Jmat.Real.lambertw, newton loop step

Average Accuracy: 78.3% → 98.0%

Time: 15.1s

Precision: binary64

Cost: 13636

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-63}:\\ \;\;\;\;\frac{x \cdot \left(1 - wj\right)}{wj + 1} + \left({wj}^{4} + \left(wj \cdot wj\right) \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (if (<= x 7e-63)
   (+
    (/ (* x (- 1.0 wj)) (+ wj 1.0))
    (+ (pow wj 4.0) (* (* wj wj) (- 1.0 wj))))
   (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 tmp;
	if (x <= 7e-63) {
		tmp = ((x * (1.0 - wj)) / (wj + 1.0)) + (pow(wj, 4.0) + ((wj * wj) * (1.0 - wj)));
	} 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)
	tmp = 0.0
	if (x <= 7e-63)
		tmp = Float64(Float64(Float64(x * Float64(1.0 - wj)) / Float64(wj + 1.0)) + Float64((wj ^ 4.0) + Float64(Float64(wj * wj) * Float64(1.0 - wj))));
	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_] := If[LessEqual[x, 7e-63], N[(N[(N[(x * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 4.0], $MachinePrecision] + N[(N[(wj * wj), $MachinePrecision] * N[(1.0 - wj), $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	78.3%
Target	79.1%
Herbie	98.0%

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

Derivation

1. Split input into 2 regimes

2. if x < 7.00000000000000006e-63

   1. Initial program 69.8%

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

   2. Simplified 70.7%

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

      [Start] 69.8	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 69.8	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-mul-1 [=>] 69.8	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      *-commutative [=>] 69.8	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
      *-commutative [<=] 69.8	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      neg-mul-1 [<=] 69.8	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-sub0 [=>] 69.8	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 69.8	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      associate--r- [=>] 69.8	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      +-commutative [=>] 69.8	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      sub0-neg [=>] 69.8	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [<=] 69.8	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   3. Applied egg-rr 84.6%

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

   4. Taylor expanded in wj around 0 98.3%

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

   5. Applied egg-rr 98.3%

      \[\leadsto \frac{\frac{x}{e^{wj}}}{wj + 1} - \left(-1 \cdot {wj}^{4} + \color{blue}{\left(wj \cdot wj\right) \cdot \left(-1 + wj\right)}\right) \]

   6. Taylor expanded in wj around 0 97.5%

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

   7. Simplified 97.5%

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

      [Start] 97.5	\[ \frac{-1 \cdot \left(wj \cdot x\right) + x}{wj + 1} - \left(-1 \cdot {wj}^{4} + \left(wj \cdot wj\right) \cdot \left(-1 + wj\right)\right) \]
      associate-*r* [=>] 97.5	\[ \frac{\color{blue}{\left(-1 \cdot wj\right) \cdot x} + x}{wj + 1} - \left(-1 \cdot {wj}^{4} + \left(wj \cdot wj\right) \cdot \left(-1 + wj\right)\right) \]
      neg-mul-1 [<=] 97.5	\[ \frac{\color{blue}{\left(-wj\right)} \cdot x + x}{wj + 1} - \left(-1 \cdot {wj}^{4} + \left(wj \cdot wj\right) \cdot \left(-1 + wj\right)\right) \]
      distribute-lft1-in [=>] 97.5	\[ \frac{\color{blue}{\left(\left(-wj\right) + 1\right) \cdot x}}{wj + 1} - \left(-1 \cdot {wj}^{4} + \left(wj \cdot wj\right) \cdot \left(-1 + wj\right)\right) \]
      +-commutative [<=] 97.5	\[ \frac{\color{blue}{\left(1 + \left(-wj\right)\right)} \cdot x}{wj + 1} - \left(-1 \cdot {wj}^{4} + \left(wj \cdot wj\right) \cdot \left(-1 + wj\right)\right) \]
      sub-neg [<=] 97.5	\[ \frac{\color{blue}{\left(1 - wj\right)} \cdot x}{wj + 1} - \left(-1 \cdot {wj}^{4} + \left(wj \cdot wj\right) \cdot \left(-1 + wj\right)\right) \]

   if 7.00000000000000006e-63 < x

   1. Initial program 98.2%

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

   2. Simplified 99.0%

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

      [Start] 98.2	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 98.2	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-mul-1 [=>] 98.2	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      *-commutative [=>] 98.2	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
      *-commutative [<=] 98.2	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      neg-mul-1 [<=] 98.2	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-sub0 [=>] 98.2	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 98.2	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      associate--r- [=>] 98.2	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      +-commutative [=>] 98.2	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      sub0-neg [=>] 98.2	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [<=] 98.2	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   3. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-63}:\\ \;\;\;\;\frac{x \cdot \left(1 - wj\right)}{wj + 1} + \left({wj}^{4} + \left(wj \cdot wj\right) \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.5%
Cost	26624

\[\frac{\frac{x}{e^{wj}}}{wj + 1} + \left({wj}^{4} + \left({wj}^{2} - {wj}^{3}\right)\right) \]

Alternative 2
Accuracy	98.5%
Cost	13952

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

Alternative 3
Accuracy	98.0%
Cost	7812

\[\begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-63}:\\ \;\;\;\;\frac{x \cdot \left(1 - wj\right)}{wj + 1} + \left({wj}^{4} + \left(wj \cdot wj\right) \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]

Alternative 4
Accuracy	97.7%
Cost	7428

\[\begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-65}:\\ \;\;\;\;\left(wj \cdot wj + \left(x + -2 \cdot \left(x \cdot wj\right)\right)\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]

Alternative 5
Accuracy	97.7%
Cost	7300

\[\begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-63}:\\ \;\;\;\;x + \left({wj}^{4} + \left(wj \cdot wj\right) \cdot \left(1 - wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]

Alternative 6
Accuracy	97.3%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-63}:\\ \;\;\;\;wj \cdot wj + \left(x + -2 \cdot \left(x \cdot wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]

Alternative 7
Accuracy	97.9%
Cost	7104

\[\frac{\frac{x}{e^{wj}}}{wj + 1} + wj \cdot wj \]

Alternative 8
Accuracy	87.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-275} \lor \neg \left(x \leq 1.55 \cdot 10^{-290}\right):\\ \;\;\;\;x + \left(wj - \frac{wj}{wj + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 9
Accuracy	85.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-274} \lor \neg \left(x \leq 2.8 \cdot 10^{-291}\right):\\ \;\;\;\;x \cdot \left(1 + wj \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 10
Accuracy	85.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.62 \cdot 10^{-273}:\\ \;\;\;\;x \cdot \left(1 + wj \cdot -2\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-292}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - wj \cdot -2}\\ \end{array} \]

Alternative 11
Accuracy	96.9%
Cost	704

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

Alternative 12
Accuracy	85.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-273}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-292}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Accuracy	4.3%
Cost	64

\[wj \]

Alternative 14
Accuracy	85.1%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023125 
(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))))))