Jmat.Real.lambertw, newton loop step

Average Error: 13.8 → 0.9

Time: 34.3s

Precision: binary64

Cost: 14980

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj \leq 0.0026:\\ \;\;\;\;x \cdot \left(-2 \cdot wj\right) + \left(\left(\left(1 + -2 \cdot \left(x \cdot -2.5\right)\right) + x \cdot -2.3333333333333335\right) \cdot \left(-{wj}^{3}\right) + \left(x + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-wj\right) \cdot \frac{wj}{-1 - wj}\\ \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0026)
   (+
    (* x (* -2.0 wj))
    (+
     (*
      (+ (+ 1.0 (* -2.0 (* x -2.5))) (* x -2.3333333333333335))
      (- (pow wj 3.0)))
     (+ x (* (- 1.0 (* x -2.5)) (pow wj 2.0)))))
   (* (- wj) (/ wj (- -1.0 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 tmp;
	if (wj <= 0.0026) {
		tmp = (x * (-2.0 * wj)) + ((((1.0 + (-2.0 * (x * -2.5))) + (x * -2.3333333333333335)) * -pow(wj, 3.0)) + (x + ((1.0 - (x * -2.5)) * pow(wj, 2.0))));
	} else {
		tmp = -wj * (wj / (-1.0 - wj));
	}
	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) :: tmp
    if (wj <= 0.0026d0) then
        tmp = (x * ((-2.0d0) * wj)) + ((((1.0d0 + ((-2.0d0) * (x * (-2.5d0)))) + (x * (-2.3333333333333335d0))) * -(wj ** 3.0d0)) + (x + ((1.0d0 - (x * (-2.5d0))) * (wj ** 2.0d0))))
    else
        tmp = -wj * (wj / ((-1.0d0) - wj))
    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 tmp;
	if (wj <= 0.0026) {
		tmp = (x * (-2.0 * wj)) + ((((1.0 + (-2.0 * (x * -2.5))) + (x * -2.3333333333333335)) * -Math.pow(wj, 3.0)) + (x + ((1.0 - (x * -2.5)) * Math.pow(wj, 2.0))));
	} else {
		tmp = -wj * (wj / (-1.0 - wj));
	}
	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):
	tmp = 0
	if wj <= 0.0026:
		tmp = (x * (-2.0 * wj)) + ((((1.0 + (-2.0 * (x * -2.5))) + (x * -2.3333333333333335)) * -math.pow(wj, 3.0)) + (x + ((1.0 - (x * -2.5)) * math.pow(wj, 2.0))))
	else:
		tmp = -wj * (wj / (-1.0 - wj))
	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)
	tmp = 0.0
	if (wj <= 0.0026)
		tmp = Float64(Float64(x * Float64(-2.0 * wj)) + Float64(Float64(Float64(Float64(1.0 + Float64(-2.0 * Float64(x * -2.5))) + Float64(x * -2.3333333333333335)) * Float64(-(wj ^ 3.0))) + Float64(x + Float64(Float64(1.0 - Float64(x * -2.5)) * (wj ^ 2.0)))));
	else
		tmp = Float64(Float64(-wj) * Float64(wj / Float64(-1.0 - wj)));
	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)
	tmp = 0.0;
	if (wj <= 0.0026)
		tmp = (x * (-2.0 * wj)) + ((((1.0 + (-2.0 * (x * -2.5))) + (x * -2.3333333333333335)) * -(wj ^ 3.0)) + (x + ((1.0 - (x * -2.5)) * (wj ^ 2.0))));
	else
		tmp = -wj * (wj / (-1.0 - wj));
	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_] := If[LessEqual[wj, 0.0026], N[(N[(x * N[(-2.0 * wj), $MachinePrecision]), $MachinePrecision] + N[(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] + N[(x + N[(N[(1.0 - N[(x * -2.5), $MachinePrecision]), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-wj) * N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	13.8
Target	13.3
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if wj < 0.0025999999999999999

   1. Initial program 13.5

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

   2. Taylor expanded in wj around 0 0.7

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

   3. Simplified 0.7

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

      [Start] 0.7	\[ -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]
      rational.json-simplify-41 [=>] 0.7	\[ -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \color{blue}{\left(-2 \cdot \left(wj \cdot x\right) + \left(x + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right)\right)} \]
      rational.json-simplify-41 [=>] 0.7	\[ \color{blue}{-2 \cdot \left(wj \cdot x\right) + \left(\left(x + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)\right)} \]
      rational.json-simplify-43 [=>] 0.7	\[ \color{blue}{wj \cdot \left(x \cdot -2\right)} + \left(\left(x + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)\right) \]
      rational.json-simplify-43 [=>] 0.7	\[ \color{blue}{x \cdot \left(-2 \cdot wj\right)} + \left(\left(x + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right) + -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right)\right) \]
      rational.json-simplify-1 [=>] 0.7	\[ x \cdot \left(-2 \cdot wj\right) + \color{blue}{\left(-1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(x + \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2}\right)\right)} \]

   if 0.0025999999999999999 < wj

   1. Initial program 31.4

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

   2. Taylor expanded in x around 0 43.8

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

   3. Simplified 43.8

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

      [Start] 43.8	\[ wj - \frac{e^{wj} \cdot wj}{e^{wj} \cdot wj + e^{wj}} \]
      rational.json-simplify-2 [=>] 43.8	\[ wj - \frac{\color{blue}{wj \cdot e^{wj}}}{e^{wj} \cdot wj + e^{wj}} \]
      rational.json-simplify-6 [<=] 43.8	\[ wj - \frac{wj \cdot e^{wj}}{e^{wj} \cdot wj + \color{blue}{1 \cdot e^{wj}}} \]
      rational.json-simplify-51 [=>] 43.8	\[ wj - \frac{wj \cdot e^{wj}}{\color{blue}{e^{wj} \cdot \left(1 + wj\right)}} \]
      rational.json-simplify-1 [<=] 43.8	\[ wj - \frac{wj \cdot e^{wj}}{e^{wj} \cdot \color{blue}{\left(wj + 1\right)}} \]

   4. Applied egg-rr 12.8

      \[\leadsto wj - \color{blue}{\left(0 + \frac{wj}{wj + 1}\right)} \]

   5. Simplified 12.8

      \[\leadsto wj - \color{blue}{\frac{wj}{wj + 1}} \]
      Proof

      [Start] 12.8	\[ wj - \left(0 + \frac{wj}{wj + 1}\right) \]
      rational.json-simplify-1 [=>] 12.8	\[ wj - \color{blue}{\left(\frac{wj}{wj + 1} + 0\right)} \]
      rational.json-simplify-4 [=>] 12.8	\[ wj - \color{blue}{\frac{wj}{wj + 1}} \]

   6. Applied egg-rr 12.6

      \[\leadsto \color{blue}{\left(-wj\right) \cdot \frac{wj}{-1 - wj}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.0026:\\ \;\;\;\;x \cdot \left(-2 \cdot wj\right) + \left(\left(\left(1 + -2 \cdot \left(x \cdot -2.5\right)\right) + x \cdot -2.3333333333333335\right) \cdot \left(-{wj}^{3}\right) + \left(x + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-wj\right) \cdot \frac{wj}{-1 - wj}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.2
Cost	7556

\[\begin{array}{l} \mathbf{if}\;wj \leq 0.001:\\ \;\;\;\;\left(x + x \cdot \left(wj \cdot -2\right)\right) + \left(1 - x \cdot -2.5\right) \cdot {wj}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(-wj\right) \cdot \frac{wj}{-1 - wj}\\ \end{array} \]

Alternative 2
Error	1.3
Cost	7172

\[\begin{array}{l} \mathbf{if}\;wj \leq 0.00096:\\ \;\;\;\;\left(x + x \cdot \left(wj \cdot -2\right)\right) + {wj}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(-wj\right) \cdot \frac{wj}{-1 - wj}\\ \end{array} \]

Alternative 3
Error	9.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;wj \leq 2.4 \cdot 10^{-48}:\\ \;\;\;\;-2 \cdot \left(wj \cdot x\right) + x\\ \mathbf{elif}\;wj \leq 8.2 \cdot 10^{-9}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 4
Error	9.2
Cost	644

\[\begin{array}{l} \mathbf{if}\;wj \leq 2.5 \cdot 10^{-45}:\\ \;\;\;\;-2 \cdot \left(wj \cdot x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(-wj\right) \cdot \frac{wj}{-1 - wj}\\ \end{array} \]

Alternative 5
Error	10.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;wj \leq 2.5 \cdot 10^{-45}:\\ \;\;\;\;-2 \cdot \left(wj \cdot x\right) + x\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 6
Error	10.2
Cost	324

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.06 \cdot 10^{-45}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 7
Error	61.2
Cost	64

\[wj \]

Alternative 8
Error	9.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023074 
(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))))))