Jmat.Real.lambertw, newton loop step

Average Accuracy: 78.4% → 99.9%

Time: 17.9s

Precision: binary64

Cost: 7104

Math

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

↓

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

FPCore

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

↓

(FPCore (wj x) :precision binary64 (/ (+ (/ x (exp wj)) (* wj wj)) (+ wj 1.0)))

C

double code(double wj, double x) {
	return wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))));
}

↓

double code(double wj, double x) {
	return ((x / exp(wj)) + (wj * wj)) / (wj + 1.0);
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))))
end function

↓

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = ((x / exp(wj)) + (wj * wj)) / (wj + 1.0d0)
end function

Java

public static double code(double wj, double x) {
	return wj - (((wj * Math.exp(wj)) - x) / (Math.exp(wj) + (wj * Math.exp(wj))));
}

↓

public static double code(double wj, double x) {
	return ((x / Math.exp(wj)) + (wj * wj)) / (wj + 1.0);
}

Python

def code(wj, x):
	return wj - (((wj * math.exp(wj)) - x) / (math.exp(wj) + (wj * math.exp(wj))))

↓

def code(wj, x):
	return ((x / math.exp(wj)) + (wj * wj)) / (wj + 1.0)

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)
	return Float64(Float64(Float64(x / exp(wj)) + Float64(wj * wj)) / Float64(wj + 1.0))
end

MATLAB

function tmp = code(wj, x)
	tmp = wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))));
end

↓

function tmp = code(wj, x)
	tmp = ((x / exp(wj)) + (wj * wj)) / (wj + 1.0);
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_] := N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] + N[(wj * wj), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]

Target

Original	78.4%
Target	79.5%
Herbie	99.9%

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

Derivation

1. Initial program 78.4%

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

2. Simplified 79.5%

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

   [Start] 78.4	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   sub-neg [=>] 78.4	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
   neg-mul-1 [=>] 78.4	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
   *-commutative [=>] 78.4	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
   *-commutative [<=] 78.4	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
   neg-mul-1 [<=] 78.4	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
   neg-sub0 [=>] 78.4	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
   div-sub [=>] 78.4	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
   associate--r- [=>] 78.4	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
   +-commutative [=>] 78.4	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
   sub0-neg [=>] 78.4	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
   sub-neg [<=] 78.4	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

3. Applied egg-rr 55.7%

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

   [Start] 79.5	\[ wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \]
   expm1-log1p-u [=>] 79.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(wj\right)\right)} + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \]
   log1p-def [<=] 75.8	\[ \mathsf{expm1}\left(\color{blue}{\log \left(1 + wj\right)}\right) + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \]
   +-commutative [<=] 75.8	\[ \mathsf{expm1}\left(\log \color{blue}{\left(wj + 1\right)}\right) + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \]
   expm1-udef [=>] 75.8	\[ \color{blue}{\left(e^{\log \left(wj + 1\right)} - 1\right)} + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \]
   add-exp-log [<=] 76.4	\[ \left(\color{blue}{\left(wj + 1\right)} - 1\right) + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \]
   associate-+l- [=>] 55.7	\[ \color{blue}{\left(wj + 1\right) - \left(1 - \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\right)} \]

4. Simplified 76.4%

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

   [Start] 55.7	\[ \left(wj + 1\right) - \left(1 - \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\right) \]
   associate--r- [=>] 76.4	\[ \color{blue}{\left(\left(wj + 1\right) - 1\right) + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]

5. Applied egg-rr 78.2%

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

   [Start] 76.4	\[ \left(\left(wj + 1\right) - 1\right) + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \]
   associate--l+ [=>] 79.5	\[ \color{blue}{\left(wj + \left(1 - 1\right)\right)} + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \]
   metadata-eval [=>] 79.5	\[ \left(wj + \color{blue}{0}\right) + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \]
   metadata-eval [<=] 79.5	\[ \left(wj + \color{blue}{\log 1}\right) + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \]
   flip-+ [=>] 78.2	\[ \color{blue}{\frac{wj \cdot wj - \log 1 \cdot \log 1}{wj - \log 1}} + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \]

6. Applied egg-rr 79.5%

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

   [Start] 78.2	\[ \frac{wj \cdot wj}{wj} + \frac{\frac{x}{e^{wj}} - wj}{wj + 1} \]
   +-commutative [=>] 78.2	\[ \color{blue}{\frac{\frac{x}{e^{wj}} - wj}{wj + 1} + \frac{wj \cdot wj}{wj}} \]
   associate-/l* [=>] 79.5	\[ \frac{\frac{x}{e^{wj}} - wj}{wj + 1} + \color{blue}{\frac{wj}{\frac{wj}{wj}}} \]
   *-inverses [=>] 79.5	\[ \frac{\frac{x}{e^{wj}} - wj}{wj + 1} + \frac{wj}{\color{blue}{1}} \]
   frac-add [=>] 79.5	\[ \color{blue}{\frac{\left(\frac{x}{e^{wj}} - wj\right) \cdot 1 + \left(wj + 1\right) \cdot wj}{\left(wj + 1\right) \cdot 1}} \]
   *-commutative [<=] 79.5	\[ \frac{\color{blue}{1 \cdot \left(\frac{x}{e^{wj}} - wj\right)} + \left(wj + 1\right) \cdot wj}{\left(wj + 1\right) \cdot 1} \]
   *-un-lft-identity [<=] 79.5	\[ \frac{\color{blue}{\left(\frac{x}{e^{wj}} - wj\right)} + \left(wj + 1\right) \cdot wj}{\left(wj + 1\right) \cdot 1} \]
   *-commutative [<=] 79.5	\[ \frac{\left(\frac{x}{e^{wj}} - wj\right) + \color{blue}{wj \cdot \left(wj + 1\right)}}{\left(wj + 1\right) \cdot 1} \]
   distribute-rgt-in [=>] 79.5	\[ \frac{\left(\frac{x}{e^{wj}} - wj\right) + \color{blue}{\left(wj \cdot wj + 1 \cdot wj\right)}}{\left(wj + 1\right) \cdot 1} \]
   *-un-lft-identity [<=] 79.5	\[ \frac{\left(\frac{x}{e^{wj}} - wj\right) + \left(wj \cdot wj + \color{blue}{wj}\right)}{\left(wj + 1\right) \cdot 1} \]
   +-commutative [<=] 79.5	\[ \frac{\left(\frac{x}{e^{wj}} - wj\right) + \color{blue}{\left(wj + wj \cdot wj\right)}}{\left(wj + 1\right) \cdot 1} \]
   *-commutative [<=] 79.5	\[ \frac{\left(\frac{x}{e^{wj}} - wj\right) + \left(wj + wj \cdot wj\right)}{\color{blue}{1 \cdot \left(wj + 1\right)}} \]
   *-un-lft-identity [<=] 79.5	\[ \frac{\left(\frac{x}{e^{wj}} - wj\right) + \left(wj + wj \cdot wj\right)}{\color{blue}{wj + 1}} \]

7. Simplified 99.9%

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

   [Start] 79.5	\[ \frac{\left(\frac{x}{e^{wj}} - wj\right) + \left(wj + wj \cdot wj\right)}{wj + 1} \]
   sub-neg [=>] 79.5	\[ \frac{\color{blue}{\left(\frac{x}{e^{wj}} + \left(-wj\right)\right)} + \left(wj + wj \cdot wj\right)}{wj + 1} \]
   associate-+l+ [=>] 89.4	\[ \frac{\color{blue}{\frac{x}{e^{wj}} + \left(\left(-wj\right) + \left(wj + wj \cdot wj\right)\right)}}{wj + 1} \]
   +-commutative [=>] 89.4	\[ \frac{\frac{x}{e^{wj}} + \color{blue}{\left(\left(wj + wj \cdot wj\right) + \left(-wj\right)\right)}}{wj + 1} \]
   distribute-rgt1-in [=>] 89.4	\[ \frac{\frac{x}{e^{wj}} + \left(\color{blue}{\left(wj + 1\right) \cdot wj} + \left(-wj\right)\right)}{wj + 1} \]
   neg-mul-1 [=>] 89.4	\[ \frac{\frac{x}{e^{wj}} + \left(\left(wj + 1\right) \cdot wj + \color{blue}{-1 \cdot wj}\right)}{wj + 1} \]
   distribute-rgt-in [<=] 89.4	\[ \frac{\frac{x}{e^{wj}} + \color{blue}{wj \cdot \left(\left(wj + 1\right) + -1\right)}}{wj + 1} \]
   associate-+l+ [=>] 99.9	\[ \frac{\frac{x}{e^{wj}} + wj \cdot \color{blue}{\left(wj + \left(1 + -1\right)\right)}}{wj + 1} \]
   metadata-eval [=>] 99.9	\[ \frac{\frac{x}{e^{wj}} + wj \cdot \left(wj + \color{blue}{0}\right)}{wj + 1} \]
   +-commutative [<=] 99.9	\[ \frac{\frac{x}{e^{wj}} + wj \cdot \color{blue}{\left(0 + wj\right)}}{wj + 1} \]
   +-lft-identity [=>] 99.9	\[ \frac{\frac{x}{e^{wj}} + wj \cdot \color{blue}{wj}}{wj + 1} \]

8. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	83.4%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-180}:\\ \;\;\;\;\frac{x - x \cdot wj}{wj + 1}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-307}:\\ \;\;\;\;\frac{wj}{1 + \frac{1}{wj}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{wj + 1} - \frac{wj}{wj + 1}\right)\\ \end{array} \]

Alternative 2
Accuracy	83.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -9.4 \cdot 10^{-181} \lor \neg \left(x \leq 4.4 \cdot 10^{-308}\right):\\ \;\;\;\;\frac{x - x \cdot wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{wj}{1 + \frac{1}{wj}}\\ \end{array} \]

Alternative 3
Accuracy	98.4%
Cost	832

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

Alternative 4
Accuracy	82.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-180} \lor \neg \left(x \leq 6.8 \cdot 10^{-307}\right):\\ \;\;\;\;x + \left(x \cdot wj\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 5
Accuracy	83.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.52 \cdot 10^{-180} \lor \neg \left(x \leq 5.8 \cdot 10^{-307}\right):\\ \;\;\;\;x + \left(x \cdot wj\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{wj}{1 + \frac{1}{wj}}\\ \end{array} \]

Alternative 6
Accuracy	82.4%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-308}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Accuracy	4.5%
Cost	64

\[wj \]

Alternative 8
Accuracy	85.0%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023137 
(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))))))