Jmat.Real.lambertw, newton loop step

Average Error: 13.9 → 1.1

Time: 13.3s

Precision: binary64

Cost: 20288

Math

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

↓

\[\frac{\frac{x}{e^{wj}}}{wj + 1} + \left({wj}^{4} + \left(wj \cdot wj - {wj}^{3}\right)\right) \]

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 1.0)) (+ (pow wj 4.0) (- (* wj wj) (pow wj 3.0)))))

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 + 1.0)) + (pow(wj, 4.0) + ((wj * wj) - pow(wj, 3.0)));
}

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 + 1.0d0)) + ((wj ** 4.0d0) + ((wj * wj) - (wj ** 3.0d0)))
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 + 1.0)) + (Math.pow(wj, 4.0) + ((wj * wj) - Math.pow(wj, 3.0)));
}

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 + 1.0)) + (math.pow(wj, 4.0) + ((wj * wj) - math.pow(wj, 3.0)))

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 + 1.0)) + Float64((wj ^ 4.0) + Float64(Float64(wj * wj) - (wj ^ 3.0))))
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 + 1.0)) + ((wj ^ 4.0) + ((wj * wj) - (wj ^ 3.0)));
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 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 4.0], $MachinePrecision] + N[(N[(wj * wj), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	13.9
Target	13.3
Herbie	1.1

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

Derivation

1. Initial program 13.9

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

2. Simplified 13.3

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

   [Start] 13.9	\[ wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   sub-neg [=>] 13.9	\[ \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
   neg-mul-1 [=>] 13.9	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
   *-commutative [=>] 13.9	\[ wj + \color{blue}{\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \cdot -1} \]
   *-commutative [<=] 13.9	\[ wj + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}} \]
   neg-mul-1 [<=] 13.9	\[ wj + \color{blue}{\left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
   neg-sub0 [=>] 13.9	\[ wj + \color{blue}{\left(0 - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
   div-sub [=>] 13.9	\[ wj + \left(0 - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
   associate--r- [=>] 13.9	\[ wj + \color{blue}{\left(\left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right) + \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]
   +-commutative [=>] 13.9	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \left(0 - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)\right)} \]
   sub0-neg [=>] 13.9	\[ wj + \left(\frac{x}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{\left(-\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]
   sub-neg [<=] 13.9	\[ wj + \color{blue}{\left(\frac{x}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}\right)} \]

3. Applied egg-rr 7.0

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

4. Taylor expanded in wj around 0 1.1

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

5. Simplified 1.1

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

   [Start] 1.1	\[ \frac{\frac{x}{e^{wj}}}{wj + 1} - \left(-1 \cdot {wj}^{4} + \left(-1 \cdot {wj}^{2} + {wj}^{3}\right)\right) \]
   +-commutative [=>] 1.1	\[ \frac{\frac{x}{e^{wj}}}{wj + 1} - \color{blue}{\left(\left(-1 \cdot {wj}^{2} + {wj}^{3}\right) + -1 \cdot {wj}^{4}\right)} \]
   mul-1-neg [=>] 1.1	\[ \frac{\frac{x}{e^{wj}}}{wj + 1} - \left(\left(-1 \cdot {wj}^{2} + {wj}^{3}\right) + \color{blue}{\left(-{wj}^{4}\right)}\right) \]
   metadata-eval [<=] 1.1	\[ \frac{\frac{x}{e^{wj}}}{wj + 1} - \left(\left(-1 \cdot {wj}^{2} + {wj}^{3}\right) + \left(-{wj}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right) \]
   pow-sqr [<=] 1.1	\[ \frac{\frac{x}{e^{wj}}}{wj + 1} - \left(\left(-1 \cdot {wj}^{2} + {wj}^{3}\right) + \left(-\color{blue}{{wj}^{2} \cdot {wj}^{2}}\right)\right) \]
   unpow2 [=>] 1.1	\[ \frac{\frac{x}{e^{wj}}}{wj + 1} - \left(\left(-1 \cdot {wj}^{2} + {wj}^{3}\right) + \left(-\color{blue}{\left(wj \cdot wj\right)} \cdot {wj}^{2}\right)\right) \]
   unpow2 [=>] 1.1	\[ \frac{\frac{x}{e^{wj}}}{wj + 1} - \left(\left(-1 \cdot {wj}^{2} + {wj}^{3}\right) + \left(-\left(wj \cdot wj\right) \cdot \color{blue}{\left(wj \cdot wj\right)}\right)\right) \]
   unsub-neg [=>] 1.1	\[ \frac{\frac{x}{e^{wj}}}{wj + 1} - \color{blue}{\left(\left(-1 \cdot {wj}^{2} + {wj}^{3}\right) - \left(wj \cdot wj\right) \cdot \left(wj \cdot wj\right)\right)} \]
   +-commutative [=>] 1.1	\[ \frac{\frac{x}{e^{wj}}}{wj + 1} - \left(\color{blue}{\left({wj}^{3} + -1 \cdot {wj}^{2}\right)} - \left(wj \cdot wj\right) \cdot \left(wj \cdot wj\right)\right) \]
   mul-1-neg [=>] 1.1	\[ \frac{\frac{x}{e^{wj}}}{wj + 1} - \left(\left({wj}^{3} + \color{blue}{\left(-{wj}^{2}\right)}\right) - \left(wj \cdot wj\right) \cdot \left(wj \cdot wj\right)\right) \]
   unpow2 [=>] 1.1	\[ \frac{\frac{x}{e^{wj}}}{wj + 1} - \left(\left({wj}^{3} + \left(-\color{blue}{wj \cdot wj}\right)\right) - \left(wj \cdot wj\right) \cdot \left(wj \cdot wj\right)\right) \]
   unsub-neg [=>] 1.1	\[ \frac{\frac{x}{e^{wj}}}{wj + 1} - \left(\color{blue}{\left({wj}^{3} - wj \cdot wj\right)} - \left(wj \cdot wj\right) \cdot \left(wj \cdot wj\right)\right) \]
   associate-*r* [=>] 1.1	\[ \frac{\frac{x}{e^{wj}}}{wj + 1} - \left(\left({wj}^{3} - wj \cdot wj\right) - \color{blue}{\left(\left(wj \cdot wj\right) \cdot wj\right) \cdot wj}\right) \]
   unpow3 [<=] 1.1	\[ \frac{\frac{x}{e^{wj}}}{wj + 1} - \left(\left({wj}^{3} - wj \cdot wj\right) - \color{blue}{{wj}^{3}} \cdot wj\right) \]
   pow-plus [=>] 1.1	\[ \frac{\frac{x}{e^{wj}}}{wj + 1} - \left(\left({wj}^{3} - wj \cdot wj\right) - \color{blue}{{wj}^{\left(3 + 1\right)}}\right) \]
   metadata-eval [=>] 1.1	\[ \frac{\frac{x}{e^{wj}}}{wj + 1} - \left(\left({wj}^{3} - wj \cdot wj\right) - {wj}^{\color{blue}{4}}\right) \]

6. Final simplification 1.1

   \[\leadsto \frac{\frac{x}{e^{wj}}}{wj + 1} + \left({wj}^{4} + \left(wj \cdot wj - {wj}^{3}\right)\right) \]

Alternatives

Alternative 1
Error	1.6
Cost	13376

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

Alternative 2
Error	1.6
Cost	7236

\[\begin{array}{l} \mathbf{if}\;wj \leq -5.5 \cdot 10^{-8}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \left(wj \cdot wj - {wj}^{3}\right)\\ \end{array} \]

Alternative 3
Error	1.6
Cost	7104

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

Alternative 4
Error	2.2
Cost	6912

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

Alternative 5
Error	2.5
Cost	6592

\[\mathsf{fma}\left(wj, wj, x\right) \]

Alternative 6
Error	9.9
Cost	456

\[\begin{array}{l} \mathbf{if}\;wj \leq 1.6 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 1.5 \cdot 10^{-23}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	9.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;wj \leq 6.2 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 3.7 \cdot 10^{-26}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x + wj\\ \end{array} \]

Alternative 8
Error	2.5
Cost	320

\[x + wj \cdot wj \]

Alternative 9
Error	61.2
Cost	64

\[wj \]

Alternative 10
Error	9.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023187 
(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))))))