Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.9% → 98.7%

Time: 8.6s

Alternatives: 7

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -2.9789748294540682e+243
  2. x = -1.704038461652682e+238

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: 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

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: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj + \frac{x - t_0}{e^{wj} + t_0} \leq 10^{-12}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t_1 + x \cdot 0.6666666666666666\right)\right)\right) + {wj}^{2} \cdot \left(1 - t_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))) (t_1 (+ (* x -4.0) (* x 1.5))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 1e-12)
     (+
      x
      (+
       (* -2.0 (* wj x))
       (+
        (*
         (pow wj 3.0)
         (- -1.0 (+ (* x -3.0) (+ (* -2.0 t_1) (* x 0.6666666666666666)))))
        (* (pow wj 2.0) (- 1.0 t_1)))))
     (- wj (/ (- wj (/ x (exp wj))) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double t_1 = (x * -4.0) + (x * 1.5);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1e-12) {
		tmp = x + ((-2.0 * (wj * x)) + ((pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))))) + (pow(wj, 2.0) * (1.0 - t_1))));
	} else {
		tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = wj * exp(wj)
    t_1 = (x * (-4.0d0)) + (x * 1.5d0)
    if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1d-12) then
        tmp = x + (((-2.0d0) * (wj * x)) + (((wj ** 3.0d0) * ((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * t_1) + (x * 0.6666666666666666d0))))) + ((wj ** 2.0d0) * (1.0d0 - t_1))))
    else
        tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double t_1 = (x * -4.0) + (x * 1.5);
	double tmp;
	if ((wj + ((x - t_0) / (Math.exp(wj) + t_0))) <= 1e-12) {
		tmp = x + ((-2.0 * (wj * x)) + ((Math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))))) + (Math.pow(wj, 2.0) * (1.0 - t_1))));
	} else {
		tmp = wj - ((wj - (x / Math.exp(wj))) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	t_1 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if (wj + ((x - t_0) / (math.exp(wj) + t_0))) <= 1e-12:
		tmp = x + ((-2.0 * (wj * x)) + ((math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))))) + (math.pow(wj, 2.0) * (1.0 - t_1))))
	else:
		tmp = wj - ((wj - (x / math.exp(wj))) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	t_1 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 1e-12)
		tmp = Float64(x + Float64(Float64(-2.0 * Float64(wj * x)) + Float64(Float64((wj ^ 3.0) * Float64(-1.0 - Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_1) + Float64(x * 0.6666666666666666))))) + Float64((wj ^ 2.0) * Float64(1.0 - t_1)))));
	else
		tmp = Float64(wj - Float64(Float64(wj - Float64(x / exp(wj))) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	t_1 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1e-12)
		tmp = x + ((-2.0 * (wj * x)) + (((wj ^ 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_1) + (x * 0.6666666666666666))))) + ((wj ^ 2.0) * (1.0 - t_1))));
	else
		tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-12], N[(x + N[(N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[wj, 3.0], $MachinePrecision] * N[(-1.0 - N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * t$95$1), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $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))))) < 9.9999999999999998e-13

   1. Initial program 65.0%

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

      1. div-sub 65.0%

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

      2. associate-/l* 65.0%

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

      3. distribute-rgt1-in 65.0%

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

      4. associate-/l* 65.0%

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

      5. *-inverses 65.0%

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

      6. /-rgt-identity 65.0%

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

      7. distribute-rgt1-in 65.6%

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

      8. associate-/l/ 65.5%

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

      9. div-sub 65.5%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in wj around 0 98.7%

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

   if 9.9999999999999998e-13 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 95.4%

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

      1. div-sub 95.4%

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

      2. associate-/l* 95.5%

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

      3. distribute-rgt1-in 95.5%

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

      4. associate-/l* 95.5%

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

      5. *-inverses 98.5%

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

      6. /-rgt-identity 98.5%

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

      7. distribute-rgt1-in 100.0%

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

      8. associate-/l/ 99.9%

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

      9. div-sub 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 10^{-12}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \]

Alternative 2: 97.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -6.2 \cdot 10^{-9}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(wj \cdot wj, 1 + x \cdot 2.5, wj \cdot \left(x \cdot -2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -6.2e-9)
   (- wj (/ (- wj (/ x (exp wj))) (+ wj 1.0)))
   (+ x (fma (* wj wj) (+ 1.0 (* x 2.5)) (* wj (* x -2.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -6.2e-9) {
		tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0));
	} else {
		tmp = x + fma((wj * wj), (1.0 + (x * 2.5)), (wj * (x * -2.0)));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -6.2e-9)
		tmp = Float64(wj - Float64(Float64(wj - Float64(x / exp(wj))) / Float64(wj + 1.0)));
	else
		tmp = Float64(x + fma(Float64(wj * wj), Float64(1.0 + Float64(x * 2.5)), Float64(wj * Float64(x * -2.0))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -6.2e-9], N[(wj - N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(wj * wj), $MachinePrecision] * N[(1.0 + N[(x * 2.5), $MachinePrecision]), $MachinePrecision] + N[(wj * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -6.2000000000000001e-9

   1. Initial program 71.4%

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

      1. div-sub 71.4%

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

      2. associate-/l* 71.8%

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

      3. distribute-rgt1-in 71.8%

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

      4. associate-/l* 71.8%

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

      5. *-inverses 71.8%

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

      6. /-rgt-identity 71.8%

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

      7. distribute-rgt1-in 89.8%

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

      8. associate-/l/ 89.8%

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

      9. div-sub 89.8%

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

   3. Simplified 89.8%

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

   if -6.2000000000000001e-9 < wj

   1. Initial program 73.0%

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

      1. div-sub 73.0%

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

      2. associate-/l* 73.1%

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

      3. distribute-rgt1-in 73.1%

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

      4. associate-/l* 73.1%

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

      5. *-inverses 73.9%

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

      6. /-rgt-identity 73.9%

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

      7. distribute-rgt1-in 73.9%

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

      8. associate-/l/ 73.9%

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

      9. div-sub 73.9%

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

   3. Simplified 73.9%

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

   4. Taylor expanded in wj around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. fma-def 98.2%

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

      3. unpow2 98.2%

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

      4. sub-neg 98.2%

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

      5. distribute-rgt-out 98.2%

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

      6. distribute-rgt-neg-in 98.2%

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

      7. metadata-eval 98.2%

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

      8. metadata-eval 98.2%

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

      9. *-commutative 98.2%

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

      10. associate-*l* 98.2%

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

   6. Simplified 98.2%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left(wj \cdot wj, 1 + x \cdot 2.5, wj \cdot \left(x \cdot -2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -6.2 \cdot 10^{-9}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(wj \cdot wj, 1 + x \cdot 2.5, wj \cdot \left(x \cdot -2\right)\right)\\ \end{array} \]

Alternative 3: 97.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.2 \cdot 10^{-11}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -2.2e-11)
   (- wj (/ (- wj (/ x (exp wj))) (+ wj 1.0)))
   (+ x (* wj wj))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -2.2e-11) {
		tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0));
	} else {
		tmp = x + (wj * 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 <= (-2.2d-11)) then
        tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0d0))
    else
        tmp = x + (wj * wj)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -2.2e-11) {
		tmp = wj - ((wj - (x / Math.exp(wj))) / (wj + 1.0));
	} else {
		tmp = x + (wj * wj);
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -2.2e-11:
		tmp = wj - ((wj - (x / math.exp(wj))) / (wj + 1.0))
	else:
		tmp = x + (wj * wj)
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -2.2e-11)
		tmp = Float64(wj - Float64(Float64(wj - Float64(x / exp(wj))) / Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64(wj * wj));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -2.2e-11)
		tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0));
	else
		tmp = x + (wj * wj);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -2.2e-11], N[(wj - N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * wj), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -2.2000000000000002e-11

   1. Initial program 73.8%

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

      1. div-sub 73.8%

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

      2. associate-/l* 74.2%

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

      3. distribute-rgt1-in 74.1%

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

      4. associate-/l* 74.1%

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

      5. *-inverses 74.1%

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

      6. /-rgt-identity 74.1%

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

      7. distribute-rgt1-in 90.7%

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

      8. associate-/l/ 90.5%

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

      9. div-sub 90.5%

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

   3. Simplified 90.5%

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

   if -2.2000000000000002e-11 < wj

   1. Initial program 72.9%

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

      1. div-sub 72.9%

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

      2. associate-/l* 72.9%

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

      3. distribute-rgt1-in 72.9%

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

      4. associate-/l* 72.9%

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

      5. *-inverses 73.8%

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

      6. /-rgt-identity 73.8%

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

      7. distribute-rgt1-in 73.8%

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

      8. associate-/l/ 73.8%

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

      9. div-sub 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in wj around 0 72.5%

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

      1. associate-*r* 72.5%

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

      2. neg-mul-1 72.5%

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

   6. Simplified 72.5%

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

   7. Taylor expanded in wj around 0 98.1%

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

      1. +-commutative 98.1%

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

      2. fma-def 98.1%

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

      3. unpow2 98.1%

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

      4. associate--l+ 98.1%

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

      5. cancel-sign-sub-inv 98.1%

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

      6. metadata-eval 98.1%

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

      7. *-lft-identity 98.1%

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

      8. sub-neg 98.1%

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

      9. mul-1-neg 98.1%

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

      10. distribute-rgt-out 98.1%

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

      11. metadata-eval 98.1%

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

   9. Simplified 98.1%

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

   10. Taylor expanded in x around 0 98.0%

       \[\leadsto x + \color{blue}{{wj}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 98.0%

          \[\leadsto x + \color{blue}{wj \cdot wj} \]

   12. Simplified 98.0%

       \[\leadsto x + \color{blue}{wj \cdot wj} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.2 \cdot 10^{-11}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot wj\\ \end{array} \]

Alternative 4: 96.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -2.8e-7) (/ x (* (exp wj) (+ wj 1.0))) (+ x (* wj wj))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -2.8e-7) {
		tmp = x / (exp(wj) * (wj + 1.0));
	} else {
		tmp = x + (wj * 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 <= (-2.8d-7)) then
        tmp = x / (exp(wj) * (wj + 1.0d0))
    else
        tmp = x + (wj * wj)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -2.8e-7) {
		tmp = x / (Math.exp(wj) * (wj + 1.0));
	} else {
		tmp = x + (wj * wj);
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -2.8e-7:
		tmp = x / (math.exp(wj) * (wj + 1.0))
	else:
		tmp = x + (wj * wj)
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -2.8e-7)
		tmp = Float64(x / Float64(exp(wj) * Float64(wj + 1.0)));
	else
		tmp = Float64(x + Float64(wj * wj));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -2.8e-7)
		tmp = x / (exp(wj) * (wj + 1.0));
	else
		tmp = x + (wj * wj);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -2.8e-7], N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(wj * wj), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -2.80000000000000019e-7

   1. Initial program 74.2%

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

      1. div-sub 74.2%

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

      2. associate-/l* 74.6%

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

      3. distribute-rgt1-in 74.6%

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

      4. associate-/l* 74.6%

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

      5. *-inverses 74.6%

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

      6. /-rgt-identity 74.6%

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

      7. distribute-rgt1-in 99.4%

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

      8. associate-/l/ 99.4%

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

      9. div-sub 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 87.4%

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

      1. +-commutative 87.4%

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

   6. Simplified 87.4%

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

   if -2.80000000000000019e-7 < wj

   1. Initial program 72.9%

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

      1. div-sub 72.9%

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

      2. associate-/l* 73.0%

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

      3. distribute-rgt1-in 72.9%

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

      4. associate-/l* 72.9%

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

      5. *-inverses 73.8%

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

      6. /-rgt-identity 73.8%

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

      7. distribute-rgt1-in 73.8%

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

      8. associate-/l/ 73.7%

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

      9. div-sub 73.7%

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

   3. Simplified 73.7%

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

   4. Taylor expanded in wj around 0 72.5%

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

      1. associate-*r* 72.5%

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

      2. neg-mul-1 72.5%

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

   6. Simplified 72.5%

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

   7. Taylor expanded in wj around 0 97.6%

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

      1. +-commutative 97.6%

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

      2. fma-def 97.6%

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

      3. unpow2 97.6%

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

      4. associate--l+ 97.6%

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

      5. cancel-sign-sub-inv 97.6%

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

      6. metadata-eval 97.6%

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

      7. *-lft-identity 97.6%

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

      8. sub-neg 97.6%

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

      9. mul-1-neg 97.6%

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

      10. distribute-rgt-out 97.6%

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

      11. metadata-eval 97.6%

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

   9. Simplified 97.6%

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

   10. Taylor expanded in x around 0 97.3%

       \[\leadsto x + \color{blue}{{wj}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 97.3%

          \[\leadsto x + \color{blue}{wj \cdot wj} \]

   12. Simplified 97.3%

       \[\leadsto x + \color{blue}{wj \cdot wj} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + wj \cdot wj\\ \end{array} \]

Alternative 5: 95.2% 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 72.9%

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

   1. div-sub 72.9%

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

   2. associate-/l* 73.0%

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

   3. distribute-rgt1-in 73.0%

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

   4. associate-/l* 73.0%

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

   5. *-inverses 73.8%

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

   6. /-rgt-identity 73.8%

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

   7. distribute-rgt1-in 74.6%

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

   8. associate-/l/ 74.5%

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

   9. div-sub 74.5%

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

3. Simplified 74.5%

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

4. Taylor expanded in wj around 0 71.9%

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

   1. associate-*r* 71.9%

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

   2. neg-mul-1 71.9%

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

6. Simplified 71.9%

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

7. Taylor expanded in wj around 0 95.9%

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

   1. +-commutative 95.9%

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

   2. fma-def 95.9%

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

   3. unpow2 95.9%

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

   4. associate--l+ 95.9%

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

   5. cancel-sign-sub-inv 95.9%

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

   6. metadata-eval 95.9%

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

   7. *-lft-identity 95.9%

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

   8. sub-neg 95.9%

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

   9. mul-1-neg 95.9%

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

   10. distribute-rgt-out 95.9%

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

   11. metadata-eval 95.9%

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

9. Simplified 95.9%

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

10. Taylor expanded in x around 0 95.1%

    \[\leadsto x + \color{blue}{{wj}^{2}} \]
11. Step-by-step derivation

    1. unpow2 95.1%

       \[\leadsto x + \color{blue}{wj \cdot wj} \]

12. Simplified 95.1%

    \[\leadsto x + \color{blue}{wj \cdot wj} \]

13. Final simplification 95.1%

    \[\leadsto x + wj \cdot wj \]

Alternative 6: 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 72.9%

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

   1. div-sub 72.9%

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

   2. associate-/l* 73.0%

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

   3. distribute-rgt1-in 73.0%

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

   4. associate-/l* 73.0%

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

   5. *-inverses 73.8%

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

   6. /-rgt-identity 73.8%

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

   7. distribute-rgt1-in 74.6%

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

   8. associate-/l/ 74.5%

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

   9. div-sub 74.5%

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

3. Simplified 74.5%

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

4. Taylor expanded in wj around inf 4.3%

   \[\leadsto \color{blue}{wj} \]

5. Final simplification 4.3%

   \[\leadsto wj \]

Alternative 7: 84.6% 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 72.9%

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

   1. div-sub 72.9%

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

   2. associate-/l* 73.0%

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

   3. distribute-rgt1-in 73.0%

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

   4. associate-/l* 73.0%

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

   5. *-inverses 73.8%

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

   6. /-rgt-identity 73.8%

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

   7. distribute-rgt1-in 74.6%

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

   8. associate-/l/ 74.5%

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

   9. div-sub 74.5%

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

3. Simplified 74.5%

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

4. Taylor expanded in wj around 0 80.9%

   \[\leadsto \color{blue}{x} \]

5. Final simplification 80.9%

   \[\leadsto x \]

Developer target: 78.9% 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 2023268 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

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