Average Error: 14.0 → 1.7
Time: 10.7s
Precision: binary64
Cost: 15552
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
\[{wj}^{3} \cdot \left(x \cdot -0.6666666666666666 - \left(x \cdot -3 + \left(1 + -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\right) + \left(\left(1 + \left(x \cdot 4 + x \cdot -1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(x \cdot wj\right)\right)\right) \]
(FPCore (wj x)
 :precision binary64
 (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))
(FPCore (wj x)
 :precision binary64
 (+
  (*
   (pow wj 3.0)
   (-
    (* x -0.6666666666666666)
    (+ (* x -3.0) (+ 1.0 (* -2.0 (+ (* x -4.0) (* x 1.5)))))))
  (+
   (* (+ 1.0 (+ (* x 4.0) (* x -1.5))) (pow wj 2.0))
   (+ x (* -2.0 (* x wj))))))
double code(double wj, double x) {
	return wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))));
}
double code(double wj, double x) {
	return (pow(wj, 3.0) * ((x * -0.6666666666666666) - ((x * -3.0) + (1.0 + (-2.0 * ((x * -4.0) + (x * 1.5))))))) + (((1.0 + ((x * 4.0) + (x * -1.5))) * pow(wj, 2.0)) + (x + (-2.0 * (x * wj))));
}
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 = ((wj ** 3.0d0) * ((x * (-0.6666666666666666d0)) - ((x * (-3.0d0)) + (1.0d0 + ((-2.0d0) * ((x * (-4.0d0)) + (x * 1.5d0))))))) + (((1.0d0 + ((x * 4.0d0) + (x * (-1.5d0)))) * (wj ** 2.0d0)) + (x + ((-2.0d0) * (x * wj))))
end function
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 (Math.pow(wj, 3.0) * ((x * -0.6666666666666666) - ((x * -3.0) + (1.0 + (-2.0 * ((x * -4.0) + (x * 1.5))))))) + (((1.0 + ((x * 4.0) + (x * -1.5))) * Math.pow(wj, 2.0)) + (x + (-2.0 * (x * wj))));
}
def code(wj, x):
	return wj - (((wj * math.exp(wj)) - x) / (math.exp(wj) + (wj * math.exp(wj))))
def code(wj, x):
	return (math.pow(wj, 3.0) * ((x * -0.6666666666666666) - ((x * -3.0) + (1.0 + (-2.0 * ((x * -4.0) + (x * 1.5))))))) + (((1.0 + ((x * 4.0) + (x * -1.5))) * math.pow(wj, 2.0)) + (x + (-2.0 * (x * wj))))
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((wj ^ 3.0) * Float64(Float64(x * -0.6666666666666666) - Float64(Float64(x * -3.0) + Float64(1.0 + Float64(-2.0 * Float64(Float64(x * -4.0) + Float64(x * 1.5))))))) + Float64(Float64(Float64(1.0 + Float64(Float64(x * 4.0) + Float64(x * -1.5))) * (wj ^ 2.0)) + Float64(x + Float64(-2.0 * Float64(x * wj)))))
end
function tmp = code(wj, x)
	tmp = wj - (((wj * exp(wj)) - x) / (exp(wj) + (wj * exp(wj))));
end
function tmp = code(wj, x)
	tmp = ((wj ^ 3.0) * ((x * -0.6666666666666666) - ((x * -3.0) + (1.0 + (-2.0 * ((x * -4.0) + (x * 1.5))))))) + (((1.0 + ((x * 4.0) + (x * -1.5))) * (wj ^ 2.0)) + (x + (-2.0 * (x * wj))));
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[(N[Power[wj, 3.0], $MachinePrecision] * N[(N[(x * -0.6666666666666666), $MachinePrecision] - N[(N[(x * -3.0), $MachinePrecision] + N[(1.0 + N[(-2.0 * N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(N[(x * 4.0), $MachinePrecision] + N[(x * -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(-2.0 * N[(x * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
{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)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.0
Target13.4
Herbie1.7
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

Derivation

  1. Initial program 14.0

    \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
  2. Simplified13.4

    \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
    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))))): 1 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)))): 0 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)))): 3 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)))))): 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)))) (/.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. Taylor expanded in wj around 0 1.7

    \[\leadsto \color{blue}{-1 \cdot \left(\left(0.6666666666666666 \cdot x + \left(-3 \cdot x + \left(1 + -2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right) \cdot {wj}^{3}\right) + \left(\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)\right)} \]
  4. Final simplification1.7

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

Alternatives

Alternative 1
Error1.8
Cost7296
\[\left(\left(x + -2 \cdot \left(x \cdot wj\right)\right) + wj \cdot wj\right) - {wj}^{3} \]
Alternative 2
Error2.0
Cost704
\[\left(x + -2 \cdot \left(x \cdot wj\right)\right) + wj \cdot wj \]
Alternative 3
Error9.2
Cost576
\[x \cdot \frac{1 - wj}{1 + wj} \]
Alternative 4
Error9.2
Cost448
\[x \cdot \left(1 + -2 \cdot wj\right) \]
Alternative 5
Error9.2
Cost448
\[x + -2 \cdot \left(x \cdot wj\right) \]
Alternative 6
Error61.2
Cost64
\[wj \]
Alternative 7
Error9.6
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022325 
(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))))))