Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.2% → 98.1%

Time: 49.3s

Alternatives: 11

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -1956178807.2542515
  2. x = 2.0650332909263575e+124

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: 98.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot t_0 + x \cdot 0.6666666666666666\right)\right)\right) + {wj}^{2} \cdot \left(1 - t_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(\frac{\frac{x}{wj + 1}}{e^{wj}} - \frac{wj}{wj + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (<= wj 6.2e-6)
     (+
      x
      (+
       (* -2.0 (* wj x))
       (+
        (*
         (pow wj 3.0)
         (- -1.0 (+ (* x -3.0) (+ (* -2.0 t_0) (* x 0.6666666666666666)))))
        (* (pow wj 2.0) (- 1.0 t_0)))))
     (+ wj (- (/ (/ x (+ wj 1.0)) (exp wj)) (/ wj (+ wj 1.0)))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= 6.2e-6) {
		tmp = x + ((-2.0 * (wj * x)) + ((pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + (pow(wj, 2.0) * (1.0 - t_0))));
	} else {
		tmp = wj + (((x / (wj + 1.0)) / exp(wj)) - (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) :: tmp
    t_0 = (x * (-4.0d0)) + (x * 1.5d0)
    if (wj <= 6.2d-6) then
        tmp = x + (((-2.0d0) * (wj * x)) + (((wj ** 3.0d0) * ((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * t_0) + (x * 0.6666666666666666d0))))) + ((wj ** 2.0d0) * (1.0d0 - t_0))))
    else
        tmp = wj + (((x / (wj + 1.0d0)) / exp(wj)) - (wj / (wj + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= 6.2e-6) {
		tmp = x + ((-2.0 * (wj * x)) + ((Math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + (Math.pow(wj, 2.0) * (1.0 - t_0))));
	} else {
		tmp = wj + (((x / (wj + 1.0)) / Math.exp(wj)) - (wj / (wj + 1.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if wj <= 6.2e-6:
		tmp = x + ((-2.0 * (wj * x)) + ((math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + (math.pow(wj, 2.0) * (1.0 - t_0))))
	else:
		tmp = wj + (((x / (wj + 1.0)) / math.exp(wj)) - (wj / (wj + 1.0)))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (wj <= 6.2e-6)
		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_0) + Float64(x * 0.6666666666666666))))) + Float64((wj ^ 2.0) * Float64(1.0 - t_0)))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x / Float64(wj + 1.0)) / exp(wj)) - Float64(wj / Float64(wj + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if (wj <= 6.2e-6)
		tmp = x + ((-2.0 * (wj * x)) + (((wj ^ 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))) + ((wj ^ 2.0) * (1.0 - t_0))));
	else
		tmp = wj + (((x / (wj + 1.0)) / exp(wj)) - (wj / (wj + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, 6.2e-6], 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$0), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[wj, 2.0], $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if wj < 6.1999999999999999e-6

   1. Initial program 76.7%

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

      1. div-sub 76.7%

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

      2. distribute-rgt1-in 76.7%

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

      3. times-frac 76.7%

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

      4. *-inverses 76.7%

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

      5. associate-*l/ 76.7%

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

      6. *-rgt-identity 76.7%

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

      7. distribute-rgt1-in 77.1%

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

      8. associate-/l/ 77.1%

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

      9. div-sub 77.1%

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

   3. Simplified 77.1%

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

   4. Taylor expanded in wj around 0 98.4%

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

   if 6.1999999999999999e-6 < wj

   1. Initial program 55.0%

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

      1. div-sub 55.0%

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

      2. distribute-rgt1-in 55.0%

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

      3. times-frac 55.0%

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

      4. *-inverses 99.5%

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

      5. associate-*l/ 99.5%

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

      6. *-rgt-identity 99.5%

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

      7. distribute-rgt1-in 99.7%

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

      8. associate-/l/ 99.5%

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

      9. div-sub 99.5%

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

   3. Simplified 99.5%

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

      1. sub-div 99.5%

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

      2. associate-/l/ 99.7%

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

      3. associate-/r* 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(\frac{\frac{x}{wj + 1}}{e^{wj}} - \frac{wj}{wj + 1}\right)\\ \end{array} \]

Alternative 2: 97.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 5.4 \cdot 10^{-7}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + wj \cdot wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(\frac{\frac{x}{wj + 1}}{e^{wj}} - \frac{wj}{wj + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 5.4e-7)
   (+
    x
    (+
     (* -2.0 (* wj x))
     (+
      (*
       (pow wj 3.0)
       (-
        -1.0
        (+
         (* x -3.0)
         (+ (* -2.0 (+ (* x -4.0) (* x 1.5))) (* x 0.6666666666666666)))))
      (* wj wj))))
   (+ wj (- (/ (/ x (+ wj 1.0)) (exp wj)) (/ wj (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 5.4e-7) {
		tmp = x + ((-2.0 * (wj * x)) + ((pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666))))) + (wj * wj)));
	} else {
		tmp = wj + (((x / (wj + 1.0)) / exp(wj)) - (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) :: tmp
    if (wj <= 5.4d-7) then
        tmp = x + (((-2.0d0) * (wj * x)) + (((wj ** 3.0d0) * ((-1.0d0) - ((x * (-3.0d0)) + (((-2.0d0) * ((x * (-4.0d0)) + (x * 1.5d0))) + (x * 0.6666666666666666d0))))) + (wj * wj)))
    else
        tmp = wj + (((x / (wj + 1.0d0)) / exp(wj)) - (wj / (wj + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 5.4e-7) {
		tmp = x + ((-2.0 * (wj * x)) + ((Math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666))))) + (wj * wj)));
	} else {
		tmp = wj + (((x / (wj + 1.0)) / Math.exp(wj)) - (wj / (wj + 1.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 5.4e-7:
		tmp = x + ((-2.0 * (wj * x)) + ((math.pow(wj, 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666))))) + (wj * wj)))
	else:
		tmp = wj + (((x / (wj + 1.0)) / math.exp(wj)) - (wj / (wj + 1.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 5.4e-7)
		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 * Float64(Float64(x * -4.0) + Float64(x * 1.5))) + Float64(x * 0.6666666666666666))))) + Float64(wj * wj))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x / Float64(wj + 1.0)) / exp(wj)) - Float64(wj / Float64(wj + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 5.4e-7)
		tmp = x + ((-2.0 * (wj * x)) + (((wj ^ 3.0) * (-1.0 - ((x * -3.0) + ((-2.0 * ((x * -4.0) + (x * 1.5))) + (x * 0.6666666666666666))))) + (wj * wj)));
	else
		tmp = wj + (((x / (wj + 1.0)) / exp(wj)) - (wj / (wj + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 5.4e-7], 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 * N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 5.40000000000000018e-7

   1. Initial program 76.6%

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

      1. div-sub 76.6%

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

      2. distribute-rgt1-in 76.6%

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

      3. times-frac 76.6%

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

      4. *-inverses 76.6%

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

      5. associate-*l/ 76.6%

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

      6. *-rgt-identity 76.6%

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

      7. distribute-rgt1-in 77.0%

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

      8. associate-/l/ 77.0%

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

      9. div-sub 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in wj around 0 98.4%

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

   5. Taylor expanded in x around 0 98.4%

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

      1. unpow2 98.4%

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

   7. Simplified 98.4%

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

   if 5.40000000000000018e-7 < wj

   1. Initial program 59.5%

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

      1. div-sub 59.5%

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

      2. distribute-rgt1-in 59.5%

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

      3. times-frac 59.5%

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

      4. *-inverses 99.5%

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

      5. associate-*l/ 99.5%

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

      6. *-rgt-identity 99.5%

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

      7. distribute-rgt1-in 99.7%

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

      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. Step-by-step derivation

      1. sub-div 99.4%

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

      2. associate-/l/ 99.7%

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

      3. associate-/r* 99.7%

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

   5. Applied egg-rr 99.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 5.4 \cdot 10^{-7}:\\ \;\;\;\;x + \left(-2 \cdot \left(wj \cdot x\right) + \left({wj}^{3} \cdot \left(-1 - \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right) + wj \cdot wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(\frac{\frac{x}{wj + 1}}{e^{wj}} - \frac{wj}{wj + 1}\right)\\ \end{array} \]

Alternative 3: 97.2% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.5e-10)
   (+ x (* wj wj))
   (- wj (- (/ wj (+ wj 1.0)) (* x (/ (exp (- wj)) (+ wj 1.0)))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.5e-10) {
		tmp = x + (wj * wj);
	} else {
		tmp = wj - ((wj / (wj + 1.0)) - (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) :: tmp
    if (wj <= 2.5d-10) then
        tmp = x + (wj * wj)
    else
        tmp = wj - ((wj / (wj + 1.0d0)) - (x * (exp(-wj) / (wj + 1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 2.5e-10) {
		tmp = x + (wj * wj);
	} else {
		tmp = wj - ((wj / (wj + 1.0)) - (x * (Math.exp(-wj) / (wj + 1.0))));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 2.5e-10:
		tmp = x + (wj * wj)
	else:
		tmp = wj - ((wj / (wj + 1.0)) - (x * (math.exp(-wj) / (wj + 1.0))))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.5e-10)
		tmp = Float64(x + Float64(wj * wj));
	else
		tmp = Float64(wj - Float64(Float64(wj / Float64(wj + 1.0)) - Float64(x * Float64(exp(Float64(-wj)) / Float64(wj + 1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 2.5e-10)
		tmp = x + (wj * wj);
	else
		tmp = wj - ((wj / (wj + 1.0)) - (x * (exp(-wj) / (wj + 1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.5e-10], N[(x + N[(wj * wj), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] - N[(x * N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.50000000000000016e-10

   1. Initial program 76.6%

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

      1. div-sub 76.6%

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

      2. distribute-rgt1-in 76.6%

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

      3. times-frac 76.6%

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

      4. *-inverses 76.6%

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

      5. associate-*l/ 76.6%

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

      6. *-rgt-identity 76.6%

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

      7. distribute-rgt1-in 77.0%

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

      8. associate-/l/ 77.0%

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

      9. div-sub 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in wj around 0 98.4%

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

      1. +-commutative 98.4%

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

      2. fma-def 98.4%

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

      3. unpow2 98.4%

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

      4. distribute-rgt-out 98.4%

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

      5. metadata-eval 98.4%

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

      6. associate-*r* 98.4%

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

      7. *-commutative 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in x around 0 98.6%

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

      1. unpow2 98.6%

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

   9. Simplified 98.6%

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

   if 2.50000000000000016e-10 < wj

   1. Initial program 64.6%

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

      1. div-sub 64.6%

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

      2. distribute-rgt1-in 64.6%

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

      3. times-frac 64.9%

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

      4. *-inverses 93.5%

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

      5. associate-*l/ 93.5%

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

      6. *-rgt-identity 93.5%

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

      7. distribute-rgt1-in 93.5%

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

      8. associate-/l/ 93.2%

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

      9. div-sub 93.2%

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

   3. Simplified 93.2%

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

      1. div-sub 93.2%

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

      2. div-inv 92.9%

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

      3. div-inv 92.9%

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

      4. *-commutative 92.9%

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

      5. associate-*l* 92.9%

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

      6. div-inv 93.2%

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

      7. *-inverses 64.7%

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

      8. times-frac 64.4%

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

      9. distribute-rgt1-in 64.4%

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

      10. associate-/l/ 64.6%

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

      11. distribute-rgt1-in 64.6%

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

      12. div-sub 64.6%

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

      13. div-inv 64.1%

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

      14. *-commutative 64.1%

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

      15. sub-neg 64.1%

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

      16. distribute-lft-in 64.1%

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

   5. Applied egg-rr 93.6%

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

4. Final simplification 98.3%

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

Alternative 4: 97.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-11}:\\ \;\;\;\;x + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + \left(\frac{\frac{x}{wj + 1}}{e^{wj}} - \frac{wj}{wj + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 7.2e-11)
   (+ x (* wj wj))
   (+ wj (- (/ (/ x (+ wj 1.0)) (exp wj)) (/ wj (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 7.2e-11) {
		tmp = x + (wj * wj);
	} else {
		tmp = wj + (((x / (wj + 1.0)) / exp(wj)) - (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) :: tmp
    if (wj <= 7.2d-11) then
        tmp = x + (wj * wj)
    else
        tmp = wj + (((x / (wj + 1.0d0)) / exp(wj)) - (wj / (wj + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 7.2e-11) {
		tmp = x + (wj * wj);
	} else {
		tmp = wj + (((x / (wj + 1.0)) / Math.exp(wj)) - (wj / (wj + 1.0)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 7.2e-11:
		tmp = x + (wj * wj)
	else:
		tmp = wj + (((x / (wj + 1.0)) / math.exp(wj)) - (wj / (wj + 1.0)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 7.2e-11)
		tmp = Float64(x + Float64(wj * wj));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x / Float64(wj + 1.0)) / exp(wj)) - Float64(wj / Float64(wj + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 7.2e-11)
		tmp = x + (wj * wj);
	else
		tmp = wj + (((x / (wj + 1.0)) / exp(wj)) - (wj / (wj + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 7.2e-11], N[(x + N[(wj * wj), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 7.19999999999999969e-11

   1. Initial program 76.6%

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

      1. div-sub 76.6%

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

      2. distribute-rgt1-in 76.6%

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

      3. times-frac 76.6%

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

      4. *-inverses 76.6%

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

      5. associate-*l/ 76.6%

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

      6. *-rgt-identity 76.6%

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

      7. distribute-rgt1-in 77.0%

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

      8. associate-/l/ 77.0%

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

      9. div-sub 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in wj around 0 98.4%

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

      1. +-commutative 98.4%

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

      2. fma-def 98.4%

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

      3. unpow2 98.4%

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

      4. distribute-rgt-out 98.4%

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

      5. metadata-eval 98.4%

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

      6. associate-*r* 98.4%

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

      7. *-commutative 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in x around 0 98.6%

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

      1. unpow2 98.6%

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

   9. Simplified 98.6%

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

   if 7.19999999999999969e-11 < wj

   1. Initial program 64.6%

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

      1. div-sub 64.6%

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

      2. distribute-rgt1-in 64.6%

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

      3. times-frac 64.9%

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

      4. *-inverses 93.5%

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

      5. associate-*l/ 93.5%

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

      6. *-rgt-identity 93.5%

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

      7. distribute-rgt1-in 93.5%

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

      8. associate-/l/ 93.2%

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

      9. div-sub 93.2%

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

   3. Simplified 93.2%

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

      1. sub-div 93.2%

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

      2. associate-/l/ 93.5%

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

      3. associate-/r* 93.5%

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

   5. Applied egg-rr 93.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-11}:\\ \;\;\;\;x + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + \left(\frac{\frac{x}{wj + 1}}{e^{wj}} - \frac{wj}{wj + 1}\right)\\ \end{array} \]

Alternative 5: 97.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-11}:\\ \;\;\;\;x + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 7.2e-11)
   (+ x (* wj wj))
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 7.2e-11) {
		tmp = x + (wj * wj);
	} else {
		tmp = wj + (((x / exp(wj)) - 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) :: tmp
    if (wj <= 7.2d-11) then
        tmp = x + (wj * wj)
    else
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 7.2e-11) {
		tmp = x + (wj * wj);
	} else {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 7.2e-11:
		tmp = x + (wj * wj)
	else:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 7.2e-11)
		tmp = Float64(x + Float64(wj * wj));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 7.2e-11)
		tmp = x + (wj * wj);
	else
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 7.2e-11], N[(x + N[(wj * wj), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 7.19999999999999969e-11

   1. Initial program 76.6%

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

      1. div-sub 76.6%

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

      2. distribute-rgt1-in 76.6%

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

      3. times-frac 76.6%

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

      4. *-inverses 76.6%

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

      5. associate-*l/ 76.6%

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

      6. *-rgt-identity 76.6%

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

      7. distribute-rgt1-in 77.0%

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

      8. associate-/l/ 77.0%

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

      9. div-sub 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in wj around 0 98.4%

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

      1. +-commutative 98.4%

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

      2. fma-def 98.4%

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

      3. unpow2 98.4%

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

      4. distribute-rgt-out 98.4%

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

      5. metadata-eval 98.4%

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

      6. associate-*r* 98.4%

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

      7. *-commutative 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in x around 0 98.6%

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

      1. unpow2 98.6%

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

   9. Simplified 98.6%

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

   if 7.19999999999999969e-11 < wj

   1. Initial program 64.6%

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

      1. div-sub 64.6%

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

      2. distribute-rgt1-in 64.6%

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

      3. times-frac 64.9%

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

      4. *-inverses 93.5%

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

      5. associate-*l/ 93.5%

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

      6. *-rgt-identity 93.5%

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

      7. distribute-rgt1-in 93.5%

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

      8. associate-/l/ 93.2%

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

      9. div-sub 93.2%

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

   3. Simplified 93.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-11}:\\ \;\;\;\;x + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]

Alternative 6: 96.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;x + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\frac{x}{-1 - wj}}{e^{wj}}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 6.8e-7) (+ x (* wj wj)) (- wj (/ (/ x (- -1.0 wj)) (exp wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 6.8e-7) {
		tmp = x + (wj * wj);
	} else {
		tmp = wj - ((x / (-1.0 - wj)) / exp(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 <= 6.8d-7) then
        tmp = x + (wj * wj)
    else
        tmp = wj - ((x / ((-1.0d0) - wj)) / exp(wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 6.8e-7) {
		tmp = x + (wj * wj);
	} else {
		tmp = wj - ((x / (-1.0 - wj)) / Math.exp(wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 6.8e-7:
		tmp = x + (wj * wj)
	else:
		tmp = wj - ((x / (-1.0 - wj)) / math.exp(wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 6.8e-7)
		tmp = Float64(x + Float64(wj * wj));
	else
		tmp = Float64(wj - Float64(Float64(x / Float64(-1.0 - wj)) / exp(wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 6.8e-7)
		tmp = x + (wj * wj);
	else
		tmp = wj - ((x / (-1.0 - wj)) / exp(wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 6.8e-7], N[(x + N[(wj * wj), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(x / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision] / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 6.79999999999999948e-7

   1. Initial program 76.6%

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

      1. div-sub 76.6%

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

      2. distribute-rgt1-in 76.6%

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

      3. times-frac 76.6%

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

      4. *-inverses 76.6%

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

      5. associate-*l/ 76.6%

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

      6. *-rgt-identity 76.6%

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

      7. distribute-rgt1-in 77.0%

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

      8. associate-/l/ 77.0%

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

      9. div-sub 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in wj around 0 97.8%

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

      1. +-commutative 97.8%

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

      2. fma-def 97.8%

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

      3. unpow2 97.8%

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

      4. distribute-rgt-out 97.8%

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

      5. metadata-eval 97.8%

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

      6. associate-*r* 97.8%

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

      7. *-commutative 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in x around 0 97.9%

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

      1. unpow2 97.9%

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

   9. Simplified 97.9%

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

   if 6.79999999999999948e-7 < wj

   1. Initial program 59.5%

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

      1. div-sub 59.5%

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

      2. distribute-rgt1-in 59.5%

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

      3. times-frac 59.5%

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

      4. *-inverses 99.5%

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

      5. associate-*l/ 99.5%

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

      6. *-rgt-identity 99.5%

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

      7. distribute-rgt1-in 99.7%

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

      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 69.8%

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

      1. distribute-rgt-in 69.7%

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

      2. *-lft-identity 69.7%

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

      3. distribute-rgt1-in 69.8%

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

      4. *-commutative 69.8%

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

      5. associate-/r* 69.5%

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

      6. associate-*r/ 69.5%

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

      7. associate-*r/ 69.5%

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

      8. neg-mul-1 69.5%

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

   6. Simplified 69.5%

      \[\leadsto wj - \color{blue}{\frac{\frac{-x}{e^{wj}}}{wj + 1}} \]
   7. Step-by-step derivation

      1. neg-mul-1 69.5%

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

      2. associate-/l/ 69.8%

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

      3. neg-mul-1 69.8%

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

      4. neg-sub0 69.8%

         \[\leadsto wj - \frac{\color{blue}{0 - x}}{\left(wj + 1\right) \cdot e^{wj}} \]

      5. div-sub 69.8%

         \[\leadsto wj - \color{blue}{\left(\frac{0}{\left(wj + 1\right) \cdot e^{wj}} - \frac{x}{\left(wj + 1\right) \cdot e^{wj}}\right)} \]

      6. sub-neg 69.8%

         \[\leadsto wj - \color{blue}{\left(\frac{0}{\left(wj + 1\right) \cdot e^{wj}} + \left(-\frac{x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)} \]

      7. div0 69.8%

         \[\leadsto wj - \left(\color{blue}{0} + \left(-\frac{x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right) \]

      8. distribute-frac-neg 69.8%

         \[\leadsto wj - \left(0 + \color{blue}{\frac{-x}{\left(wj + 1\right) \cdot e^{wj}}}\right) \]

      9. neg-mul-1 69.8%

         \[\leadsto wj - \left(0 + \frac{\color{blue}{-1 \cdot x}}{\left(wj + 1\right) \cdot e^{wj}}\right) \]

      10. associate-/l/ 69.5%

          \[\leadsto wj - \left(0 + \color{blue}{\frac{\frac{-1 \cdot x}{e^{wj}}}{wj + 1}}\right) \]

      11. neg-mul-1 69.5%

          \[\leadsto wj - \left(0 + \frac{\frac{\color{blue}{-x}}{e^{wj}}}{wj + 1}\right) \]

      12. frac-2neg 69.5%

          \[\leadsto wj - \left(0 + \color{blue}{\frac{-\frac{-x}{e^{wj}}}{-\left(wj + 1\right)}}\right) \]

      13. distribute-frac-neg 69.5%

          \[\leadsto wj - \left(0 + \frac{-\color{blue}{\left(-\frac{x}{e^{wj}}\right)}}{-\left(wj + 1\right)}\right) \]

      14. remove-double-neg 69.5%

          \[\leadsto wj - \left(0 + \frac{\color{blue}{\frac{x}{e^{wj}}}}{-\left(wj + 1\right)}\right) \]

      15. +-commutative 69.5%

          \[\leadsto wj - \left(0 + \frac{\frac{x}{e^{wj}}}{-\color{blue}{\left(1 + wj\right)}}\right) \]

      16. distribute-neg-in 69.5%

          \[\leadsto wj - \left(0 + \frac{\frac{x}{e^{wj}}}{\color{blue}{\left(-1\right) + \left(-wj\right)}}\right) \]

      17. metadata-eval 69.5%

          \[\leadsto wj - \left(0 + \frac{\frac{x}{e^{wj}}}{\color{blue}{-1} + \left(-wj\right)}\right) \]

      18. unsub-neg 69.5%

          \[\leadsto wj - \left(0 + \frac{\frac{x}{e^{wj}}}{\color{blue}{-1 - wj}}\right) \]

   8. Applied egg-rr 69.5%

      \[\leadsto wj - \color{blue}{\left(0 + \frac{\frac{x}{e^{wj}}}{-1 - wj}\right)} \]
   9. Step-by-step derivation

      1. +-lft-identity 69.5%

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

      2. *-rgt-identity 69.5%

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

      3. associate-*l/ 69.5%

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

      4. associate-*r/ 69.5%

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

      5. associate-*l/ 69.8%

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

      6. associate-*r/ 69.8%

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

      7. *-rgt-identity 69.8%

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

   10. Simplified 69.8%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;x + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\frac{x}{-1 - wj}}{e^{wj}}\\ \end{array} \]

Alternative 7: 96.2% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.55:\\ \;\;\;\;x + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.55) (+ x (* wj wj)) (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.55) {
		tmp = x + (wj * wj);
	} else {
		tmp = wj - (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) :: tmp
    if (wj <= 1.55d0) then
        tmp = x + (wj * wj)
    else
        tmp = wj - (wj / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.55) {
		tmp = x + (wj * wj);
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.55:
		tmp = x + (wj * wj)
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.55)
		tmp = Float64(x + Float64(wj * wj));
	else
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.55)
		tmp = x + (wj * wj);
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.55], N[(x + N[(wj * wj), $MachinePrecision]), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.55000000000000004

   1. Initial program 76.9%

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

      1. div-sub 76.9%

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

      2. distribute-rgt1-in 76.9%

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

      3. times-frac 76.9%

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

      4. *-inverses 76.9%

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

      5. associate-*l/ 76.9%

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

      6. *-rgt-identity 76.9%

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

      7. distribute-rgt1-in 77.3%

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

      8. associate-/l/ 77.3%

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

      9. div-sub 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in wj around 0 97.2%

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

      1. +-commutative 97.2%

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

      2. fma-def 97.2%

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

      3. unpow2 97.2%

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

      4. distribute-rgt-out 97.2%

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

      5. metadata-eval 97.2%

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

      6. associate-*r* 97.2%

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

      7. *-commutative 97.2%

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

   6. Simplified 97.2%

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

   7. Taylor expanded in x around 0 97.0%

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

      1. unpow2 97.0%

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

   9. Simplified 97.0%

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

   if 1.55000000000000004 < wj

   1. Initial program 42.4%

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

      1. div-sub 42.4%

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

      2. distribute-rgt1-in 42.4%

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

      3. times-frac 42.4%

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

      4. *-inverses 99.6%

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

      5. associate-*l/ 99.6%

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

      6. *-rgt-identity 99.6%

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

      7. distribute-rgt1-in 99.8%

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

      8. associate-/l/ 99.6%

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

      9. div-sub 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 71.7%

      \[\leadsto \color{blue}{wj - \frac{wj}{1 + wj}} \]
   5. Step-by-step derivation

      1. +-commutative 71.7%

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

   6. Simplified 71.7%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.55:\\ \;\;\;\;x + wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 8: 83.9% accurate, 43.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.9 \cdot 10^{-71}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 3.1 \cdot 10^{-63}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.9e-71) x (if (<= wj 3.1e-63) (* wj wj) x)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.9e-71) {
		tmp = x;
	} else if (wj <= 3.1e-63) {
		tmp = wj * wj;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 2.9d-71) then
        tmp = x
    else if (wj <= 3.1d-63) then
        tmp = wj * wj
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 2.9e-71) {
		tmp = x;
	} else if (wj <= 3.1e-63) {
		tmp = wj * wj;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 2.9e-71:
		tmp = x
	elif wj <= 3.1e-63:
		tmp = wj * wj
	else:
		tmp = x
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.9e-71)
		tmp = x;
	elseif (wj <= 3.1e-63)
		tmp = Float64(wj * wj);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 2.9e-71)
		tmp = x;
	elseif (wj <= 3.1e-63)
		tmp = wj * wj;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.9e-71], x, If[LessEqual[wj, 3.1e-63], N[(wj * wj), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.8999999999999999e-71 or 3.09999999999999984e-63 < wj

   1. Initial program 77.6%

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

      1. div-sub 77.6%

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

      2. distribute-rgt1-in 77.6%

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

      3. times-frac 77.6%

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

      4. *-inverses 79.2%

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

      5. associate-*l/ 79.2%

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

      6. *-rgt-identity 79.2%

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

      7. distribute-rgt1-in 79.6%

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

      8. associate-/l/ 79.5%

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

      9. div-sub 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in wj around 0 84.8%

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

   if 2.8999999999999999e-71 < wj < 3.09999999999999984e-63

   1. Initial program 18.0%

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

      1. div-sub 18.0%

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

      2. distribute-rgt1-in 18.0%

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

      3. times-frac 18.0%

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

      4. *-inverses 18.0%

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

      5. associate-*l/ 18.0%

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

      6. *-rgt-identity 18.0%

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

      7. distribute-rgt1-in 18.0%

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

      8. associate-/l/ 18.0%

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

      9. div-sub 18.0%

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

   3. Simplified 18.0%

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

      1. clear-num 18.2%

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

      2. inv-pow 18.2%

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

   5. Applied egg-rr 18.2%

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

      1. unpow-1 18.2%

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

   7. Simplified 18.2%

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

   8. Taylor expanded in x around 0 4.3%

      \[\leadsto wj - \frac{1}{\color{blue}{\frac{1 + wj}{wj}}} \]
   9. Step-by-step derivation

      1. +-commutative 4.3%

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

   10. Simplified 4.3%

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

   11. Taylor expanded in wj around 0 86.0%

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

       1. unpow2 86.0%

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

   13. Simplified 86.0%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.9 \cdot 10^{-71}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 3.1 \cdot 10^{-63}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 95.4% 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 75.9%

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

   1. div-sub 75.9%

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

   2. distribute-rgt1-in 75.9%

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

   3. times-frac 75.9%

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

   4. *-inverses 77.5%

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

   5. associate-*l/ 77.5%

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

   6. *-rgt-identity 77.5%

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

   7. distribute-rgt1-in 77.9%

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

   8. associate-/l/ 77.9%

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

   9. div-sub 77.9%

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

3. Simplified 77.9%

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

4. Taylor expanded in wj around 0 94.8%

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

   1. +-commutative 94.8%

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

   2. fma-def 94.8%

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

   3. unpow2 94.8%

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

   4. distribute-rgt-out 94.8%

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

   5. metadata-eval 94.8%

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

   6. associate-*r* 94.8%

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

   7. *-commutative 94.8%

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

6. Simplified 94.8%

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

7. Taylor expanded in x around 0 94.6%

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

   1. unpow2 94.6%

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

9. Simplified 94.6%

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

10. Final simplification 94.6%

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

Alternative 10: 4.3% 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 75.9%

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

   1. div-sub 75.9%

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

   2. distribute-rgt1-in 75.9%

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

   3. times-frac 75.9%

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

   4. *-inverses 77.5%

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

   5. associate-*l/ 77.5%

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

   6. *-rgt-identity 77.5%

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

   7. distribute-rgt1-in 77.9%

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

   8. associate-/l/ 77.9%

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

   9. div-sub 77.9%

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

3. Simplified 77.9%

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

4. Taylor expanded in wj around inf 4.5%

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

5. Final simplification 4.5%

   \[\leadsto wj \]

Alternative 11: 84.5% 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 75.9%

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

   1. div-sub 75.9%

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

   2. distribute-rgt1-in 75.9%

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

   3. times-frac 75.9%

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

   4. *-inverses 77.5%

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

   5. associate-*l/ 77.5%

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

   6. *-rgt-identity 77.5%

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

   7. distribute-rgt1-in 77.9%

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

   8. associate-/l/ 77.9%

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

   9. div-sub 77.9%

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

3. Simplified 77.9%

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

4. Taylor expanded in wj around 0 83.0%

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

5. Final simplification 83.0%

   \[\leadsto x \]

Developer target: 79.0% 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 2023297 
(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))))))