf{else}:\\ \;\;\;\;wj - 1\\ \end{array} \]

Alternative 6
Error	8.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;wj \leq 8.8 \cdot 10^{-9}:\\ \;\;\;\;-2 \cdot \left(wj \cdot x\right) + x\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 7
Error	8.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;wj \leq 8.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 8
Error	9.0
Cost	324

\[\begin{array}{l} \mathbf{if}\;wj \leq 7.2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj - 1\\ \end{array} \]

Alternative 9
Error	61.2
Cost	64

\[wj \]

Alternative 10
Error	9.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023053 
(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))))))