Jmat.Real.lambertw, newton loop step

Average Error: 14.0 → 1.8

Time: 13.3s

Precision: binary64

Cost: 15552

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot 4 + x \cdot -1.5\\ {wj}^{3} \cdot \left(x \cdot -0.6666666666666666 + \left(x \cdot 3 + \left(-1 + -2 \cdot t_0\right)\right)\right) + \left(\left(1 + t_0\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(x \cdot wj\right)\right)\right) \end{array} \]

FPCore

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

↓

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x 4.0) (* x -1.5))))
   (+
    (*
     (pow wj 3.0)
     (+ (* x -0.6666666666666666) (+ (* x 3.0) (+ -1.0 (* -2.0 t_0)))))
    (+ (* (+ 1.0 t_0) (pow wj 2.0)) (+ x (* -2.0 (* x 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 = (x * 4.0) + (x * -1.5);
	return (pow(wj, 3.0) * ((x * -0.6666666666666666) + ((x * 3.0) + (-1.0 + (-2.0 * t_0))))) + (((1.0 + t_0) * pow(wj, 2.0)) + (x + (-2.0 * (x * wj))));
}

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
    real(8) :: t_0
    t_0 = (x * 4.0d0) + (x * (-1.5d0))
    code = ((wj ** 3.0d0) * ((x * (-0.6666666666666666d0)) + ((x * 3.0d0) + ((-1.0d0) + ((-2.0d0) * t_0))))) + (((1.0d0 + t_0) * (wj ** 2.0d0)) + (x + ((-2.0d0) * (x * wj))))
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) {
	double t_0 = (x * 4.0) + (x * -1.5);
	return (Math.pow(wj, 3.0) * ((x * -0.6666666666666666) + ((x * 3.0) + (-1.0 + (-2.0 * t_0))))) + (((1.0 + t_0) * Math.pow(wj, 2.0)) + (x + (-2.0 * (x * wj))));
}

Python

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

↓

def code(wj, x):
	t_0 = (x * 4.0) + (x * -1.5)
	return (math.pow(wj, 3.0) * ((x * -0.6666666666666666) + ((x * 3.0) + (-1.0 + (-2.0 * t_0))))) + (((1.0 + t_0) * math.pow(wj, 2.0)) + (x + (-2.0 * (x * wj))))

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(Float64(x * 4.0) + Float64(x * -1.5))
	return Float64(Float64((wj ^ 3.0) * Float64(Float64(x * -0.6666666666666666) + Float64(Float64(x * 3.0) + Float64(-1.0 + Float64(-2.0 * t_0))))) + Float64(Float64(Float64(1.0 + t_0) * (wj ^ 2.0)) + Float64(x + Float64(-2.0 * Float64(x * wj)))))
end

MATLAB

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

↓

function tmp = code(wj, x)
	t_0 = (x * 4.0) + (x * -1.5);
	tmp = ((wj ^ 3.0) * ((x * -0.6666666666666666) + ((x * 3.0) + (-1.0 + (-2.0 * t_0))))) + (((1.0 + t_0) * (wj ^ 2.0)) + (x + (-2.0 * (x * wj))));
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[(N[(x * 4.0), $MachinePrecision] + N[(x * -1.5), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[wj, 3.0], $MachinePrecision] * N[(N[(x * -0.6666666666666666), $MachinePrecision] + N[(N[(x * 3.0), $MachinePrecision] + N[(-1.0 + N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + t$95$0), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(-2.0 * N[(x * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	14.0
Target	13.5
Herbie	1.8

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

Derivation

1. Initial program 14.0

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

2. Simplified 13.5

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
   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)))): 1 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)))): 6 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, 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)))))): 1 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, 0 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. Taylor expanded in wj around 0 1.8

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

4. Final simplification 1.8

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

Alternatives

Alternative 1
Error	1.8
Cost	7296

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

Alternative 2
Error	10.5
Cost	7048

\[\begin{array}{l} t_0 := wj \cdot \mathsf{fma}\left(wj, -wj, wj\right)\\ \mathbf{if}\;wj \leq -1.630136685715376 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;wj \leq 2.8488990407509116 \cdot 10^{-36}:\\ \;\;\;\;x + -2 \cdot \left(x \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	10.5
Cost	7048

\[\begin{array}{l} \mathbf{if}\;wj \leq -1.630136685715376 \cdot 10^{-44}:\\ \;\;\;\;wj \cdot wj - {wj}^{3}\\ \mathbf{elif}\;wj \leq 2.8488990407509116 \cdot 10^{-36}:\\ \;\;\;\;x + -2 \cdot \left(x \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot \mathsf{fma}\left(wj, -wj, wj\right)\\ \end{array} \]

Alternative 4
Error	2.1
Cost	6912

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

Alternative 5
Error	8.5
Cost	2124

\[\begin{array}{l} t_0 := wj + \frac{\left(\left(x - x \cdot wj\right) + \left(wj \cdot wj\right) \cdot \left(x \cdot 0.5 - wj \cdot \left(x \cdot 0.16666666666666666\right)\right)\right) - wj}{1 + wj}\\ \mathbf{if}\;wj \leq -2.6845691898550104 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;wj \leq -1.630136685715376 \cdot 10^{-44}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq 3.724637995429443 \cdot 10^{-40}:\\ \;\;\;\;x + -2 \cdot \left(x \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	8.5
Cost	1612

\[\begin{array}{l} t_0 := wj + \frac{\left(x - x \cdot \left(wj + wj \cdot \left(wj \cdot -0.5\right)\right)\right) - wj}{1 + wj}\\ \mathbf{if}\;wj \leq -2.6845691898550104 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;wj \leq -1.630136685715376 \cdot 10^{-44}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq 3.724637995429443 \cdot 10^{-40}:\\ \;\;\;\;x + -2 \cdot \left(x \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	8.6
Cost	1228

\[\begin{array}{l} t_0 := wj + \frac{\left(x - x \cdot wj\right) - wj}{1 + wj}\\ \mathbf{if}\;wj \leq -2.6845691898550104 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;wj \leq -1.630136685715376 \cdot 10^{-44}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq 3.724637995429443 \cdot 10^{-40}:\\ \;\;\;\;x + -2 \cdot \left(x \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	10.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;wj \leq -1.630136685715376 \cdot 10^{-44}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq 2.8488990407509116 \cdot 10^{-36}:\\ \;\;\;\;x + -2 \cdot \left(x \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 9
Error	10.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;wj \leq -1.630136685715376 \cdot 10^{-44}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq 2.8488990407509116 \cdot 10^{-36}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 10
Error	61.2
Cost	64

\[wj \]

Alternative 11
Error	9.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022294 
(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))))))