Average Error: 13.8 → 0.3
Time: 14.1s
Precision: binary64
Cost: 33860
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t_0}{e^{wj} + t_0} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;wj \cdot \left(wj - wj \cdot wj\right) + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;wj + \left(wj + -1\right) \cdot \frac{\frac{x}{e^{wj}} - wj}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \]
(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-17)
     (+ (* wj (- wj (* wj wj))) (/ x (* (exp wj) (+ wj 1.0))))
     (+ wj (* (+ wj -1.0) (/ (- (/ x (exp wj)) wj) (fma wj wj -1.0)))))))
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-17) {
		tmp = (wj * (wj - (wj * wj))) + (x / (exp(wj) * (wj + 1.0)));
	} else {
		tmp = wj + ((wj + -1.0) * (((x / exp(wj)) - wj) / fma(wj, wj, -1.0)));
	}
	return tmp;
}
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-17)
		tmp = Float64(Float64(wj * Float64(wj - Float64(wj * wj))) + Float64(x / Float64(exp(wj) * Float64(wj + 1.0))));
	else
		tmp = Float64(wj + Float64(Float64(wj + -1.0) * Float64(Float64(Float64(x / exp(wj)) - wj) / fma(wj, wj, -1.0))));
	end
	return tmp
end
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-17], N[(N[(wj * N[(wj - N[(wj * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(wj + -1.0), $MachinePrecision] * N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]
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^{-17}:\\
\;\;\;\;wj \cdot \left(wj - wj \cdot wj\right) + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\

\mathbf{else}:\\
\;\;\;\;wj + \left(wj + -1\right) \cdot \frac{\frac{x}{e^{wj}} - wj}{\mathsf{fma}\left(wj, wj, -1\right)}\\


\end{array}

Error

Target

Original13.8
Target13.2
Herbie0.3
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

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))))) < 4.9999999999999999e-17

    1. Initial program 18.3

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
    2. Simplified18.3

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
      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)))): 5 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)))): 0 points increase in error, 3 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)))): 2 points increase in error, 1 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)))))): 3 points increase in error, 1 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, 1 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)))))): 0 points increase in error, 0 points decrease in error
    3. Applied egg-rr9.3

      \[\leadsto \color{blue}{\left(wj - \frac{wj}{wj + 1}\right) + \frac{x}{\left(wj + 1\right) \cdot e^{wj}}} \]
    4. Taylor expanded in wj around 0 0.2

      \[\leadsto \color{blue}{\left({wj}^{2} + -1 \cdot {wj}^{3}\right)} + \frac{x}{\left(wj + 1\right) \cdot e^{wj}} \]
    5. Simplified0.2

      \[\leadsto \color{blue}{\left(wj \cdot wj - {wj}^{3}\right)} + \frac{x}{\left(wj + 1\right) \cdot e^{wj}} \]
      Proof
      (-.f64 (*.f64 wj wj) (pow.f64 wj 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 wj 2)) (pow.f64 wj 3)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (pow.f64 wj 2) (neg.f64 (pow.f64 wj 3)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 wj 2) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 wj 3)))): 0 points increase in error, 0 points decrease in error
    6. Applied egg-rr0.2

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

    if 4.9999999999999999e-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. Simplified0.6

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
      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)))): 5 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)))): 0 points increase in error, 3 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)))): 2 points increase in error, 1 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)))))): 3 points increase in error, 1 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, 1 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)))))): 0 points increase in error, 0 points decrease in error
    3. Applied egg-rr0.6

      \[\leadsto wj - \color{blue}{\frac{wj - \frac{x}{e^{wj}}}{\mathsf{fma}\left(wj, wj, -1\right)} \cdot \left(wj + -1\right)} \]
  3. Recombined 2 regimes into one program.
  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;wj \cdot \left(wj - wj \cdot wj\right) + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;wj + \left(wj + -1\right) \cdot \frac{\frac{x}{e^{wj}} - wj}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error0.5
Cost7492
\[\begin{array}{l} \mathbf{if}\;wj \leq 5 \cdot 10^{-11}:\\ \;\;\;\;wj \cdot \left(wj - wj \cdot wj\right) + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;wj + \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{-1}{wj + 1}\\ \end{array} \]
Alternative 2
Error0.7
Cost7364
\[\begin{array}{l} \mathbf{if}\;wj \leq 5 \cdot 10^{-11}:\\ \;\;\;\;wj \cdot wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;wj + \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{-1}{wj + 1}\\ \end{array} \]
Alternative 3
Error1.0
Cost7236
\[\begin{array}{l} \mathbf{if}\;wj \leq 0.00078:\\ \;\;\;\;wj \cdot wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]
Alternative 4
Error0.7
Cost7236
\[\begin{array}{l} \mathbf{if}\;wj \leq 5 \cdot 10^{-11}:\\ \;\;\;\;wj \cdot wj + \frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]
Alternative 5
Error1.1
Cost1092
\[\begin{array}{l} \mathbf{if}\;wj \leq 0.00044:\\ \;\;\;\;wj \cdot \left(wj - wj \cdot wj\right) + \frac{x}{1 + \left(wj + wj\right)}\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]
Alternative 6
Error1.4
Cost836
\[\begin{array}{l} \mathbf{if}\;wj \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]
Alternative 7
Error1.4
Cost708
\[\begin{array}{l} \mathbf{if}\;wj \leq 0.00019:\\ \;\;\;\;x + wj \cdot \left(wj - wj \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]
Alternative 8
Error1.7
Cost580
\[\begin{array}{l} \mathbf{if}\;wj \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;x + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]
Alternative 9
Error9.9
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-262}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-254}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 10
Error2.6
Cost320
\[x + wj \cdot wj \]
Alternative 11
Error61.2
Cost64
\[wj \]
Alternative 12
Error9.6
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022328 
(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))))))