Jmat.Real.lambertw, newton loop step

Average Error: 13.9 → 0.6

Time: 14.4s

Precision: binary64

Cost: 7428

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;\left(wj \cdot wj + \left(x + -2 \cdot \left(x \cdot wj\right)\right)\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{1 + wj}\\ \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (if (<= wj 5.5e-7)
   (- (+ (* wj wj) (+ x (* -2.0 (* x wj)))) (pow wj 3.0))
   (+ wj (/ (- (/ x (exp 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 <= 5.5e-7) {
		tmp = ((wj * wj) + (x + (-2.0 * (x * wj)))) - pow(wj, 3.0);
	} else {
		tmp = wj + (((x / exp(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 <= 5.5d-7) then
        tmp = ((wj * wj) + (x + ((-2.0d0) * (x * wj)))) - (wj ** 3.0d0)
    else
        tmp = wj + (((x / exp(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 <= 5.5e-7) {
		tmp = ((wj * wj) + (x + (-2.0 * (x * wj)))) - Math.pow(wj, 3.0);
	} else {
		tmp = wj + (((x / Math.exp(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 <= 5.5e-7:
		tmp = ((wj * wj) + (x + (-2.0 * (x * wj)))) - math.pow(wj, 3.0)
	else:
		tmp = wj + (((x / math.exp(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 <= 5.5e-7)
		tmp = Float64(Float64(Float64(wj * wj) + Float64(x + Float64(-2.0 * Float64(x * wj)))) - (wj ^ 3.0));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - 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 <= 5.5e-7)
		tmp = ((wj * wj) + (x + (-2.0 * (x * wj)))) - (wj ^ 3.0);
	else
		tmp = wj + (((x / exp(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, 5.5e-7], N[(N[(N[(wj * wj), $MachinePrecision] + N[(x + N[(-2.0 * N[(x * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	13.9
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 wj < 5.5000000000000003e-7

   1. Initial program 13.4

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

   2. Simplified 13.4

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

      [Start] 13.4	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 13.4	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-mul-1 [=>] 13.4	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      *-commutative [=>] 13.4	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
      *-commutative [<=] 13.4	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      neg-mul-1 [<=] 13.4	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-sub0 [=>] 13.4	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 13.4	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      associate--r- [=>] 13.4	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      +-commutative [=>] 13.4	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      sub0-neg [=>] 13.4	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [<=] 13.4	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   3. Taylor expanded in wj around 0 0.6

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

   4. Taylor expanded in x around 0 0.7

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

   5. Simplified 0.7

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

      [Start] 0.7	\[ -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left({wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]
      unpow2 [=>] 0.7	\[ -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(\color{blue}{wj \cdot wj} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]

   6. Taylor expanded in x around 0 0.6

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

   if 5.5000000000000003e-7 < wj

   1. Initial program 32.4

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

   2. Simplified 1.7

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

      [Start] 32.4	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 32.4	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-mul-1 [=>] 32.4	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      *-commutative [=>] 32.4	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
      *-commutative [<=] 32.4	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      neg-mul-1 [<=] 32.4	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-sub0 [=>] 32.4	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 32.5	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      associate--r- [=>] 32.5	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      +-commutative [=>] 32.5	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      sub0-neg [=>] 32.5	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [<=] 32.5	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;\left(wj \cdot wj + \left(x + -2 \cdot \left(x \cdot wj\right)\right)\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{1 + wj}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.8
Cost	9088

\[\left(\left(wj \cdot wj - \left(wj \cdot wj\right) \cdot \left(x \cdot -2.5\right)\right) + \left(x + -2 \cdot \left(x \cdot wj\right)\right)\right) - \left(0.6666666666666666 \cdot x + \left(x \cdot -3 + \left(1 - -2 \cdot \left(x \cdot 4 + x \cdot -1.5\right)\right)\right)\right) \cdot {wj}^{3} \]

Alternative 2
Error	1.8
Cost	8960

\[\left(\left(x + -2 \cdot \left(x \cdot wj\right)\right) - \left(wj \cdot wj\right) \cdot \left(-1 + x \cdot -2.5\right)\right) - \left(0.6666666666666666 \cdot x + \left(x \cdot -3 + \left(1 - -2 \cdot \left(x \cdot 4 + x \cdot -1.5\right)\right)\right)\right) \cdot {wj}^{3} \]

Alternative 3
Error	1.7
Cost	7808

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

Alternative 4
Error	0.9
Cost	7492

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

Alternative 5
Error	0.9
Cost	7236

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

Alternative 6
Error	8.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.22 \cdot 10^{-36}:\\ \;\;\;\;x + -2 \cdot \left(x \cdot wj\right)\\ \mathbf{elif}\;wj \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;\left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \]

Alternative 7
Error	8.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-38}:\\ \;\;\;\;\left(1 - wj\right) \cdot \frac{x}{1 + wj}\\ \mathbf{elif}\;wj \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;\left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \]

Alternative 8
Error	8.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.34 \cdot 10^{-36}:\\ \;\;\;\;\frac{x \cdot \left(1 - wj\right)}{1 + wj}\\ \mathbf{elif}\;wj \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;\left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \]

Alternative 9
Error	8.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.4 \cdot 10^{-37}:\\ \;\;\;\;x + \left(x \cdot -2\right) \cdot \left(wj - wj \cdot wj\right)\\ \mathbf{elif}\;wj \leq 8.5 \cdot 10^{-6}:\\ \;\;\;\;\left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \]

Alternative 10
Error	9.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;wj \leq 3.1 \cdot 10^{-37} \lor \neg \left(wj \leq 1.45 \cdot 10^{-10}\right):\\ \;\;\;\;x + -2 \cdot \left(x \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 11
Error	8.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;wj \leq 2.7 \cdot 10^{-37}:\\ \;\;\;\;x + -2 \cdot \left(x \cdot wj\right)\\ \mathbf{elif}\;wj \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \]

Alternative 12
Error	2.0
Cost	704

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

Alternative 13
Error	10.0
Cost	324

\[\begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-38}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 14
Error	61.2
Cost	64

\[wj \]

Alternative 15
Error	9.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023054 
(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))))))