?

Average Error: 21.9% → 0.06%
Time: 15.7s
Precision: binary64
Cost: 13376

?

\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
↓
\[\frac{\mathsf{fma}\left(wj, wj, \frac{x}{e^{wj}}\right)}{wj + 1} \]
(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))
↓
(FPCore (wj x) :precision binary64 (/ (fma wj wj (/ x (exp 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) {
	return fma(wj, wj, (x / exp(wj))) / (wj + 1.0);
}

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)
	return Float64(fma(wj, wj, Float64(x / exp(wj))) / Float64(wj + 1.0))
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_] := N[(N[(wj * wj + N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]

wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{\mathsf{fma}\left(wj, wj, \frac{x}{e^{wj}}\right)}{wj + 1}

Error?

Target

Original21.9%
Target20.98%
Herbie0.06%
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

Derivation?

  1. Initial program 21.9

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

  2. Simplified20.98

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

    [Start]21.9

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

    sub-neg [=>]21.9

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

    neg-mul-1 [=>]21.9

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

    *-commutative [=>]21.9

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

    *-commutative [<=]21.9

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

    neg-mul-1 [<=]21.9

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

    neg-sub0 [=>]21.9

    \[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]

    div-sub [=>]21.9

    \[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]

    associate--r- [=>]21.9

    \[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]

    +-commutative [=>]21.9

    \[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]

    sub0-neg [=>]21.9

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

    sub-neg [<=]21.9

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

  3. Applied egg-rr44.81

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

  4. Simplified23.89

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

    [Start]44.81

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

    associate--r- [=>]23.89

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

  5. Applied egg-rr22.1

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

  6. Applied egg-rr20.97

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

  7. Simplified0.06

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

    [Start]20.97

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

    sub-neg [=>]20.97

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

    associate-+l+ [=>]11.12

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

    +-commutative [=>]11.12

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

    distribute-rgt1-in [=>]11.14

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

    neg-mul-1 [=>]11.14

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

    distribute-rgt-in [<=]11.12

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

    associate-+l+ [=>]0.06

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

    metadata-eval [=>]0.06

    \[ \frac{\frac{x}{e^{wj}} + wj \cdot \left(wj + \color{blue}{0}\right)}{wj + 1} \]

    +-commutative [<=]0.06

    \[ \frac{\frac{x}{e^{wj}} + wj \cdot \color{blue}{\left(0 + wj\right)}}{wj + 1} \]

    +-lft-identity [=>]0.06

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

  8. Taylor expanded in x around 0 0.06

    \[\leadsto \frac{\color{blue}{{wj}^{2} + \frac{x}{e^{wj}}}}{wj + 1} \]

  9. Simplified0.06

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

    [Start]0.06

    \[ \frac{{wj}^{2} + \frac{x}{e^{wj}}}{wj + 1} \]

    unpow2 [=>]0.06

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

    fma-udef [<=]0.06

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

  10. Final simplification0.06

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

Alternatives

Alternative 1
Error0.06%
Cost7104
\[\frac{\frac{x}{e^{wj}} + wj \cdot wj}{wj + 1} \]
Alternative 2
Error1.19%
Cost1344
\[\frac{wj \cdot wj + \left(\left(x - wj \cdot x\right) + \left(wj \cdot wj\right) \cdot \left(x \cdot 0.5\right)\right)}{wj + 1} \]
Alternative 3
Error14.68%
Cost1096
\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-299}:\\ \;\;\;\;\frac{x - wj \cdot x}{wj + 1}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-298}:\\ \;\;\;\;\frac{wj}{\frac{wj + 1}{wj}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{wj + 1} - \frac{wj}{wj + 1}\right)\\ \end{array} \]
Alternative 4
Error14.79%
Cost841
\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-290} \lor \neg \left(x \leq 5.4 \cdot 10^{-296}\right):\\ \;\;\;\;\frac{x - wj \cdot x}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{wj}{\frac{wj + 1}{wj}}\\ \end{array} \]
Alternative 5
Error1.4%
Cost832
\[\frac{wj \cdot wj + \left(x - wj \cdot x\right)}{wj + 1} \]
Alternative 6
Error14.9%
Cost713
\[\begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-290} \lor \neg \left(x \leq 3 \cdot 10^{-298}\right):\\ \;\;\;\;x \cdot \left(1 + wj \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]
Alternative 7
Error14.81%
Cost712
\[\begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-298}:\\ \;\;\;\;x + \left(wj \cdot x\right) \cdot -2\\ \mathbf{elif}\;x \leq 10^{-296}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + wj \cdot -2\right)\\ \end{array} \]
Alternative 8
Error14.75%
Cost712
\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-294}:\\ \;\;\;\;x + \left(wj \cdot x\right) \cdot -2\\ \mathbf{elif}\;x \leq 10^{-295}:\\ \;\;\;\;\frac{wj}{\frac{wj + 1}{wj}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + wj \cdot -2\right)\\ \end{array} \]
Alternative 9
Error15.35%
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-294}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-297}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 10
Error95.6%
Cost64
\[wj \]
Alternative 11
Error15.34%
Cost64
\[x \]

Error

Reproduce?

herbie shell --seed 2023115 
(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))))))