Jmat.Real.lambertw, newton loop step

Average Error: 13.5 → 0.4

Time: 20.5s

Precision: binary64

Cost: 34180

Math

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

↓

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t_0 - x}{e^{wj} + t_0} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\left(x \cdot \left(-2 \cdot wj\right) + \left(x + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\right)\right) + \left(-{wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(wj + 1\right) \cdot e^{wj}} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 2e-10)
     (+
      (+ (* x (* -2.0 wj)) (+ x (* (- 1.0 (* x -2.5)) (pow wj 2.0))))
      (- (pow wj 3.0)))
     (+ (/ x (* (+ wj 1.0) (exp wj))) (- wj (/ 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 = wj * exp(wj);
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 2e-10) {
		tmp = ((x * (-2.0 * wj)) + (x + ((1.0 - (x * -2.5)) * pow(wj, 2.0)))) + -pow(wj, 3.0);
	} else {
		tmp = (x / ((wj + 1.0) * exp(wj))) + (wj - (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) :: tmp
    t_0 = wj * exp(wj)
    if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 2d-10) then
        tmp = ((x * ((-2.0d0) * wj)) + (x + ((1.0d0 - (x * (-2.5d0))) * (wj ** 2.0d0)))) + -(wj ** 3.0d0)
    else
        tmp = (x / ((wj + 1.0d0) * exp(wj))) + (wj - (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 = wj * Math.exp(wj);
	double tmp;
	if ((wj - ((t_0 - x) / (Math.exp(wj) + t_0))) <= 2e-10) {
		tmp = ((x * (-2.0 * wj)) + (x + ((1.0 - (x * -2.5)) * Math.pow(wj, 2.0)))) + -Math.pow(wj, 3.0);
	} else {
		tmp = (x / ((wj + 1.0) * Math.exp(wj))) + (wj - (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 = wj * math.exp(wj)
	tmp = 0
	if (wj - ((t_0 - x) / (math.exp(wj) + t_0))) <= 2e-10:
		tmp = ((x * (-2.0 * wj)) + (x + ((1.0 - (x * -2.5)) * math.pow(wj, 2.0)))) + -math.pow(wj, 3.0)
	else:
		tmp = (x / ((wj + 1.0) * math.exp(wj))) + (wj - (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(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 2e-10)
		tmp = Float64(Float64(Float64(x * Float64(-2.0 * wj)) + Float64(x + Float64(Float64(1.0 - Float64(x * -2.5)) * (wj ^ 2.0)))) + Float64(-(wj ^ 3.0)));
	else
		tmp = Float64(Float64(x / Float64(Float64(wj + 1.0) * exp(wj))) + Float64(wj - 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 = wj * exp(wj);
	tmp = 0.0;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 2e-10)
		tmp = ((x * (-2.0 * wj)) + (x + ((1.0 - (x * -2.5)) * (wj ^ 2.0)))) + -(wj ^ 3.0);
	else
		tmp = (x / ((wj + 1.0) * exp(wj))) + (wj - (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[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-10], N[(N[(N[(x * N[(-2.0 * wj), $MachinePrecision]), $MachinePrecision] + N[(x + N[(N[(1.0 - N[(x * -2.5), $MachinePrecision]), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[Power[wj, 3.0], $MachinePrecision])), $MachinePrecision], N[(N[(x / N[(N[(wj + 1.0), $MachinePrecision] * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	13.5
Target	12.9
Herbie	0.4

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

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))))) < 2.00000000000000007e-10

   1. Initial program 17.7

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

   2. Taylor expanded in wj around 0 0.5

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

   3. Simplified 0.5

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

      [Start] 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	\[ \color{blue}{\left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)} \]
      rational.json-simplify-41 [=>] 0.5	\[ \color{blue}{\left(-2 \cdot \left(wj \cdot x\right) + \left(x + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right)\right)} + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \]
      rational.json-simplify-43 [=>] 0.5	\[ \left(\color{blue}{wj \cdot \left(x \cdot -2\right)} + \left(x + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right)\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \]
      rational.json-simplify-43 [=>] 0.5	\[ \left(\color{blue}{x \cdot \left(-2 \cdot wj\right)} + \left(x + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right)\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \]
      rational.json-simplify-2 [=>] 0.5	\[ \left(x \cdot \left(-2 \cdot wj\right) + \left(x + \left(1 - \left(\color{blue}{x \cdot -4} + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right)\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \]
      rational.json-simplify-47 [=>] 0.5	\[ \left(x \cdot \left(-2 \cdot wj\right) + \left(x + \left(1 - \color{blue}{x \cdot \left(1.5 + -4\right)}\right) \cdot {wj}^{2}\right)\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \]
      metadata-eval [=>] 0.5	\[ \left(x \cdot \left(-2 \cdot wj\right) + \left(x + \left(1 - x \cdot \color{blue}{-2.5}\right) \cdot {wj}^{2}\right)\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) \]
      rational.json-simplify-43 [=>] 0.5	\[ \left(x \cdot \left(-2 \cdot wj\right) + \left(x + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\right)\right) + \color{blue}{\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot \left({wj}^{3} \cdot -1\right)} \]

   4. Taylor expanded in x around 0 0.5

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

   5. Simplified 0.5

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

      [Start] 0.5	\[ \left(x \cdot \left(-2 \cdot wj\right) + \left(x + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\right)\right) + -1 \cdot {wj}^{3} \]
      rational.json-simplify-2 [=>] 0.5	\[ \left(x \cdot \left(-2 \cdot wj\right) + \left(x + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\right)\right) + \color{blue}{{wj}^{3} \cdot -1} \]
      rational.json-simplify-9 [=>] 0.5	\[ \left(x \cdot \left(-2 \cdot wj\right) + \left(x + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\right)\right) + \color{blue}{\left(-{wj}^{3}\right)} \]

   if 2.00000000000000007e-10 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 2.5

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

   2. Applied egg-rr 2.5

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

   3. Taylor expanded in x around 0 0.2

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

   4. Taylor expanded in x around 0 0.2

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

   5. Simplified 0.2

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

      [Start] 0.2	\[ \left(wj + \frac{x}{e^{wj} \cdot \left(1 + wj\right)}\right) - \frac{wj}{1 + wj} \]
      rational.json-simplify-2 [<=] 0.2	\[ \left(wj + \frac{x}{\color{blue}{\left(1 + wj\right) \cdot e^{wj}}}\right) - \frac{wj}{1 + wj} \]
      rational.json-simplify-5 [<=] 0.2	\[ \left(wj + \frac{x}{\left(1 + wj\right) \cdot e^{wj}}\right) - \color{blue}{\left(\frac{wj}{1 + wj} - 0\right)} \]
      rational.json-simplify-44 [<=] 0.2	\[ \color{blue}{0 - \left(\frac{wj}{1 + wj} - \left(wj + \frac{x}{\left(1 + wj\right) \cdot e^{wj}}\right)\right)} \]
      rational.json-simplify-45 [<=] 0.2	\[ 0 - \color{blue}{\left(\left(\frac{wj}{1 + wj} - wj\right) - \frac{x}{\left(1 + wj\right) \cdot e^{wj}}\right)} \]
      metadata-eval [<=] 0.2	\[ \color{blue}{\left(-1 - -1\right)} - \left(\left(\frac{wj}{1 + wj} - wj\right) - \frac{x}{\left(1 + wj\right) \cdot e^{wj}}\right) \]
      rational.json-simplify-42 [<=] 0.8	\[ \color{blue}{\left(-1 - \left(\left(\frac{wj}{1 + wj} - wj\right) - \frac{x}{\left(1 + wj\right) \cdot e^{wj}}\right)\right) - -1} \]
      rational.json-simplify-42 [=>] 0.8	\[ \left(-1 - \color{blue}{\left(\left(\frac{wj}{1 + wj} - \frac{x}{\left(1 + wj\right) \cdot e^{wj}}\right) - wj\right)}\right) - -1 \]
      rational.json-simplify-44 [<=] 0.8	\[ \color{blue}{\left(wj - \left(\left(\frac{wj}{1 + wj} - \frac{x}{\left(1 + wj\right) \cdot e^{wj}}\right) - -1\right)\right)} - -1 \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\left(x \cdot \left(-2 \cdot wj\right) + \left(x + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\right)\right) + \left(-{wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(wj + 1\right) \cdot e^{wj}} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	13764

\[\begin{array}{l} t_0 := \frac{wj}{1 + wj}\\ t_1 := t_0 \cdot t_0\\ \mathbf{if}\;wj \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;\left(x \cdot \left(-2 \cdot wj\right) + {wj}^{2}\right) - \left({wj}^{3} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{\left(1 + wj\right) \cdot e^{wj}} + wj\right) - \left(t_0 \cdot \left(t_0 \cdot t_1\right)\right) \cdot \frac{\frac{1}{t_0}}{t_1}\\ \end{array} \]

Alternative 2
Error	0.9
Cost	9924

\[\begin{array}{l} t_0 := \frac{wj}{1 + wj}\\ t_1 := t_0 \cdot t_0\\ \mathbf{if}\;wj \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(-2 \cdot wj\right) + \left(x + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{\left(1 + wj\right) \cdot e^{wj}} + wj\right) - \left(t_0 \cdot \left(t_0 \cdot t_1\right)\right) \cdot \frac{\frac{1}{t_0}}{t_1}\\ \end{array} \]

Alternative 3
Error	0.9
Cost	7492

\[\begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \left(-2 \cdot wj\right) + \left(x + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(wj + 1\right) \cdot e^{wj}} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array} \]

Alternative 4
Error	1.3
Cost	7172

\[\begin{array}{l} \mathbf{if}\;wj \leq 0.00087:\\ \;\;\;\;x \cdot \left(-2 \cdot wj\right) + \left(x + {wj}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 5
Error	1.6
Cost	6788

\[\begin{array}{l} \mathbf{if}\;wj \leq 0.000125:\\ \;\;\;\;{wj}^{2} + x\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 6
Error	8.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;wj \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;-2 \cdot \left(wj \cdot x\right) + x\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 7
Error	8.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;wj \leq 0.000145:\\ \;\;\;\;\frac{x}{wj + \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 8
Error	8.9
Cost	448

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

Alternative 9
Error	61.2
Cost	64

\[wj \]

Alternative 10
Error	9.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023077 
(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))))))