Jmat.Real.lambertw, newton loop step

Average Error: 13.5 → 0.7

Time: 11.6s

Precision: binary64

Cost: 33604

Math

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

↓

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t_0}{e^{wj} + t_0} \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\left(wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right) - wj \cdot \left(wj \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj - \frac{x}{e^{wj}}, \frac{-1}{wj + 1}, wj\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 5e-18)
     (- (+ (* wj wj) (+ x (* -2.0 (* wj x)))) (* wj (* wj wj)))
     (fma (- wj (/ x (exp 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 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 5e-18) {
		tmp = ((wj * wj) + (x + (-2.0 * (wj * x)))) - (wj * (wj * wj));
	} else {
		tmp = fma((wj - (x / exp(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(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 5e-18)
		tmp = Float64(Float64(Float64(wj * wj) + Float64(x + Float64(-2.0 * Float64(wj * x)))) - Float64(wj * Float64(wj * wj)));
	else
		tmp = fma(Float64(wj - Float64(x / exp(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[(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], 5e-18], N[(N[(N[(wj * wj), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(wj * N[(wj * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]]]

Target

Original	13.5
Target	12.9
Herbie	0.7

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

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))))) < 5.00000000000000036e-18

   1. Initial program 17.9

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

   2. Taylor expanded in wj around 0 0.7

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

   3. Taylor expanded in x around 0 0.9

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

   4. Simplified 0.9

      \[\leadsto -1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(\color{blue}{wj \cdot wj} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right) \]
      Proof
      (*.f64 wj wj): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unpow2_binary64 (pow.f64 wj 2)): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in x around 0 0.8

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

   6. Applied egg-rr 0.8

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

   if 5.00000000000000036e-18 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 2.5

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

   2. Simplified 0.5

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
      Proof
      (-.f64 wj (/.f64 (-.f64 wj (/.f64 x (exp.f64 wj))) (+.f64 wj 1))): 0 points increase in error, 0 points decrease in error
      (-.f64 wj (Rewrite=> div-sub_binary64 (-.f64 (/.f64 wj (+.f64 wj 1)) (/.f64 (/.f64 x (exp.f64 wj)) (+.f64 wj 1))))): 0 points increase in error, 0 points decrease in error
      (-.f64 wj (-.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (/.f64 wj (+.f64 wj 1)) 1)) (/.f64 (/.f64 x (exp.f64 wj)) (+.f64 wj 1)))): 0 points increase in error, 0 points decrease in error
      (-.f64 wj (-.f64 (*.f64 (/.f64 wj (+.f64 wj 1)) (Rewrite<= *-inverses_binary64 (/.f64 (exp.f64 wj) (exp.f64 wj)))) (/.f64 (/.f64 x (exp.f64 wj)) (+.f64 wj 1)))): 2 points increase in error, 0 points decrease in error
      (-.f64 wj (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 wj (exp.f64 wj)) (*.f64 (+.f64 wj 1) (exp.f64 wj)))) (/.f64 (/.f64 x (exp.f64 wj)) (+.f64 wj 1)))): 2 points increase in error, 0 points decrease in error
      (-.f64 wj (-.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) (/.f64 (/.f64 x (exp.f64 wj)) (+.f64 wj 1)))): 0 points increase in error, 0 points decrease in error
      (-.f64 wj (-.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (Rewrite=> associate-/l/_binary64 (/.f64 x (*.f64 (+.f64 wj 1) (exp.f64 wj)))))): 2 points increase in error, 2 points decrease in error
      (-.f64 wj (-.f64 (/.f64 (*.f64 wj (exp.f64 wj)) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))) (/.f64 x (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))))): 0 points increase in error, 2 points decrease in error
      (-.f64 wj (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))): 1 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.5

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\left(wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right) - wj \cdot \left(wj \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj - \frac{x}{e^{wj}}, \frac{-1}{wj + 1}, wj\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	7236

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.597802182809226 \cdot 10^{-9}:\\ \;\;\;\;\left(wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right) - wj \cdot \left(wj \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]

Alternative 2
Error	8.6
Cost	1088

\[\frac{x \cdot \left(1 - wj\right) + \left(wj \cdot \left(wj \cdot x\right)\right) \cdot 0.5}{wj + 1} \]

Alternative 3
Error	1.8
Cost	1088

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

Alternative 4
Error	8.6
Cost	960

\[\frac{x + x \cdot \left(wj \cdot \left(wj \cdot 0.5\right) - wj\right)}{wj + 1} \]

Alternative 5
Error	8.6
Cost	704

\[x + x \cdot \left(wj \cdot \left(-2 + wj \cdot 2.5\right)\right) \]

Alternative 6
Error	8.7
Cost	576

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

Alternative 7
Error	8.7
Cost	576

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

Alternative 8
Error	8.8
Cost	448

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

Alternative 9
Error	9.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022300 
(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))))))