Jmat.Real.lambertw, newton loop step

Average Error: 14.2 → 0.5

Time: 18.8s

Precision: binary64

Cost: 33860

Math

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

↓

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t_0}{e^{wj} + t_0} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\left(wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{e^{wj}} - wj}{1 - wj \cdot wj}, 1 - wj, wj\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 2e-17)
     (- (+ (* wj wj) (+ x (* -2.0 (* wj x)))) (pow wj 3.0))
     (fma (/ (- (/ x (exp wj)) wj) (- 1.0 (* wj wj))) (- 1.0 wj) 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 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 2e-17) {
		tmp = ((wj * wj) + (x + (-2.0 * (wj * x)))) - pow(wj, 3.0);
	} else {
		tmp = fma((((x / exp(wj)) - wj) / (1.0 - (wj * wj))), (1.0 - wj), 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(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 2e-17)
		tmp = Float64(Float64(Float64(wj * wj) + Float64(x + Float64(-2.0 * Float64(wj * x)))) - (wj ^ 3.0));
	else
		tmp = fma(Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(1.0 - Float64(wj * wj))), Float64(1.0 - wj), 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[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-17], N[(N[(N[(wj * wj), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(1.0 - N[(wj * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - wj), $MachinePrecision] + wj), $MachinePrecision]]]

Target

Original	14.2
Target	13.5
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 2.00000000000000014e-17

   1. Initial program 18.6

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

   2. Simplified 18.6

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

      [Start] 18.6	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 18.6	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-mul-1 [=>] 18.6	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      *-commutative [=>] 18.6	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
      *-commutative [<=] 18.6	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      neg-mul-1 [<=] 18.6	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-sub0 [=>] 18.6	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 18.6	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      associate--r- [=>] 18.6	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      +-commutative [=>] 18.6	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      sub0-neg [=>] 18.6	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [<=] 18.6	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   3. Taylor expanded in wj around 0 0.4

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

   4. Taylor expanded in x around 0 0.6

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

   5. Simplified 0.6

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

      [Start] 0.6	\[ -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left({wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]
      unpow2 [=>] 0.6	\[ -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(\color{blue}{wj \cdot wj} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]

   6. Taylor expanded in x around 0 0.5

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

   if 2.00000000000000014e-17 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 2.9

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

   2. Simplified 0.6

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

      [Start] 2.9	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
      sub-neg [=>] 2.9	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-mul-1 [=>] 2.9	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      *-commutative [=>] 2.9	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
      *-commutative [<=] 2.9	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
      neg-mul-1 [<=] 2.9	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      neg-sub0 [=>] 2.9	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      div-sub [=>] 2.9	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      associate--r- [=>] 2.9	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
      +-commutative [=>] 2.9	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
      sub0-neg [=>] 2.9	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
      sub-neg [<=] 2.9	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   3. Applied egg-rr 0.6

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

   4. Simplified 0.6

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

      [Start] 0.6	\[ wj + \frac{\frac{x}{e^{wj}} - wj}{-\mathsf{fma}\left(wj, wj, -1\right)} \cdot \left(-\left(wj + -1\right)\right) \]
      distribute-neg-in [=>] 0.6	\[ wj + \frac{\frac{x}{e^{wj}} - wj}{-\mathsf{fma}\left(wj, wj, -1\right)} \cdot \color{blue}{\left(\left(-wj\right) + \left(--1\right)\right)} \]
      metadata-eval [=>] 0.6	\[ wj + \frac{\frac{x}{e^{wj}} - wj}{-\mathsf{fma}\left(wj, wj, -1\right)} \cdot \left(\left(-wj\right) + \color{blue}{1}\right) \]
      +-commutative [<=] 0.6	\[ wj + \frac{\frac{x}{e^{wj}} - wj}{-\mathsf{fma}\left(wj, wj, -1\right)} \cdot \color{blue}{\left(1 + \left(-wj\right)\right)} \]
      sub-neg [<=] 0.6	\[ wj + \frac{\frac{x}{e^{wj}} - wj}{-\mathsf{fma}\left(wj, wj, -1\right)} \cdot \color{blue}{\left(1 - wj\right)} \]
      associate-*l/ [=>] 0.6	\[ wj + \color{blue}{\frac{\left(\frac{x}{e^{wj}} - wj\right) \cdot \left(1 - wj\right)}{-\mathsf{fma}\left(wj, wj, -1\right)}} \]
      fma-udef [=>] 0.6	\[ wj + \frac{\left(\frac{x}{e^{wj}} - wj\right) \cdot \left(1 - wj\right)}{-\color{blue}{\left(wj \cdot wj + -1\right)}} \]
      distribute-neg-in [=>] 0.6	\[ wj + \frac{\left(\frac{x}{e^{wj}} - wj\right) \cdot \left(1 - wj\right)}{\color{blue}{\left(-wj \cdot wj\right) + \left(--1\right)}} \]
      metadata-eval [=>] 0.6	\[ wj + \frac{\left(\frac{x}{e^{wj}} - wj\right) \cdot \left(1 - wj\right)}{\left(-wj \cdot wj\right) + \color{blue}{1}} \]
      +-commutative [<=] 0.6	\[ wj + \frac{\left(\frac{x}{e^{wj}} - wj\right) \cdot \left(1 - wj\right)}{\color{blue}{1 + \left(-wj \cdot wj\right)}} \]
      sub-neg [<=] 0.6	\[ wj + \frac{\left(\frac{x}{e^{wj}} - wj\right) \cdot \left(1 - wj\right)}{\color{blue}{1 - wj \cdot wj}} \]
      associate-/l* [=>] 0.6	\[ wj + \color{blue}{\frac{\frac{x}{e^{wj}} - wj}{\frac{1 - wj \cdot wj}{1 - wj}}} \]

   5. Applied egg-rr 0.6

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\left(wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{e^{wj}} - wj}{1 - wj \cdot wj}, 1 - wj, wj\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.9
Cost	8576

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

Alternative 2
Error	1.8
Cost	7296

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

Alternative 3
Error	2.1
Cost	7040

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

Alternative 4
Error	8.4
Cost	1737

\[\begin{array}{l} \mathbf{if}\;wj \leq -3.9 \cdot 10^{-28} \lor \neg \left(wj \leq -3.2 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{x - wj \cdot \left(x + \left(wj \cdot x\right) \cdot -0.5\right)}{wj + 1} + \left(wj - \frac{wj}{wj + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 5
Error	8.6
Cost	1476

\[\begin{array}{l} \mathbf{if}\;wj \leq -2.7 \cdot 10^{-28}:\\ \;\;\;\;wj + \frac{\left(\left(wj \cdot wj\right) \cdot \left(x \cdot 0.5\right) - \left(wj \cdot x - x\right)\right) - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq -1 \cdot 10^{-46}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq 7.8 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \left(1 + wj \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(wj \cdot wj\right) \cdot \left(-2 - wj\right)}{\left(wj + 1\right) \cdot \left(-2 - wj\right)}\\ \end{array} \]

Alternative 6
Error	8.7
Cost	1356

\[\begin{array}{l} \mathbf{if}\;wj \leq -1.65 \cdot 10^{-29}:\\ \;\;\;\;wj + \frac{\left(x - wj \cdot x\right) - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq -3.2 \cdot 10^{-46}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq 8.5 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \left(1 + wj \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(wj \cdot wj\right) \cdot \left(-2 - wj\right)}{\left(wj + 1\right) \cdot \left(-2 - wj\right)}\\ \end{array} \]

Alternative 7
Error	8.6
Cost	1228

\[\begin{array}{l} t_0 := wj + \frac{\left(x - wj \cdot x\right) - wj}{wj + 1}\\ \mathbf{if}\;wj \leq -1.12 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;wj \leq -3.2 \cdot 10^{-46}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq 6.2 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	9.1
Cost	844

\[\begin{array}{l} \mathbf{if}\;wj \leq -7.2 \cdot 10^{-27}:\\ \;\;\;\;x \cdot \left(1 - wj \cdot 2\right)\\ \mathbf{elif}\;wj \leq -1.2 \cdot 10^{-49}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq 9.2 \cdot 10^{-10}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 9
Error	9.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;wj \leq -5.4 \cdot 10^{-29} \lor \neg \left(wj \leq -3.2 \cdot 10^{-46}\right):\\ \;\;\;\;x \cdot \left(1 - wj \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 10
Error	10.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;wj \leq -3.3 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \left(1 - wj \cdot 2\right)\\ \mathbf{elif}\;wj \leq -6.2 \cdot 10^{-48}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \end{array} \]

Alternative 11
Error	61.2
Cost	64

\[wj \]

Alternative 12
Error	10.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023016 
(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))))))