Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.2% → 98.6%

Time: 14.8s

Alternatives: 12

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -3.513273790779189e+224
  2. x = -1.2220762302192277e-167

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

real(8) function code(wj, x)
    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: 78.2% 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

real(8) function code(wj, x)
    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: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + \left(1 + \left(-1 - wj \cdot \left(-2 \cdot \left(x \cdot -2.5\right) + \left(x \cdot 0.6666666666666666 - \left(-1 - x \cdot -3\right)\right)\right)\right)\right)\right) + x \cdot 2.5\right) + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\frac{x}{e^{wj}} - wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 2e-12)
     (+
      x
      (*
       wj
       (+
        (*
         wj
         (+
          (+
           1.0
           (+
            1.0
            (-
             -1.0
             (*
              wj
              (+
               (* -2.0 (* x -2.5))
               (- (* x 0.6666666666666666) (- -1.0 (* x -3.0))))))))
          (* x 2.5)))
        (* x -2.0))))
     (- wj (/ (- (/ x (exp wj)) wj) (- -1.0 wj))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 2e-12) {
		tmp = x + (wj * ((wj * ((1.0 + (1.0 + (-1.0 - (wj * ((-2.0 * (x * -2.5)) + ((x * 0.6666666666666666) - (-1.0 - (x * -3.0)))))))) + (x * 2.5))) + (x * -2.0)));
	} else {
		tmp = wj - (((x / exp(wj)) - wj) / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = wj * exp(wj)
    if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 2d-12) then
        tmp = x + (wj * ((wj * ((1.0d0 + (1.0d0 + ((-1.0d0) - (wj * (((-2.0d0) * (x * (-2.5d0))) + ((x * 0.6666666666666666d0) - ((-1.0d0) - (x * (-3.0d0))))))))) + (x * 2.5d0))) + (x * (-2.0d0))))
    else
        tmp = wj - (((x / exp(wj)) - wj) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (Math.exp(wj) + t_0))) <= 2e-12) {
		tmp = x + (wj * ((wj * ((1.0 + (1.0 + (-1.0 - (wj * ((-2.0 * (x * -2.5)) + ((x * 0.6666666666666666) - (-1.0 - (x * -3.0)))))))) + (x * 2.5))) + (x * -2.0)));
	} else {
		tmp = wj - (((x / Math.exp(wj)) - wj) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	tmp = 0
	if (wj + ((x - t_0) / (math.exp(wj) + t_0))) <= 2e-12:
		tmp = x + (wj * ((wj * ((1.0 + (1.0 + (-1.0 - (wj * ((-2.0 * (x * -2.5)) + ((x * 0.6666666666666666) - (-1.0 - (x * -3.0)))))))) + (x * 2.5))) + (x * -2.0)))
	else:
		tmp = wj - (((x / math.exp(wj)) - wj) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 2e-12)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(Float64(1.0 + Float64(1.0 + Float64(-1.0 - Float64(wj * Float64(Float64(-2.0 * Float64(x * -2.5)) + Float64(Float64(x * 0.6666666666666666) - Float64(-1.0 - Float64(x * -3.0)))))))) + Float64(x * 2.5))) + Float64(x * -2.0))));
	else
		tmp = Float64(wj - Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	tmp = 0.0;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 2e-12)
		tmp = x + (wj * ((wj * ((1.0 + (1.0 + (-1.0 - (wj * ((-2.0 * (x * -2.5)) + ((x * 0.6666666666666666) - (-1.0 - (x * -3.0)))))))) + (x * 2.5))) + (x * -2.0)));
	else
		tmp = wj - (((x / exp(wj)) - wj) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-12], N[(x + N[(wj * N[(N[(wj * N[(N[(1.0 + N[(1.0 + N[(-1.0 - N[(wj * N[(N[(-2.0 * N[(x * -2.5), $MachinePrecision]), $MachinePrecision] + N[(N[(x * 0.6666666666666666), $MachinePrecision] - N[(-1.0 - N[(x * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 1.99999999999999996e-12

   1. Initial program 73.1%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. sub-neg 73.1%

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

      2. distribute-neg-frac 73.1%

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

      3. distribute-rgt1-in 73.7%

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

      4. associate-/l/ 73.6%

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

      5. sub-neg 73.6%

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

      6. +-commutative 73.6%

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

      7. distribute-neg-in 73.6%

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

      8. remove-double-neg 73.6%

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

      9. sub-neg 73.6%

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

      10. div-sub 73.1%

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

      11. associate-/l* 73.1%

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

      12. *-inverses 73.6%

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

      13. *-rgt-identity 73.6%

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

   3. Simplified 73.6%

      \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 98.8%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 98.8%

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

      2. metadata-eval 98.8%

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

   7. Simplified 98.8%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 - wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right) + x \cdot 2.5\right) + x \cdot -2\right)} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 93.1%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right)\right)}\right) + x \cdot 2.5\right) + x \cdot -2\right) \]

      2. expm1-undefine 93.1%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 - \color{blue}{\left(e^{\mathsf{log1p}\left(wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right)} - 1\right)}\right) + x \cdot 2.5\right) + x \cdot -2\right) \]

      3. log1p-undefine 93.1%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 - \left(e^{\color{blue}{\log \left(1 + wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right)}} - 1\right)\right) + x \cdot 2.5\right) + x \cdot -2\right) \]

      4. add-exp-log 98.8%

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

      5. *-commutative 98.8%

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

      6. associate-+l+ 98.8%

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

   9. Applied egg-rr 98.8%

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

   if 1.99999999999999996e-12 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 93.5%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. sub-neg 93.5%

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

      2. distribute-neg-frac 93.5%

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

      3. distribute-rgt1-in 95.1%

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

      4. associate-/l/ 95.1%

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

      5. sub-neg 95.1%

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

      6. +-commutative 95.1%

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

      7. distribute-neg-in 95.1%

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

      8. remove-double-neg 95.1%

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

      9. sub-neg 95.1%

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

      10. div-sub 93.5%

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

      11. associate-/l* 93.5%

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

      12. *-inverses 99.9%

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

      13. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(\left(1 + \left(1 + \left(-1 - wj \cdot \left(-2 \cdot \left(x \cdot -2.5\right) + \left(x \cdot 0.6666666666666666 - \left(-1 - x \cdot -3\right)\right)\right)\right)\right)\right) + x \cdot 2.5\right) + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\frac{x}{e^{wj}} - wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.9% accurate, 8.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + (wj * ((wj * ((1.0d0 + (1.0d0 + ((-1.0d0) - (wj * (((-2.0d0) * (x * (-2.5d0))) + ((x * 0.6666666666666666d0) - ((-1.0d0) - (x * (-3.0d0))))))))) + (x * 2.5d0))) + (x * (-2.0d0))))
end function

Java

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

Python

def code(wj, x):
	return x + (wj * ((wj * ((1.0 + (1.0 + (-1.0 - (wj * ((-2.0 * (x * -2.5)) + ((x * 0.6666666666666666) - (-1.0 - (x * -3.0)))))))) + (x * 2.5))) + (x * -2.0)))

Julia

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

MATLAB

function tmp = code(wj, x)
	tmp = x + (wj * ((wj * ((1.0 + (1.0 + (-1.0 - (wj * ((-2.0 * (x * -2.5)) + ((x * 0.6666666666666666) - (-1.0 - (x * -3.0)))))))) + (x * 2.5))) + (x * -2.0)));
end

Wolfram

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

Derivation

1. Initial program 78.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. sub-neg 78.1%

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

   2. distribute-neg-frac 78.1%

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

   3. distribute-rgt1-in 78.9%

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

   4. associate-/l/ 78.8%

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

   5. sub-neg 78.8%

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

   6. +-commutative 78.8%

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

   7. distribute-neg-in 78.8%

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

   8. remove-double-neg 78.8%

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

   9. sub-neg 78.8%

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

   10. div-sub 78.0%

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

   11. associate-/l* 78.0%

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

   12. *-inverses 80.0%

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

   13. *-rgt-identity 80.0%

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

3. Simplified 80.0%

   \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 96.8%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 96.8%

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

   2. metadata-eval 96.8%

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

7. Simplified 96.8%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 - wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right) + x \cdot 2.5\right) + x \cdot -2\right)} \]
8. Step-by-step derivation

   1. expm1-log1p-u 87.4%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right)\right)}\right) + x \cdot 2.5\right) + x \cdot -2\right) \]

   2. expm1-undefine 87.4%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 - \color{blue}{\left(e^{\mathsf{log1p}\left(wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right)} - 1\right)}\right) + x \cdot 2.5\right) + x \cdot -2\right) \]

   3. log1p-undefine 87.4%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 - \left(e^{\color{blue}{\log \left(1 + wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right)}} - 1\right)\right) + x \cdot 2.5\right) + x \cdot -2\right) \]

   4. add-exp-log 96.8%

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

   5. *-commutative 96.8%

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

   6. associate-+l+ 96.8%

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

9. Applied egg-rr 96.8%

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

10. Final simplification 96.8%

    \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 + \left(1 + \left(-1 - wj \cdot \left(-2 \cdot \left(x \cdot -2.5\right) + \left(x \cdot 0.6666666666666666 - \left(-1 - x \cdot -3\right)\right)\right)\right)\right)\right) + x \cdot 2.5\right) + x \cdot -2\right) \]
11. Add Preprocessing

Alternative 3: 95.9% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x - (wj * ((wj * (((wj * ((1.0d0 + (x * (-3.0d0))) + (((-2.0d0) * (x * (-2.5d0))) + (x * 0.6666666666666666d0)))) + (-1.0d0)) - (x * 2.5d0))) - (x * (-2.0d0))))
end function

Java

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

Python

def code(wj, x):
	return x - (wj * ((wj * (((wj * ((1.0 + (x * -3.0)) + ((-2.0 * (x * -2.5)) + (x * 0.6666666666666666)))) + -1.0) - (x * 2.5))) - (x * -2.0)))

Julia

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

MATLAB

function tmp = code(wj, x)
	tmp = x - (wj * ((wj * (((wj * ((1.0 + (x * -3.0)) + ((-2.0 * (x * -2.5)) + (x * 0.6666666666666666)))) + -1.0) - (x * 2.5))) - (x * -2.0)));
end

Wolfram

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

Derivation

1. Initial program 78.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. sub-neg 78.1%

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

   2. distribute-neg-frac 78.1%

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

   3. distribute-rgt1-in 78.9%

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

   4. associate-/l/ 78.8%

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

   5. sub-neg 78.8%

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

   6. +-commutative 78.8%

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

   7. distribute-neg-in 78.8%

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

   8. remove-double-neg 78.8%

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

   9. sub-neg 78.8%

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

   10. div-sub 78.0%

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

   11. associate-/l* 78.0%

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

   12. *-inverses 80.0%

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

   13. *-rgt-identity 80.0%

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

3. Simplified 80.0%

   \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 96.8%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 96.8%

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

   2. metadata-eval 96.8%

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

7. Simplified 96.8%

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

8. Final simplification 96.8%

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

Alternative 4: 95.9% accurate, 16.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + (wj * ((x * (-2.0d0)) - (wj * (1.0d0 - (2.0d0 + ((x * 2.5d0) - wj))))))
end function

Java

public static double code(double wj, double x) {
	return x + (wj * ((x * -2.0) - (wj * (1.0 - (2.0 + ((x * 2.5) - wj))))));
}

Python

def code(wj, x):
	return x + (wj * ((x * -2.0) - (wj * (1.0 - (2.0 + ((x * 2.5) - wj))))))

Julia

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

MATLAB

function tmp = code(wj, x)
	tmp = x + (wj * ((x * -2.0) - (wj * (1.0 - (2.0 + ((x * 2.5) - wj))))));
end

Wolfram

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

Derivation

1. Initial program 78.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. sub-neg 78.1%

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

   2. distribute-neg-frac 78.1%

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

   3. distribute-rgt1-in 78.9%

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

   4. associate-/l/ 78.8%

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

   5. sub-neg 78.8%

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

   6. +-commutative 78.8%

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

   7. distribute-neg-in 78.8%

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

   8. remove-double-neg 78.8%

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

   9. sub-neg 78.8%

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

   10. div-sub 78.0%

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

   11. associate-/l* 78.0%

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

   12. *-inverses 80.0%

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

   13. *-rgt-identity 80.0%

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

3. Simplified 80.0%

   \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 96.8%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 96.8%

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

   2. metadata-eval 96.8%

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

7. Simplified 96.8%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 - wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right) + x \cdot 2.5\right) + x \cdot -2\right)} \]
8. Step-by-step derivation

   1. expm1-log1p-u 87.4%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right)\right)}\right) + x \cdot 2.5\right) + x \cdot -2\right) \]

   2. expm1-undefine 87.4%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 - \color{blue}{\left(e^{\mathsf{log1p}\left(wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right)} - 1\right)}\right) + x \cdot 2.5\right) + x \cdot -2\right) \]

   3. log1p-undefine 87.4%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 - \left(e^{\color{blue}{\log \left(1 + wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right)}} - 1\right)\right) + x \cdot 2.5\right) + x \cdot -2\right) \]

   4. add-exp-log 96.8%

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

   5. *-commutative 96.8%

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

   6. associate-+l+ 96.8%

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

9. Applied egg-rr 96.8%

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

10. Taylor expanded in x around 0 96.7%

    \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 - \left(\left(1 + \color{blue}{1} \cdot wj\right) - 1\right)\right) + x \cdot 2.5\right) + x \cdot -2\right) \]
11. Step-by-step derivation

    1. expm1-log1p-u 71.7%

       \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - \left(\left(1 + 1 \cdot wj\right) - 1\right)\right) + x \cdot 2.5\right)\right)} + x \cdot -2\right) \]

    2. expm1-undefine 71.7%

       \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(1 - \left(\left(1 + 1 \cdot wj\right) - 1\right)\right) + x \cdot 2.5\right)} - 1\right)} + x \cdot -2\right) \]

    3. *-un-lft-identity 71.7%

       \[\leadsto x + wj \cdot \left(wj \cdot \left(e^{\mathsf{log1p}\left(\left(1 - \left(\left(1 + \color{blue}{wj}\right) - 1\right)\right) + x \cdot 2.5\right)} - 1\right) + x \cdot -2\right) \]

    4. add-exp-log 71.7%

       \[\leadsto x + wj \cdot \left(wj \cdot \left(e^{\mathsf{log1p}\left(\left(1 - \left(\color{blue}{e^{\log \left(1 + wj\right)}} - 1\right)\right) + x \cdot 2.5\right)} - 1\right) + x \cdot -2\right) \]

    5. expm1-define 71.7%

       \[\leadsto x + wj \cdot \left(wj \cdot \left(e^{\mathsf{log1p}\left(\left(1 - \color{blue}{\mathsf{expm1}\left(\log \left(1 + wj\right)\right)}\right) + x \cdot 2.5\right)} - 1\right) + x \cdot -2\right) \]

    6. log1p-define 71.7%

       \[\leadsto x + wj \cdot \left(wj \cdot \left(e^{\mathsf{log1p}\left(\left(1 - \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(wj\right)}\right)\right) + x \cdot 2.5\right)} - 1\right) + x \cdot -2\right) \]

    7. expm1-log1p-u 71.7%

       \[\leadsto x + wj \cdot \left(wj \cdot \left(e^{\mathsf{log1p}\left(\left(1 - \color{blue}{wj}\right) + x \cdot 2.5\right)} - 1\right) + x \cdot -2\right) \]

12. Applied egg-rr 71.7%

    \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(1 - wj\right) + x \cdot 2.5\right)} - 1\right)} + x \cdot -2\right) \]
13. Step-by-step derivation

    1. sub-neg 71.7%

       \[\leadsto x + wj \cdot \left(wj \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(1 - wj\right) + x \cdot 2.5\right)} + \left(-1\right)\right)} + x \cdot -2\right) \]

    2. log1p-undefine 71.7%

       \[\leadsto x + wj \cdot \left(wj \cdot \left(e^{\color{blue}{\log \left(1 + \left(\left(1 - wj\right) + x \cdot 2.5\right)\right)}} + \left(-1\right)\right) + x \cdot -2\right) \]

    3. rem-exp-log 96.7%

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

    4. sub-neg 96.7%

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

    5. *-commutative 96.7%

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

    6. associate-+r+ 96.7%

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

    7. mul-1-neg 96.7%

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

    8. associate-+r+ 96.7%

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

    9. metadata-eval 96.7%

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

    10. mul-1-neg 96.7%

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

    11. +-commutative 96.7%

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

    12. fma-define 96.7%

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

    13. fma-neg 96.7%

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

    14. metadata-eval 96.7%

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

14. Simplified 96.7%

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

15. Final simplification 96.7%

    \[\leadsto x + wj \cdot \left(x \cdot -2 - wj \cdot \left(1 - \left(2 + \left(x \cdot 2.5 - wj\right)\right)\right)\right) \]
16. Add Preprocessing

Alternative 5: 96.4% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 3.8 \cdot 10^{-10}:\\ \;\;\;\;x + \left(1 - wj\right) \cdot \left(wj \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\left(x - wj \cdot x\right) - wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 3.8e-10)
   (+ x (* (- 1.0 wj) (* wj wj)))
   (- wj (/ (- (- x (* wj x)) wj) (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 3.8e-10) {
		tmp = x + ((1.0 - wj) * (wj * wj));
	} else {
		tmp = wj - (((x - (wj * x)) - wj) / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 3.8d-10) then
        tmp = x + ((1.0d0 - wj) * (wj * wj))
    else
        tmp = wj - (((x - (wj * x)) - wj) / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 3.8e-10) {
		tmp = x + ((1.0 - wj) * (wj * wj));
	} else {
		tmp = wj - (((x - (wj * x)) - wj) / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 3.8e-10:
		tmp = x + ((1.0 - wj) * (wj * wj))
	else:
		tmp = wj - (((x - (wj * x)) - wj) / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 3.8e-10)
		tmp = Float64(x + Float64(Float64(1.0 - wj) * Float64(wj * wj)));
	else
		tmp = Float64(wj - Float64(Float64(Float64(x - Float64(wj * x)) - wj) / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 3.8e-10)
		tmp = x + ((1.0 - wj) * (wj * wj));
	else
		tmp = wj - (((x - (wj * x)) - wj) / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 3.8e-10], N[(x + N[(N[(1.0 - wj), $MachinePrecision] * N[(wj * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(N[(x - N[(wj * x), $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 3.7999999999999998e-10

   1. Initial program 78.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. sub-neg 78.7%

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

      2. distribute-neg-frac 78.7%

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

      3. distribute-rgt1-in 79.5%

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

      4. associate-/l/ 79.5%

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

      5. sub-neg 79.5%

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

      6. +-commutative 79.5%

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

      7. distribute-neg-in 79.5%

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

      8. remove-double-neg 79.5%

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

      9. sub-neg 79.5%

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

      10. div-sub 78.7%

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

      11. associate-/l* 78.7%

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

      12. *-inverses 79.5%

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

      13. *-rgt-identity 79.5%

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

   3. Simplified 79.5%

      \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 98.3%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 98.3%

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

      2. metadata-eval 98.3%

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

   7. Simplified 98.3%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 - wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right) + x \cdot 2.5\right) + x \cdot -2\right)} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 89.0%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right)\right)}\right) + x \cdot 2.5\right) + x \cdot -2\right) \]

      2. expm1-undefine 89.0%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 - \color{blue}{\left(e^{\mathsf{log1p}\left(wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right)} - 1\right)}\right) + x \cdot 2.5\right) + x \cdot -2\right) \]

      3. log1p-undefine 89.0%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 - \left(e^{\color{blue}{\log \left(1 + wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right)}} - 1\right)\right) + x \cdot 2.5\right) + x \cdot -2\right) \]

      4. add-exp-log 98.3%

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

      5. *-commutative 98.3%

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

      6. associate-+l+ 98.3%

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

   9. Applied egg-rr 98.3%

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

   10. Taylor expanded in x around 0 98.2%

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

   11. Taylor expanded in x around 0 97.6%

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

       1. unpow2 97.6%

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

   13. Simplified 97.6%

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

   if 3.7999999999999998e-10 < wj

   1. Initial program 58.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. sub-neg 58.3%

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

      2. distribute-neg-frac 58.3%

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

      3. distribute-rgt1-in 58.3%

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

      4. associate-/l/ 58.4%

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

      5. sub-neg 58.4%

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

      6. +-commutative 58.4%

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

      7. distribute-neg-in 58.4%

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

      8. remove-double-neg 58.4%

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

      9. sub-neg 58.4%

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

      10. div-sub 58.4%

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

      11. associate-/l* 58.4%

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

      12. *-inverses 95.9%

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

      13. *-rgt-identity 95.9%

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

   3. Simplified 95.9%

      \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0 84.7%

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

      1. mul-1-neg 84.7%

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

      2. unsub-neg 84.7%

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

      3. *-commutative 84.7%

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

   7. Simplified 84.7%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 3.8 \cdot 10^{-10}:\\ \;\;\;\;x + \left(1 - wj\right) \cdot \left(wj \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\left(x - wj \cdot x\right) - wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.8% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - wj\right)\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (+ x (* wj (+ (* x -2.0) (* wj (- 1.0 wj))))))

C

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

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + (wj * ((x * (-2.0d0)) + (wj * (1.0d0 - wj))))
end function

Java

public static double code(double wj, double x) {
	return x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
}

Python

def code(wj, x):
	return x + (wj * ((x * -2.0) + (wj * (1.0 - wj))))

Julia

function code(wj, x)
	return Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(1.0 - wj)))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + (wj * ((x * -2.0) + (wj * (1.0 - wj))));
end

Wolfram

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

Derivation

1. Initial program 78.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. sub-neg 78.1%

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

   2. distribute-neg-frac 78.1%

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

   3. distribute-rgt1-in 78.9%

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

   4. associate-/l/ 78.8%

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

   5. sub-neg 78.8%

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

   6. +-commutative 78.8%

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

   7. distribute-neg-in 78.8%

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

   8. remove-double-neg 78.8%

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

   9. sub-neg 78.8%

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

   10. div-sub 78.0%

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

   11. associate-/l* 78.0%

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

   12. *-inverses 80.0%

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

   13. *-rgt-identity 80.0%

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

3. Simplified 80.0%

   \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 96.8%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 96.8%

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

   2. metadata-eval 96.8%

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

7. Simplified 96.8%

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

8. Taylor expanded in x around 0 96.5%

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

9. Final simplification 96.5%

   \[\leadsto x + wj \cdot \left(x \cdot -2 + wj \cdot \left(1 - wj\right)\right) \]
10. Add Preprocessing

Alternative 7: 95.3% accurate, 34.8× speedup

Math

\[\begin{array}{l} \\ x + \left(1 - wj\right) \cdot \left(wj \cdot wj\right) \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (+ x (* (- 1.0 wj) (* wj wj))))

C

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

Fortran

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

Java

public static double code(double wj, double x) {
	return x + ((1.0 - wj) * (wj * wj));
}

Python

def code(wj, x):
	return x + ((1.0 - wj) * (wj * wj))

Julia

function code(wj, x)
	return Float64(x + Float64(Float64(1.0 - wj) * Float64(wj * wj)))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + ((1.0 - wj) * (wj * wj));
end

Wolfram

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

Derivation

1. Initial program 78.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. sub-neg 78.1%

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

   2. distribute-neg-frac 78.1%

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

   3. distribute-rgt1-in 78.9%

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

   4. associate-/l/ 78.8%

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

   5. sub-neg 78.8%

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

   6. +-commutative 78.8%

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

   7. distribute-neg-in 78.8%

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

   8. remove-double-neg 78.8%

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

   9. sub-neg 78.8%

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

   10. div-sub 78.0%

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

   11. associate-/l* 78.0%

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

   12. *-inverses 80.0%

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

   13. *-rgt-identity 80.0%

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

3. Simplified 80.0%

   \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 96.8%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 96.8%

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

   2. metadata-eval 96.8%

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

7. Simplified 96.8%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 - wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right) + x \cdot 2.5\right) + x \cdot -2\right)} \]
8. Step-by-step derivation

   1. expm1-log1p-u 87.4%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right)\right)}\right) + x \cdot 2.5\right) + x \cdot -2\right) \]

   2. expm1-undefine 87.4%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 - \color{blue}{\left(e^{\mathsf{log1p}\left(wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right)} - 1\right)}\right) + x \cdot 2.5\right) + x \cdot -2\right) \]

   3. log1p-undefine 87.4%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(\left(1 - \left(e^{\color{blue}{\log \left(1 + wj \cdot \left(\left(-2 \cdot \left(x \cdot -2.5\right) + x \cdot 0.6666666666666666\right) + \left(x \cdot -3 + 1\right)\right)\right)}} - 1\right)\right) + x \cdot 2.5\right) + x \cdot -2\right) \]

   4. add-exp-log 96.8%

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

   5. *-commutative 96.8%

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

   6. associate-+l+ 96.8%

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

9. Applied egg-rr 96.8%

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

10. Taylor expanded in x around 0 96.7%

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

11. Taylor expanded in x around 0 95.6%

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

    1. unpow2 95.6%

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

13. Simplified 95.6%

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

14. Final simplification 95.6%

    \[\leadsto x + \left(1 - wj\right) \cdot \left(wj \cdot wj\right) \]
15. Add Preprocessing

Alternative 8: 94.9% accurate, 44.7× speedup

Math

\[\begin{array}{l} \\ x + wj \cdot \left(wj - x\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return x + (wj * (wj - x))

Julia

function code(wj, x)
	return Float64(x + Float64(wj * Float64(wj - x)))
end

MATLAB

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

Wolfram

code[wj_, x_] := N[(x + N[(wj * N[(wj - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. sub-neg 78.1%

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

   2. distribute-neg-frac 78.1%

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

   3. distribute-rgt1-in 78.9%

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

   4. associate-/l/ 78.8%

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

   5. sub-neg 78.8%

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

   6. +-commutative 78.8%

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

   7. distribute-neg-in 78.8%

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

   8. remove-double-neg 78.8%

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

   9. sub-neg 78.8%

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

   10. div-sub 78.0%

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

   11. associate-/l* 78.0%

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

   12. *-inverses 80.0%

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

   13. *-rgt-identity 80.0%

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

3. Simplified 80.0%

   \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 77.5%

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

6. Taylor expanded in wj around 0 95.2%

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

   1. sub-neg 95.2%

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

   2. neg-mul-1 95.2%

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

   3. remove-double-neg 95.2%

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

   4. +-commutative 95.2%

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

8. Simplified 95.2%

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

9. Taylor expanded in x around 0 95.2%

   \[\leadsto x + wj \cdot \left(\color{blue}{wj} - x\right) \]
10. Add Preprocessing

Alternative 9: 94.9% accurate, 62.6× speedup

Math

\[\begin{array}{l} \\ x + wj \cdot wj \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (+ x (* wj wj)))

C

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

Fortran

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

Java

public static double code(double wj, double x) {
	return x + (wj * wj);
}

Python

def code(wj, x):
	return x + (wj * wj)

Julia

function code(wj, x)
	return Float64(x + Float64(wj * wj))
end

MATLAB

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

Wolfram

code[wj_, x_] := N[(x + N[(wj * wj), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. sub-neg 78.1%

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

   2. distribute-neg-frac 78.1%

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

   3. distribute-rgt1-in 78.9%

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

   4. associate-/l/ 78.8%

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

   5. sub-neg 78.8%

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

   6. +-commutative 78.8%

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

   7. distribute-neg-in 78.8%

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

   8. remove-double-neg 78.8%

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

   9. sub-neg 78.8%

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

   10. div-sub 78.0%

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

   11. associate-/l* 78.0%

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

   12. *-inverses 80.0%

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

   13. *-rgt-identity 80.0%

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

3. Simplified 80.0%

   \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 77.5%

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

6. Taylor expanded in wj around 0 95.2%

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

   1. sub-neg 95.2%

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

   2. neg-mul-1 95.2%

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

   3. remove-double-neg 95.2%

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

   4. +-commutative 95.2%

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

8. Simplified 95.2%

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

9. Taylor expanded in x around 0 95.1%

   \[\leadsto x + wj \cdot \color{blue}{wj} \]
10. Add Preprocessing

Alternative 10: 83.8% accurate, 62.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - wj\right) \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (* x (- 1.0 wj)))

C

double code(double wj, double x) {
	return x * (1.0 - wj);
}

Fortran

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

Java

public static double code(double wj, double x) {
	return x * (1.0 - wj);
}

Python

def code(wj, x):
	return x * (1.0 - wj)

Julia

function code(wj, x)
	return Float64(x * Float64(1.0 - wj))
end

MATLAB

function tmp = code(wj, x)
	tmp = x * (1.0 - wj);
end

Wolfram

code[wj_, x_] := N[(x * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. sub-neg 78.1%

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

   2. distribute-neg-frac 78.1%

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

   3. distribute-rgt1-in 78.9%

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

   4. associate-/l/ 78.8%

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

   5. sub-neg 78.8%

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

   6. +-commutative 78.8%

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

   7. distribute-neg-in 78.8%

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

   8. remove-double-neg 78.8%

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

   9. sub-neg 78.8%

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

   10. div-sub 78.0%

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

   11. associate-/l* 78.0%

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

   12. *-inverses 80.0%

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

   13. *-rgt-identity 80.0%

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

3. Simplified 80.0%

   \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 77.5%

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

6. Taylor expanded in wj around 0 85.9%

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

   1. *-lft-identity 85.9%

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

   2. mul-1-neg 85.9%

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

   3. distribute-lft-neg-in 85.9%

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

   4. distribute-rgt-out 85.9%

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

   5. sub-neg 85.9%

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

8. Simplified 85.9%

   \[\leadsto \color{blue}{x \cdot \left(1 - wj\right)} \]
9. Add Preprocessing

Alternative 11: 83.8% accurate, 313.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 x)

C

double code(double wj, double x) {
	return x;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double wj, double x) {
	return x;
}

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

function tmp = code(wj, x)
	tmp = x;
end

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 78.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. sub-neg 78.1%

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

   2. distribute-neg-frac 78.1%

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

   3. distribute-rgt1-in 78.9%

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

   4. associate-/l/ 78.8%

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

   5. sub-neg 78.8%

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

   6. +-commutative 78.8%

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

   7. distribute-neg-in 78.8%

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

   8. remove-double-neg 78.8%

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

   9. sub-neg 78.8%

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

   10. div-sub 78.0%

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

   11. associate-/l* 78.0%

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

   12. *-inverses 80.0%

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

   13. *-rgt-identity 80.0%

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

3. Simplified 80.0%

   \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0 85.9%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Alternative 12: 4.4% accurate, 313.0× speedup

Math

\[\begin{array}{l} \\ wj \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 wj)

C

double code(double wj, double x) {
	return wj;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = wj
end function

Java

public static double code(double wj, double x) {
	return wj;
}

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

function tmp = code(wj, x)
	tmp = wj;
end

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 78.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. sub-neg 78.1%

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

   2. distribute-neg-frac 78.1%

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

   3. distribute-rgt1-in 78.9%

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

   4. associate-/l/ 78.8%

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

   5. sub-neg 78.8%

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

   6. +-commutative 78.8%

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

   7. distribute-neg-in 78.8%

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

   8. remove-double-neg 78.8%

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

   9. sub-neg 78.8%

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

   10. div-sub 78.0%

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

   11. associate-/l* 78.0%

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

   12. *-inverses 80.0%

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

   13. *-rgt-identity 80.0%

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

3. Simplified 80.0%

   \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}} \]
4. Add Preprocessing

5. Taylor expanded in wj around inf 4.7%

   \[\leadsto \color{blue}{wj} \]
6. Add Preprocessing

Developer target: 79.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(wj, x)
	return Float64(wj - Float64(Float64(wj / Float64(wj + 1.0)) - Float64(x / Float64(exp(wj) + Float64(wj * exp(wj))))))
end

MATLAB

function tmp = code(wj, x)
	tmp = wj - ((wj / (wj + 1.0)) - (x / (exp(wj) + (wj * exp(wj)))));
end

Wolfram

code[wj_, x_] := N[(wj - N[(N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[Exp[wj], $MachinePrecision] + N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024107 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :alt
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

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