Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.9% → 97.5%

Time: 17.0s

Alternatives: 3

Speedup: N/A×

• could not determine a ground truth

  1. wj = 1.422998719770901e+294
  2. x = 2.632162353825493e-280

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ wj - \frac{t\_0 - x}{e^{wj} + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj)))) (- wj (/ (- t_0 x) (+ (exp wj) t_0)))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	return wj - ((t_0 - x) / (exp(wj) + t_0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(wj, x)
use fmin_fmax_functions
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = wj * exp(wj)
    code = wj - ((t_0 - x) / (exp(wj) + t_0))
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	return wj - ((t_0 - x) / (Math.exp(wj) + t_0));
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	return wj - ((t_0 - x) / (math.exp(wj) + t_0))

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	return Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0)))
end

MATLAB

function tmp = code(wj, x)
	t_0 = wj * exp(wj);
	tmp = wj - ((t_0 - x) / (exp(wj) + t_0));
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ wj - \frac{t\_0 - x}{e^{wj} + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj)))) (- wj (/ (- t_0 x) (+ (exp wj) t_0)))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	return wj - ((t_0 - x) / (exp(wj) + t_0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(wj, x)
use fmin_fmax_functions
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = wj * exp(wj)
    code = wj - ((t_0 - x) / (exp(wj) + t_0))
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	return wj - ((t_0 - x) / (Math.exp(wj) + t_0));
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	return wj - ((t_0 - x) / (math.exp(wj) + t_0))

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	return Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0)))
end

MATLAB

function tmp = code(wj, x)
	t_0 = wj * exp(wj);
	tmp = wj - ((t_0 - x) / (exp(wj) + t_0));
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 97.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{e^{wj}}\\ t_1 := t\_0 - -1\\ \mathbf{if}\;wj \leq 0.65:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(0.6666666666666666, x, \left(x \cdot -2.5\right) \cdot -2\right)\right) - -1\right) \cdot wj, -1, 1\right) - x \cdot -2.5, wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\frac{t\_1}{wj} - t\_1}{-wj}, -1, t\_0\right) - -1}{wj}, -1, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ x (exp wj))) (t_1 (- t_0 -1.0)))
   (if (<= wj 0.65)
     (fma
      (fma
       (-
        (fma
         (*
          (- (fma -3.0 x (fma 0.6666666666666666 x (* (* x -2.5) -2.0))) -1.0)
          wj)
         -1.0
         1.0)
        (* x -2.5))
       wj
       (* -2.0 x))
      wj
      x)
     (-
      wj
      (fma
       (/ (- (fma (/ (- (/ t_1 wj) t_1) (- wj)) -1.0 t_0) -1.0) wj)
       -1.0
       1.0)))))

C

double code(double wj, double x) {
	double t_0 = x / exp(wj);
	double t_1 = t_0 - -1.0;
	double tmp;
	if (wj <= 0.65) {
		tmp = fma(fma((fma(((fma(-3.0, x, fma(0.6666666666666666, x, ((x * -2.5) * -2.0))) - -1.0) * wj), -1.0, 1.0) - (x * -2.5)), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - fma(((fma((((t_1 / wj) - t_1) / -wj), -1.0, t_0) - -1.0) / wj), -1.0, 1.0);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(x / exp(wj))
	t_1 = Float64(t_0 - -1.0)
	tmp = 0.0
	if (wj <= 0.65)
		tmp = fma(fma(Float64(fma(Float64(Float64(fma(-3.0, x, fma(0.6666666666666666, x, Float64(Float64(x * -2.5) * -2.0))) - -1.0) * wj), -1.0, 1.0) - Float64(x * -2.5)), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - fma(Float64(Float64(fma(Float64(Float64(Float64(t_1 / wj) - t_1) / Float64(-wj)), -1.0, t_0) - -1.0) / wj), -1.0, 1.0));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - -1.0), $MachinePrecision]}, If[LessEqual[wj, 0.65], N[(N[(N[(N[(N[(N[(N[(-3.0 * x + N[(0.6666666666666666 * x + N[(N[(x * -2.5), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] * wj), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision] - N[(x * -2.5), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(N[(N[(N[(N[(N[(t$95$1 / wj), $MachinePrecision] - t$95$1), $MachinePrecision] / (-wj)), $MachinePrecision] * -1.0 + t$95$0), $MachinePrecision] - -1.0), $MachinePrecision] / wj), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.650000000000000022

   1. Initial program 75.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(0.6666666666666666, x, \left(x \cdot -2.5\right) \cdot -2\right)\right) + 1\right) \cdot wj, -1, 1\right) - x \cdot -2.5, wj, -2 \cdot x\right), wj, x\right)} \]

   if 0.650000000000000022 < wj

   1. Initial program 33.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around -inf

      \[\leadsto wj - \color{blue}{\left(1 + -1 \cdot \frac{1 + \left(-1 \cdot \frac{-1 \cdot \frac{1 + \frac{x}{e^{wj}}}{wj} - -1 \cdot \left(1 + \frac{x}{e^{wj}}\right)}{wj} + \frac{x}{e^{wj}}\right)}{wj}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 80.8%

      \[\leadsto wj - \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\left(-\frac{\frac{x}{e^{wj}} + 1}{wj}\right) - \left(-\left(\frac{x}{e^{wj}} + 1\right)\right)}{wj}, -1, \frac{x}{e^{wj}}\right) + 1}{wj}, -1, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.65:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(0.6666666666666666, x, \left(x \cdot -2.5\right) \cdot -2\right)\right) - -1\right) \cdot wj, -1, 1\right) - x \cdot -2.5, wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\frac{\frac{x}{e^{wj}} - -1}{wj} - \left(\frac{x}{e^{wj}} - -1\right)}{-wj}, -1, \frac{x}{e^{wj}}\right) - -1}{wj}, -1, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(0.6666666666666666, x, \left(x \cdot -2.5\right) \cdot -2\right)\right) - -1\right) \cdot wj, -1, 1\right) - x \cdot -2.5, wj, -2 \cdot x\right), wj, x\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (fma
  (fma
   (-
    (fma
     (*
      (- (fma -3.0 x (fma 0.6666666666666666 x (* (* x -2.5) -2.0))) -1.0)
      wj)
     -1.0
     1.0)
    (* x -2.5))
   wj
   (* -2.0 x))
  wj
  x))

C

double code(double wj, double x) {
	return fma(fma((fma(((fma(-3.0, x, fma(0.6666666666666666, x, ((x * -2.5) * -2.0))) - -1.0) * wj), -1.0, 1.0) - (x * -2.5)), wj, (-2.0 * x)), wj, x);
}

Julia

function code(wj, x)
	return fma(fma(Float64(fma(Float64(Float64(fma(-3.0, x, fma(0.6666666666666666, x, Float64(Float64(x * -2.5) * -2.0))) - -1.0) * wj), -1.0, 1.0) - Float64(x * -2.5)), wj, Float64(-2.0 * x)), wj, x)
end

Wolfram

code[wj_, x_] := N[(N[(N[(N[(N[(N[(N[(-3.0 * x + N[(0.6666666666666666 * x + N[(N[(x * -2.5), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] * wj), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision] - N[(x * -2.5), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision]

Derivation

1. Initial program 74.4%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around 0

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 96.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(0.6666666666666666, x, \left(x \cdot -2.5\right) \cdot -2\right)\right) + 1\right) \cdot wj, -1, 1\right) - x \cdot -2.5, wj, -2 \cdot x\right), wj, x\right)} \]

6. Final simplification 96.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(0.6666666666666666, x, \left(x \cdot -2.5\right) \cdot -2\right)\right) - -1\right) \cdot wj, -1, 1\right) - x \cdot -2.5, wj, -2 \cdot x\right), wj, x\right) \]
7. Add Preprocessing

Alternative 3: 96.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(0.6666666666666666, x, \left(x \cdot -2.5\right) \cdot -2\right)\right) - -1\right) \cdot wj, -1, 1\right) - x \cdot -2.5, wj, -2 \cdot x\right) \cdot wj + x \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (+
  (*
   (fma
    (-
     (fma
      (*
       (- (fma -3.0 x (fma 0.6666666666666666 x (* (* x -2.5) -2.0))) -1.0)
       wj)
      -1.0
      1.0)
     (* x -2.5))
    wj
    (* -2.0 x))
   wj)
  x))

C

double code(double wj, double x) {
	return (fma((fma(((fma(-3.0, x, fma(0.6666666666666666, x, ((x * -2.5) * -2.0))) - -1.0) * wj), -1.0, 1.0) - (x * -2.5)), wj, (-2.0 * x)) * wj) + x;
}

Julia

function code(wj, x)
	return Float64(Float64(fma(Float64(fma(Float64(Float64(fma(-3.0, x, fma(0.6666666666666666, x, Float64(Float64(x * -2.5) * -2.0))) - -1.0) * wj), -1.0, 1.0) - Float64(x * -2.5)), wj, Float64(-2.0 * x)) * wj) + x)
end

Wolfram

code[wj_, x_] := N[(N[(N[(N[(N[(N[(N[(N[(-3.0 * x + N[(0.6666666666666666 * x + N[(N[(x * -2.5), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] * wj), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision] - N[(x * -2.5), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 74.4%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around 0

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 96.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(0.6666666666666666, x, \left(x \cdot -2.5\right) \cdot -2\right)\right) + 1\right) \cdot wj, -1, 1\right) - x \cdot -2.5, wj, -2 \cdot x\right), wj, x\right)} \]

7. Applied rewrites 96.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(0.6666666666666666, x, \left(x \cdot -2.5\right) \cdot -2\right)\right) + 1\right) \cdot wj, -1, 1\right) - x \cdot -2.5, wj, -2 \cdot x\right) \cdot wj + \color{blue}{x} \]

8. Final simplification 96.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(0.6666666666666666, x, \left(x \cdot -2.5\right) \cdot -2\right)\right) - -1\right) \cdot wj, -1, 1\right) - x \cdot -2.5, wj, -2 \cdot x\right) \cdot wj + x \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :alt
  (! :herbie-platform herbie20 (let ((ew (exp wj))) (- wj (- (/ wj (+ wj 1)) (/ x (+ ew (* wj ew)))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))