Jmat.Real.lambertw, newton loop step

Average Accuracy: 79.1% → 99.0%

Time: 17.8s

Precision: binary64

Cost: 15684

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot 4 + x \cdot -1.5\\ \mathbf{if}\;wj \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;{wj}^{3} \cdot \left(x \cdot -0.6666666666666666 + \left(x \cdot 3 + \left(-1 + -2 \cdot t_0\right)\right)\right) + \left(\left(1 + t_0\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x 4.0) (* x -1.5))))
   (if (<= wj 4.8e-6)
     (+
      (*
       (pow wj 3.0)
       (+ (* x -0.6666666666666666) (+ (* x 3.0) (+ -1.0 (* -2.0 t_0)))))
      (+ (* (+ 1.0 t_0) (pow wj 2.0)) (+ x (* -2.0 (* wj x)))))
     (fma (- (/ x (exp wj)) wj) (/ 1.0 (+ 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 t_0 = (x * 4.0) + (x * -1.5);
	double tmp;
	if (wj <= 4.8e-6) {
		tmp = (pow(wj, 3.0) * ((x * -0.6666666666666666) + ((x * 3.0) + (-1.0 + (-2.0 * t_0))))) + (((1.0 + t_0) * pow(wj, 2.0)) + (x + (-2.0 * (wj * x))));
	} else {
		tmp = fma(((x / exp(wj)) - wj), (1.0 / (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)
	t_0 = Float64(Float64(x * 4.0) + Float64(x * -1.5))
	tmp = 0.0
	if (wj <= 4.8e-6)
		tmp = Float64(Float64((wj ^ 3.0) * Float64(Float64(x * -0.6666666666666666) + Float64(Float64(x * 3.0) + Float64(-1.0 + Float64(-2.0 * t_0))))) + Float64(Float64(Float64(1.0 + t_0) * (wj ^ 2.0)) + Float64(x + Float64(-2.0 * Float64(wj * x)))));
	else
		tmp = fma(Float64(Float64(x / exp(wj)) - wj), Float64(1.0 / Float64(wj + 1.0)), wj);
	end
	return 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]}, If[LessEqual[wj, 4.8e-6], N[(N[(N[Power[wj, 3.0], $MachinePrecision] * 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]), $MachinePrecision] + N[(N[(N[(1.0 + t$95$0), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]]]

Target

Original	79.1%
Target	80.1%
Herbie	99.0%

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

Derivation

1. Split input into 2 regimes

2. if wj < 4.7999999999999998e-6

   1. Initial program 79.7%

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

   2. Simplified 79.7%

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

      [Start] 79.7	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 79.7	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-mul-1 [=>] 79.7	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      *-commutative [=>] 79.7	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
      *-commutative [<=] 79.7	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      neg-mul-1 [<=] 79.7	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-sub0 [=>] 79.7	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 79.7	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      associate--r- [=>] 79.7	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      +-commutative [=>] 79.7	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      sub0-neg [=>] 79.7	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [<=] 79.7	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   3. Taylor expanded in wj around 0 99.0%

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

   if 4.7999999999999998e-6 < wj

   1. Initial program 52.4%

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

   2. Simplified 97.2%

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

      [Start] 52.4	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 52.4	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-mul-1 [=>] 52.4	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      *-commutative [=>] 52.4	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
      *-commutative [<=] 52.4	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      neg-mul-1 [<=] 52.4	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-sub0 [=>] 52.4	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 52.4	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      associate--r- [=>] 52.4	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      +-commutative [=>] 52.4	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      sub0-neg [=>] 52.4	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [<=] 52.4	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   3. Applied egg-rr 97.3%

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

      [Start] 97.2	\[ wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \]
      +-commutative [=>] 97.2	\[ \color{blue}{\frac{\frac{x}{e^{wj}} - wj}{wj + 1} + wj} \]
      div-inv [=>] 97.2	\[ \color{blue}{\left(\frac{x}{e^{wj}} - wj\right) \cdot \frac{1}{wj + 1}} + wj \]
      fma-def [=>] 97.3	\[ \color{blue}{\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;{wj}^{3} \cdot \left(x \cdot -0.6666666666666666 + \left(x \cdot 3 + \left(-1 + -2 \cdot \left(x \cdot 4 + x \cdot -1.5\right)\right)\right)\right) + \left(\left(1 + \left(x \cdot 4 + x \cdot -1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.0%
Cost	7812

\[\begin{array}{l} \mathbf{if}\;wj \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}} + \left({wj}^{4} + \left(wj \cdot wj\right) \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(wj + -1\right) \cdot \frac{wj - \frac{x}{e^{wj}}}{1 - wj \cdot wj}\\ \end{array} \]

Alternative 2
Accuracy	98.9%
Cost	7620

\[\begin{array}{l} \mathbf{if}\;wj \leq 3.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}} + \left(wj \cdot wj - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(wj + -1\right) \cdot \frac{wj - \frac{x}{e^{wj}}}{1 - wj \cdot wj}\\ \end{array} \]

Alternative 3
Accuracy	98.9%
Cost	7556

\[\begin{array}{l} \mathbf{if}\;wj \leq 4.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}} + \left(wj \cdot wj - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \]

Alternative 4
Accuracy	98.6%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;wj \leq 7.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}} + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \]

Alternative 5
Accuracy	97.6%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;wj \leq 8.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{\frac{wj + 1}{1 - wj}} + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\frac{x}{e^{wj}}}{-1 - wj}\\ \end{array} \]

Alternative 6
Accuracy	96.9%
Cost	704

\[\frac{x}{1 + wj \cdot 2} + wj \cdot wj \]

Alternative 7
Accuracy	96.4%
Cost	320

\[x + wj \cdot wj \]

Alternative 8
Accuracy	4.4%
Cost	64

\[wj \]

Alternative 9
Accuracy	86.1%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023130 
(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))))))