Jmat.Real.lambertw, newton loop step

Average Accuracy: 79.3% → 97.5%

Time: 16.1s

Precision: binary64

Cost: 7492

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj \leq -5.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{wj + 1} - \left(\frac{wj}{wj + 1} - wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj + \left(x + -2 \cdot \left(x \cdot wj\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (if (<= wj -5.5e-9)
   (- (/ (/ x (exp wj)) (+ wj 1.0)) (- (/ wj (+ wj 1.0)) wj))
   (+ (* wj wj) (+ 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 tmp;
	if (wj <= -5.5e-9) {
		tmp = ((x / exp(wj)) / (wj + 1.0)) - ((wj / (wj + 1.0)) - wj);
	} else {
		tmp = (wj * wj) + (x + (-2.0 * (x * 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-9)) then
        tmp = ((x / exp(wj)) / (wj + 1.0d0)) - ((wj / (wj + 1.0d0)) - wj)
    else
        tmp = (wj * wj) + (x + ((-2.0d0) * (x * 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-9) {
		tmp = ((x / Math.exp(wj)) / (wj + 1.0)) - ((wj / (wj + 1.0)) - wj);
	} else {
		tmp = (wj * wj) + (x + (-2.0 * (x * 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-9:
		tmp = ((x / math.exp(wj)) / (wj + 1.0)) - ((wj / (wj + 1.0)) - wj)
	else:
		tmp = (wj * wj) + (x + (-2.0 * (x * 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-9)
		tmp = Float64(Float64(Float64(x / exp(wj)) / Float64(wj + 1.0)) - Float64(Float64(wj / Float64(wj + 1.0)) - wj));
	else
		tmp = Float64(Float64(wj * wj) + Float64(x + Float64(-2.0 * Float64(x * 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-9)
		tmp = ((x / exp(wj)) / (wj + 1.0)) - ((wj / (wj + 1.0)) - wj);
	else
		tmp = (wj * wj) + (x + (-2.0 * (x * 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-9], N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision], N[(N[(wj * wj), $MachinePrecision] + N[(x + N[(-2.0 * N[(x * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	79.3%
Target	80.3%
Herbie	97.5%

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

Derivation

1. Split input into 2 regimes

2. if wj < -5.4999999999999996e-9

   1. Initial program 91.5%

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

   2. Simplified 91.5%

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

      [Start] 91.5	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 91.5	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-mul-1 [=>] 91.5	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      *-commutative [=>] 91.5	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
      *-commutative [<=] 91.5	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      neg-mul-1 [<=] 91.5	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-sub0 [=>] 91.5	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 91.4	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      associate--r- [=>] 91.4	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      +-commutative [=>] 91.4	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      sub0-neg [=>] 91.4	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [<=] 91.4	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   3. Applied egg-rr 91.5%

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

      [Start] 91.5	\[ wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \]
      +-commutative [=>] 91.5	\[ \color{blue}{\frac{\frac{x}{e^{wj}} - wj}{wj + 1} + wj} \]
      div-sub [=>] 91.5	\[ \color{blue}{\left(\frac{\frac{x}{e^{wj}}}{wj + 1} - \frac{wj}{wj + 1}\right)} + wj \]
      associate-+l- [=>] 91.5	\[ \color{blue}{\frac{\frac{x}{e^{wj}}}{wj + 1} - \left(\frac{wj}{wj + 1} - wj\right)} \]

   if -5.4999999999999996e-9 < wj

   1. Initial program 79.0%

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

   2. Simplified 80.0%

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

      [Start] 79.0	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 79.0	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-mul-1 [=>] 79.0	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      *-commutative [=>] 79.0	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
      *-commutative [<=] 79.0	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      neg-mul-1 [<=] 79.0	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-sub0 [=>] 79.0	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 79.0	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      associate--r- [=>] 79.0	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      +-commutative [=>] 79.0	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      sub0-neg [=>] 79.0	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [<=] 79.0	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   3. Taylor expanded in wj around 0 78.9%

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

   4. Simplified 78.9%

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

      [Start] 78.9	\[ wj + \frac{\left(-1 \cdot \left(wj \cdot x\right) + x\right) - wj}{wj + 1} \]
      +-commutative [=>] 78.9	\[ wj + \frac{\color{blue}{\left(x + -1 \cdot \left(wj \cdot x\right)\right)} - wj}{wj + 1} \]
      mul-1-neg [=>] 78.9	\[ wj + \frac{\left(x + \color{blue}{\left(-wj \cdot x\right)}\right) - wj}{wj + 1} \]
      unsub-neg [=>] 78.9	\[ wj + \frac{\color{blue}{\left(x - wj \cdot x\right)} - wj}{wj + 1} \]
      *-commutative [<=] 78.9	\[ wj + \frac{\left(x - \color{blue}{x \cdot wj}\right) - wj}{wj + 1} \]

   5. Taylor expanded in wj around 0 97.6%

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

   6. Taylor expanded in x around 0 97.7%

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

   7. Simplified 97.7%

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

      [Start] 97.7	\[ {wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right) \]
      unpow2 [=>] 97.7	\[ \color{blue}{wj \cdot wj} + \left(-2 \cdot \left(wj \cdot x\right) + x\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -5.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{wj + 1} - \left(\frac{wj}{wj + 1} - wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj + \left(x + -2 \cdot \left(x \cdot wj\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.1%
Cost	14272

\[\frac{x}{\frac{wj + 1}{1 - wj}} + \left(\left({wj}^{4} - {wj}^{5}\right) - wj \cdot \left(wj \cdot wj - wj\right)\right) \]

Alternative 2
Accuracy	97.1%
Cost	14016

\[\frac{x}{\frac{wj + 1}{1 - wj}} + \left({wj}^{4} + \left(wj \cdot wj - {wj}^{3}\right)\right) \]

Alternative 3
Accuracy	96.8%
Cost	7424

\[\frac{x}{\frac{wj + 1}{1 - wj}} + \left(wj \cdot wj - {wj}^{3}\right) \]

Alternative 4
Accuracy	97.5%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;wj \leq -5.4 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj + \left(x + -2 \cdot \left(x \cdot wj\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	96.3%
Cost	704

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

Alternative 6
Accuracy	87.2%
Cost	576

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

Alternative 7
Accuracy	85.5%
Cost	448

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

Alternative 8
Accuracy	4.3%
Cost	64

\[wj \]

Alternative 9
Accuracy	85.0%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023135 
(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))))))