Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.7% → 99.5%

Time: 10.1s

Alternatives: 12

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -1.2909651177491228e+203
  2. x = 1.82625818702026e-222

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: 78.7% 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: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := wj + \frac{x - t_0}{e^{wj} + t_0}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot -3\right) - x \cdot 0.6666666666666666\right) + \left(wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))) (t_1 (+ wj (/ (- x t_0) (+ (exp wj) t_0)))))
   (if (<= t_1 -2e+57)
     (/ (/ x (exp wj)) (+ wj 1.0))
     (if (<= t_1 2e-14)
       (+
        (*
         (pow wj 3.0)
         (-
          (- (- -1.0 (* -2.0 (+ (* x -4.0) (* x 1.5)))) (* x -3.0))
          (* x 0.6666666666666666)))
        (+ (* wj wj) (+ x (* -2.0 (* wj x)))))
       (+ wj (/ (- (* x (exp (- wj))) wj) (+ wj 1.0)))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double t_1 = wj + ((x - t_0) / (exp(wj) + t_0));
	double tmp;
	if (t_1 <= -2e+57) {
		tmp = (x / exp(wj)) / (wj + 1.0);
	} else if (t_1 <= 2e-14) {
		tmp = (pow(wj, 3.0) * (((-1.0 - (-2.0 * ((x * -4.0) + (x * 1.5)))) - (x * -3.0)) - (x * 0.6666666666666666))) + ((wj * wj) + (x + (-2.0 * (wj * x))));
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = wj * exp(wj)
    t_1 = wj + ((x - t_0) / (exp(wj) + t_0))
    if (t_1 <= (-2d+57)) then
        tmp = (x / exp(wj)) / (wj + 1.0d0)
    else if (t_1 <= 2d-14) then
        tmp = ((wj ** 3.0d0) * ((((-1.0d0) - ((-2.0d0) * ((x * (-4.0d0)) + (x * 1.5d0)))) - (x * (-3.0d0))) - (x * 0.6666666666666666d0))) + ((wj * wj) + (x + ((-2.0d0) * (wj * x))))
    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 t_0 = wj * Math.exp(wj);
	double t_1 = wj + ((x - t_0) / (Math.exp(wj) + t_0));
	double tmp;
	if (t_1 <= -2e+57) {
		tmp = (x / Math.exp(wj)) / (wj + 1.0);
	} else if (t_1 <= 2e-14) {
		tmp = (Math.pow(wj, 3.0) * (((-1.0 - (-2.0 * ((x * -4.0) + (x * 1.5)))) - (x * -3.0)) - (x * 0.6666666666666666))) + ((wj * wj) + (x + (-2.0 * (wj * x))));
	} else {
		tmp = wj + (((x * Math.exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	t_1 = wj + ((x - t_0) / (math.exp(wj) + t_0))
	tmp = 0
	if t_1 <= -2e+57:
		tmp = (x / math.exp(wj)) / (wj + 1.0)
	elif t_1 <= 2e-14:
		tmp = (math.pow(wj, 3.0) * (((-1.0 - (-2.0 * ((x * -4.0) + (x * 1.5)))) - (x * -3.0)) - (x * 0.6666666666666666))) + ((wj * wj) + (x + (-2.0 * (wj * x))))
	else:
		tmp = wj + (((x * math.exp(-wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	t_1 = Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0)))
	tmp = 0.0
	if (t_1 <= -2e+57)
		tmp = Float64(Float64(x / exp(wj)) / Float64(wj + 1.0));
	elseif (t_1 <= 2e-14)
		tmp = Float64(Float64((wj ^ 3.0) * Float64(Float64(Float64(-1.0 - Float64(-2.0 * Float64(Float64(x * -4.0) + Float64(x * 1.5)))) - Float64(x * -3.0)) - Float64(x * 0.6666666666666666))) + Float64(Float64(wj * wj) + Float64(x + Float64(-2.0 * Float64(wj * x)))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x * exp(Float64(-wj))) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	t_1 = wj + ((x - t_0) / (exp(wj) + t_0));
	tmp = 0.0;
	if (t_1 <= -2e+57)
		tmp = (x / exp(wj)) / (wj + 1.0);
	elseif (t_1 <= 2e-14)
		tmp = ((wj ^ 3.0) * (((-1.0 - (-2.0 * ((x * -4.0) + (x * 1.5)))) - (x * -3.0)) - (x * 0.6666666666666666))) + ((wj * wj) + (x + (-2.0 * (wj * x))));
	else
		tmp = wj + (((x * exp(-wj)) - 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[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+57], N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-14], N[(N[(N[Power[wj, 3.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[(-2.0 * N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * -3.0), $MachinePrecision]), $MachinePrecision] - N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(wj * wj), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < -2.0000000000000001e57

   1. Initial program 96.7%

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

      1. sub-neg 96.7%

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

      2. div-sub 96.7%

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

      3. sub-neg 96.7%

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

      4. +-commutative 96.7%

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

      5. distribute-neg-in 96.7%

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

      6. remove-double-neg 96.7%

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

      7. sub-neg 96.7%

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

      8. div-sub 96.7%

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

      9. distribute-rgt1-in 100.0%

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

      10. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. div-inv 99.9%

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

      3. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. associate-/l/ 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   if -2.0000000000000001e57 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 2e-14

   1. Initial program 64.3%

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

      1. sub-neg 64.3%

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

      2. div-sub 64.3%

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

      3. sub-neg 64.3%

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

      4. +-commutative 64.3%

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

      5. distribute-neg-in 64.3%

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

      6. remove-double-neg 64.3%

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

      7. sub-neg 64.3%

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

      8. div-sub 64.3%

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

      9. distribute-rgt1-in 64.2%

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

      10. associate-/l/ 64.4%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in wj around 0 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. unpow2 99.9%

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

   7. Simplified 99.9%

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

   if 2e-14 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 93.8%

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

      1. sub-neg 93.8%

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

      2. div-sub 93.8%

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

      3. sub-neg 93.8%

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

      4. +-commutative 93.8%

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

      5. distribute-neg-in 93.8%

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

      6. remove-double-neg 93.8%

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

      7. sub-neg 93.8%

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

      8. div-sub 93.8%

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

      9. distribute-rgt1-in 93.9%

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

      10. associate-/l/ 93.8%

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

   3. Simplified 97.9%

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

      1. clear-num 97.6%

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

      2. associate-/r/ 97.9%

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

      3. rec-exp 97.9%

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

   5. Applied egg-rr 97.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq -2 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{elif}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot -3\right) - x \cdot 0.6666666666666666\right) + \left(wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \]

Alternative 2: 99.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -3.15 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -3.15e-9)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (if (<= wj 7.2e-9)
     (+ (* wj wj) (+ x (* -2.0 (* wj x))))
     (+ wj (/ (- (* x (exp (- wj))) wj) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -3.15e-9) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else if (wj <= 7.2e-9) {
		tmp = (wj * wj) + (x + (-2.0 * (wj * x)));
	} 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 <= (-3.15d-9)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else if (wj <= 7.2d-9) then
        tmp = (wj * wj) + (x + ((-2.0d0) * (wj * x)))
    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 <= -3.15e-9) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else if (wj <= 7.2e-9) {
		tmp = (wj * wj) + (x + (-2.0 * (wj * x)));
	} else {
		tmp = wj + (((x * Math.exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -3.15e-9:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	elif wj <= 7.2e-9:
		tmp = (wj * wj) + (x + (-2.0 * (wj * x)))
	else:
		tmp = wj + (((x * math.exp(-wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -3.15e-9)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	elseif (wj <= 7.2e-9)
		tmp = Float64(Float64(wj * wj) + Float64(x + Float64(-2.0 * Float64(wj * x))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x * exp(Float64(-wj))) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -3.15e-9)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	elseif (wj <= 7.2e-9)
		tmp = (wj * wj) + (x + (-2.0 * (wj * x)));
	else
		tmp = wj + (((x * exp(-wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -3.15e-9], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 7.2e-9], N[(N[(wj * wj), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if wj < -3.1500000000000001e-9

   1. Initial program 60.2%

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

      1. sub-neg 60.2%

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

      2. div-sub 60.2%

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

      3. sub-neg 60.2%

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

      4. +-commutative 60.2%

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

      5. distribute-neg-in 60.2%

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

      6. remove-double-neg 60.2%

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

      7. sub-neg 60.2%

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

      8. div-sub 60.2%

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

      9. distribute-rgt1-in 86.5%

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

      10. associate-/l/ 86.5%

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

   3. Simplified 86.5%

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

   if -3.1500000000000001e-9 < wj < 7.2e-9

   1. Initial program 82.3%

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

      1. distribute-rgt1-in 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in wj around 0 82.3%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{1 + 2 \cdot wj}} \]
   5. Step-by-step derivation

      1. count-2 82.3%

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

   6. Simplified 82.3%

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

   7. Taylor expanded in wj around 0 99.5%

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

   8. Taylor expanded in x around 0 99.9%

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

      1. unpow2 99.9%

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

   10. Simplified 99.9%

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

   if 7.2e-9 < wj

   1. Initial program 62.1%

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

      1. sub-neg 62.1%

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

      2. div-sub 62.1%

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

      3. sub-neg 62.1%

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

      4. +-commutative 62.1%

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

      5. distribute-neg-in 62.1%

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

      6. remove-double-neg 62.1%

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

      7. sub-neg 62.1%

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

      8. div-sub 62.1%

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

      9. distribute-rgt1-in 61.9%

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

      10. associate-/l/ 61.7%

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

   3. Simplified 91.7%

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

      1. clear-num 91.7%

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

      2. associate-/r/ 91.7%

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

      3. rec-exp 91.8%

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

   5. Applied egg-rr 91.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.15 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \]

Alternative 3: 99.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.9 \cdot 10^{-9} \lor \neg \left(wj \leq 8.2 \cdot 10^{-9}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -2.9e-9) (not (<= wj 8.2e-9)))
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+ (* wj wj) (+ x (* -2.0 (* wj x))))))

C

double code(double wj, double x) {
	double tmp;
	if ((wj <= -2.9e-9) || !(wj <= 8.2e-9)) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = (wj * wj) + (x + (-2.0 * (wj * 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-9)) .or. (.not. (wj <= 8.2d-9))) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = (wj * wj) + (x + ((-2.0d0) * (wj * x)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((wj <= -2.9e-9) || !(wj <= 8.2e-9)) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = (wj * wj) + (x + (-2.0 * (wj * x)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (wj <= -2.9e-9) or not (wj <= 8.2e-9):
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = (wj * wj) + (x + (-2.0 * (wj * x)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((wj <= -2.9e-9) || !(wj <= 8.2e-9))
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(Float64(wj * wj) + Float64(x + Float64(-2.0 * Float64(wj * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((wj <= -2.9e-9) || ~((wj <= 8.2e-9)))
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = (wj * wj) + (x + (-2.0 * (wj * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[wj, -2.9e-9], N[Not[LessEqual[wj, 8.2e-9]], $MachinePrecision]], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(wj * wj), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -2.89999999999999991e-9 or 8.2000000000000006e-9 < wj

   1. Initial program 61.3%

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

      1. sub-neg 61.3%

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

      2. div-sub 61.3%

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

      3. sub-neg 61.3%

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

      4. +-commutative 61.3%

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

      5. distribute-neg-in 61.3%

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

      6. remove-double-neg 61.3%

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

      7. sub-neg 61.3%

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

      8. div-sub 61.3%

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

      9. distribute-rgt1-in 72.8%

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

      10. associate-/l/ 72.7%

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

   3. Simplified 89.4%

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

   if -2.89999999999999991e-9 < wj < 8.2000000000000006e-9

   1. Initial program 82.3%

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

      1. distribute-rgt1-in 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in wj around 0 82.3%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{1 + 2 \cdot wj}} \]
   5. Step-by-step derivation

      1. count-2 82.3%

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

   6. Simplified 82.3%

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

   7. Taylor expanded in wj around 0 99.5%

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

   8. Taylor expanded in x around 0 99.9%

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

      1. unpow2 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.9 \cdot 10^{-9} \lor \neg \left(wj \leq 8.2 \cdot 10^{-9}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \end{array} \]

Alternative 4: 96.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-37}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x -1.35e-37)
   (+ wj (/ (/ x (exp wj)) (+ wj 1.0)))
   (+ (* wj wj) (+ x (* -2.0 (* wj x))))))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -1.35e-37) {
		tmp = wj + ((x / exp(wj)) / (wj + 1.0));
	} else {
		tmp = (wj * wj) + (x + (-2.0 * (wj * x)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.35d-37)) then
        tmp = wj + ((x / exp(wj)) / (wj + 1.0d0))
    else
        tmp = (wj * wj) + (x + ((-2.0d0) * (wj * x)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -1.35e-37) {
		tmp = wj + ((x / Math.exp(wj)) / (wj + 1.0));
	} else {
		tmp = (wj * wj) + (x + (-2.0 * (wj * x)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -1.35e-37:
		tmp = wj + ((x / math.exp(wj)) / (wj + 1.0))
	else:
		tmp = (wj * wj) + (x + (-2.0 * (wj * x)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -1.35e-37)
		tmp = Float64(wj + Float64(Float64(x / exp(wj)) / Float64(wj + 1.0)));
	else
		tmp = Float64(Float64(wj * wj) + Float64(x + Float64(-2.0 * Float64(wj * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -1.35e-37)
		tmp = wj + ((x / exp(wj)) / (wj + 1.0));
	else
		tmp = (wj * wj) + (x + (-2.0 * (wj * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -1.35e-37], N[(wj + N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(wj * wj), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000008e-37

   1. Initial program 95.9%

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

      1. distribute-rgt1-in 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in x around inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. neg-mul-1 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-/r* 99.5%

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

   6. Simplified 99.5%

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

   if -1.35000000000000008e-37 < x

   1. Initial program 74.5%

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

      1. distribute-rgt1-in 74.6%

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

   3. Simplified 74.6%

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

   4. Taylor expanded in wj around 0 72.2%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{1 + 2 \cdot wj}} \]
   5. Step-by-step derivation

      1. count-2 72.2%

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

   6. Simplified 72.2%

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

   7. Taylor expanded in wj around 0 94.9%

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

   8. Taylor expanded in x around 0 95.3%

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

      1. unpow2 95.3%

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

   10. Simplified 95.3%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-37}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \end{array} \]

Alternative 5: 98.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.00082:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{elif}\;wj \leq 6 \cdot 10^{-6}:\\ \;\;\;\;wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.00082)
   (/ x (* (exp wj) (+ wj 1.0)))
   (if (<= wj 6e-6)
     (+ (* wj wj) (+ x (* -2.0 (* wj x))))
     (- wj (/ wj (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.00082) {
		tmp = x / (exp(wj) * (wj + 1.0));
	} else if (wj <= 6e-6) {
		tmp = (wj * wj) + (x + (-2.0 * (wj * x)));
	} 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 <= (-0.00082d0)) then
        tmp = x / (exp(wj) * (wj + 1.0d0))
    else if (wj <= 6d-6) then
        tmp = (wj * wj) + (x + ((-2.0d0) * (wj * x)))
    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 <= -0.00082) {
		tmp = x / (Math.exp(wj) * (wj + 1.0));
	} else if (wj <= 6e-6) {
		tmp = (wj * wj) + (x + (-2.0 * (wj * x)));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -0.00082:
		tmp = x / (math.exp(wj) * (wj + 1.0))
	elif wj <= 6e-6:
		tmp = (wj * wj) + (x + (-2.0 * (wj * x)))
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.00082)
		tmp = Float64(x / Float64(exp(wj) * Float64(wj + 1.0)));
	elseif (wj <= 6e-6)
		tmp = Float64(Float64(wj * wj) + Float64(x + Float64(-2.0 * Float64(wj * x))));
	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 <= -0.00082)
		tmp = x / (exp(wj) * (wj + 1.0));
	elseif (wj <= 6e-6)
		tmp = (wj * wj) + (x + (-2.0 * (wj * x)));
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.00082], N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 6e-6], N[(N[(wj * wj), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -8.1999999999999998e-4

   1. Initial program 39.4%

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

      1. sub-neg 39.4%

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

      2. div-sub 39.4%

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

      3. sub-neg 39.4%

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

      4. +-commutative 39.4%

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

      5. distribute-neg-in 39.4%

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

      6. remove-double-neg 39.4%

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

      7. sub-neg 39.4%

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

      8. div-sub 39.4%

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

      9. distribute-rgt1-in 81.4%

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

      10. associate-/l/ 81.4%

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

   3. Simplified 81.4%

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

   4. Taylor expanded in x around inf 81.4%

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

   if -8.1999999999999998e-4 < wj < 6.0000000000000002e-6

   1. Initial program 82.4%

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

      1. distribute-rgt1-in 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in wj around 0 82.0%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{1 + 2 \cdot wj}} \]
   5. Step-by-step derivation

      1. count-2 82.0%

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

   6. Simplified 82.0%

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

   7. Taylor expanded in wj around 0 98.9%

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

   8. Taylor expanded in x around 0 99.2%

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

      1. unpow2 99.2%

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

   10. Simplified 99.2%

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

   if 6.0000000000000002e-6 < wj

   1. Initial program 56.8%

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

      1. sub-neg 56.8%

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

      2. div-sub 56.8%

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

      3. sub-neg 56.8%

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

      4. +-commutative 56.8%

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

      5. distribute-neg-in 56.8%

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

      6. remove-double-neg 56.8%

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

      7. sub-neg 56.8%

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

      8. div-sub 56.8%

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

      9. distribute-rgt1-in 57.0%

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

      10. associate-/l/ 56.7%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in x around 0 69.4%

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

      1. +-commutative 69.4%

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

   6. Simplified 69.4%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.00082:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{elif}\;wj \leq 6 \cdot 10^{-6}:\\ \;\;\;\;wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 6: 97.0% accurate, 24.0× speedup

Math

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

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 6e-6)
   (+ (* wj wj) (+ x (* -2.0 (* wj x))))
   (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 6e-6) {
		tmp = (wj * wj) + (x + (-2.0 * (wj * x)));
	} 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 <= 6d-6) then
        tmp = (wj * wj) + (x + ((-2.0d0) * (wj * x)))
    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 <= 6e-6) {
		tmp = (wj * wj) + (x + (-2.0 * (wj * x)));
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 6e-6:
		tmp = (wj * wj) + (x + (-2.0 * (wj * x)))
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 6e-6)
		tmp = Float64(Float64(wj * wj) + Float64(x + Float64(-2.0 * Float64(wj * x))));
	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 <= 6e-6)
		tmp = (wj * wj) + (x + (-2.0 * (wj * x)));
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 6e-6], N[(N[(wj * wj), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 6.0000000000000002e-6

   1. Initial program 81.6%

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

      1. distribute-rgt1-in 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in wj around 0 80.5%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{1 + 2 \cdot wj}} \]
   5. Step-by-step derivation

      1. count-2 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in wj around 0 96.9%

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

   8. Taylor expanded in x around 0 97.2%

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

      1. unpow2 97.2%

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

   10. Simplified 97.2%

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

   if 6.0000000000000002e-6 < wj

   1. Initial program 56.8%

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

      1. sub-neg 56.8%

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

      2. div-sub 56.8%

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

      3. sub-neg 56.8%

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

      4. +-commutative 56.8%

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

      5. distribute-neg-in 56.8%

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

      6. remove-double-neg 56.8%

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

      7. sub-neg 56.8%

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

      8. div-sub 56.8%

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

      9. distribute-rgt1-in 57.0%

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

      10. associate-/l/ 56.7%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in x around 0 69.4%

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

      1. +-commutative 69.4%

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

   6. Simplified 69.4%

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

4. Final simplification 96.4%

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

Alternative 7: 86.0% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 3.1 \cdot 10^{-30}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{elif}\;wj \leq 9.8 \cdot 10^{-9}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 3.1e-30)
   (+ x (* -2.0 (* wj x)))
   (if (<= wj 9.8e-9) (* wj wj) (- wj (/ wj (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 3.1e-30) {
		tmp = x + (-2.0 * (wj * x));
	} else if (wj <= 9.8e-9) {
		tmp = 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 <= 3.1d-30) then
        tmp = x + ((-2.0d0) * (wj * x))
    else if (wj <= 9.8d-9) then
        tmp = 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 <= 3.1e-30) {
		tmp = x + (-2.0 * (wj * x));
	} else if (wj <= 9.8e-9) {
		tmp = wj * wj;
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 3.1e-30:
		tmp = x + (-2.0 * (wj * x))
	elif wj <= 9.8e-9:
		tmp = wj * wj
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 3.1e-30)
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	elseif (wj <= 9.8e-9)
		tmp = 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 <= 3.1e-30)
		tmp = x + (-2.0 * (wj * x));
	elseif (wj <= 9.8e-9)
		tmp = wj * wj;
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 3.1e-30], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 9.8e-9], N[(wj * wj), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < 3.09999999999999991e-30

   1. Initial program 83.8%

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

      1. sub-neg 83.8%

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

      2. div-sub 83.8%

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

      3. sub-neg 83.8%

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

      4. +-commutative 83.8%

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

      5. distribute-neg-in 83.8%

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

      6. remove-double-neg 83.8%

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

      7. sub-neg 83.8%

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

      8. div-sub 83.8%

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

      9. distribute-rgt1-in 84.7%

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

      10. associate-/l/ 84.8%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in wj around 0 92.2%

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

   if 3.09999999999999991e-30 < wj < 9.80000000000000007e-9

   1. Initial program 27.9%

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

      1. distribute-rgt1-in 27.9%

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

   3. Simplified 27.9%

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

   4. Taylor expanded in wj around 0 28.0%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{1 + 2 \cdot wj}} \]
   5. Step-by-step derivation

      1. count-2 28.0%

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

   6. Simplified 28.0%

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

   7. Taylor expanded in x around 0 8.9%

      \[\leadsto \color{blue}{wj - \frac{e^{wj} \cdot wj}{1 + 2 \cdot wj}} \]
   8. Step-by-step derivation

      1. associate-/l* 9.8%

         \[\leadsto wj - \color{blue}{\frac{e^{wj}}{\frac{1 + 2 \cdot wj}{wj}}} \]

      2. count-2 9.8%

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

   9. Simplified 9.8%

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

   10. Taylor expanded in wj around 0 77.6%

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

       1. unpow2 77.6%

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

   12. Simplified 77.6%

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

   if 9.80000000000000007e-9 < wj

   1. Initial program 62.1%

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

      1. sub-neg 62.1%

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

      2. div-sub 62.1%

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

      3. sub-neg 62.1%

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

      4. +-commutative 62.1%

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

      5. distribute-neg-in 62.1%

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

      6. remove-double-neg 62.1%

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

      7. sub-neg 62.1%

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

      8. div-sub 62.1%

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

      9. distribute-rgt1-in 61.9%

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

      10. associate-/l/ 61.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in x around 0 62.4%

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

      1. +-commutative 62.4%

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

   6. Simplified 62.4%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 3.1 \cdot 10^{-30}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{elif}\;wj \leq 9.8 \cdot 10^{-9}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 8: 86.0% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 3.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{1 + \left(wj + wj\right)}\\ \mathbf{elif}\;wj \leq 9.8 \cdot 10^{-9}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 3.5e-30)
   (/ x (+ 1.0 (+ wj wj)))
   (if (<= wj 9.8e-9) (* wj wj) (- wj (/ wj (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 3.5e-30) {
		tmp = x / (1.0 + (wj + wj));
	} else if (wj <= 9.8e-9) {
		tmp = 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 <= 3.5d-30) then
        tmp = x / (1.0d0 + (wj + wj))
    else if (wj <= 9.8d-9) then
        tmp = 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 <= 3.5e-30) {
		tmp = x / (1.0 + (wj + wj));
	} else if (wj <= 9.8e-9) {
		tmp = wj * wj;
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 3.5e-30:
		tmp = x / (1.0 + (wj + wj))
	elif wj <= 9.8e-9:
		tmp = wj * wj
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 3.5e-30)
		tmp = Float64(x / Float64(1.0 + Float64(wj + wj)));
	elseif (wj <= 9.8e-9)
		tmp = 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 <= 3.5e-30)
		tmp = x / (1.0 + (wj + wj));
	elseif (wj <= 9.8e-9)
		tmp = wj * wj;
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 3.5e-30], N[(x / N[(1.0 + N[(wj + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 9.8e-9], N[(wj * wj), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < 3.5000000000000003e-30

   1. Initial program 83.8%

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

      1. distribute-rgt1-in 84.7%

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

   3. Simplified 84.7%

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

   4. Taylor expanded in wj around 0 82.8%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{1 + 2 \cdot wj}} \]
   5. Step-by-step derivation

      1. count-2 82.8%

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

   6. Simplified 82.8%

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

   7. Taylor expanded in x around inf 92.3%

      \[\leadsto \color{blue}{\frac{x}{1 + 2 \cdot wj}} \]
   8. Step-by-step derivation

      1. count-2 92.3%

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

   9. Simplified 92.3%

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

   if 3.5000000000000003e-30 < wj < 9.80000000000000007e-9

   1. Initial program 27.9%

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

      1. distribute-rgt1-in 27.9%

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

   3. Simplified 27.9%

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

   4. Taylor expanded in wj around 0 28.0%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{1 + 2 \cdot wj}} \]
   5. Step-by-step derivation

      1. count-2 28.0%

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

   6. Simplified 28.0%

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

   7. Taylor expanded in x around 0 8.9%

      \[\leadsto \color{blue}{wj - \frac{e^{wj} \cdot wj}{1 + 2 \cdot wj}} \]
   8. Step-by-step derivation

      1. associate-/l* 9.8%

         \[\leadsto wj - \color{blue}{\frac{e^{wj}}{\frac{1 + 2 \cdot wj}{wj}}} \]

      2. count-2 9.8%

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

   9. Simplified 9.8%

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

   10. Taylor expanded in wj around 0 77.6%

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

       1. unpow2 77.6%

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

   12. Simplified 77.6%

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

   if 9.80000000000000007e-9 < wj

   1. Initial program 62.1%

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

      1. sub-neg 62.1%

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

      2. div-sub 62.1%

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

      3. sub-neg 62.1%

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

      4. +-commutative 62.1%

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

      5. distribute-neg-in 62.1%

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

      6. remove-double-neg 62.1%

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

      7. sub-neg 62.1%

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

      8. div-sub 62.1%

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

      9. distribute-rgt1-in 61.9%

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

      10. associate-/l/ 61.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in x around 0 62.4%

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

      1. +-commutative 62.4%

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

   6. Simplified 62.4%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 3.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{1 + \left(wj + wj\right)}\\ \mathbf{elif}\;wj \leq 9.8 \cdot 10^{-9}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 9: 84.7% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.3 \cdot 10^{-30}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.3e-30) (+ x (* -2.0 (* wj x))) (* wj wj)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.3e-30) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = 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 <= 2.3d-30) then
        tmp = x + ((-2.0d0) * (wj * x))
    else
        tmp = wj * wj
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 2.3e-30) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 2.3e-30:
		tmp = x + (-2.0 * (wj * x))
	else:
		tmp = wj * wj
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.3e-30)
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	else
		tmp = Float64(wj * wj);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 2.3e-30)
		tmp = x + (-2.0 * (wj * x));
	else
		tmp = wj * wj;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.3e-30], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj * wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.29999999999999984e-30

   1. Initial program 83.8%

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

      1. sub-neg 83.8%

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

      2. div-sub 83.8%

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

      3. sub-neg 83.8%

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

      4. +-commutative 83.8%

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

      5. distribute-neg-in 83.8%

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

      6. remove-double-neg 83.8%

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

      7. sub-neg 83.8%

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

      8. div-sub 83.8%

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

      9. distribute-rgt1-in 84.7%

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

      10. associate-/l/ 84.8%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in wj around 0 92.2%

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

   if 2.29999999999999984e-30 < wj

   1. Initial program 45.0%

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

      1. distribute-rgt1-in 44.9%

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

   3. Simplified 44.9%

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

   4. Taylor expanded in wj around 0 26.1%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{1 + 2 \cdot wj}} \]
   5. Step-by-step derivation

      1. count-2 26.1%

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

   6. Simplified 26.1%

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

   7. Taylor expanded in x around 0 12.3%

      \[\leadsto \color{blue}{wj - \frac{e^{wj} \cdot wj}{1 + 2 \cdot wj}} \]
   8. Step-by-step derivation

      1. associate-/l* 12.7%

         \[\leadsto wj - \color{blue}{\frac{e^{wj}}{\frac{1 + 2 \cdot wj}{wj}}} \]

      2. count-2 12.7%

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

   9. Simplified 12.7%

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

   10. Taylor expanded in wj around 0 48.7%

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

       1. unpow2 48.7%

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

   12. Simplified 48.7%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.3 \cdot 10^{-30}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 10: 84.4% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 3.5 \cdot 10^{-30}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (if (<= wj 3.5e-30) x (* wj wj)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 3.5e-30) {
		tmp = x;
	} else {
		tmp = 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 <= 3.5d-30) then
        tmp = x
    else
        tmp = wj * wj
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 3.5e-30) {
		tmp = x;
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 3.5e-30:
		tmp = x
	else:
		tmp = wj * wj
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 3.5e-30)
		tmp = x;
	else
		tmp = Float64(wj * wj);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 3.5e-30)
		tmp = x;
	else
		tmp = wj * wj;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 3.5e-30], x, N[(wj * wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 3.5000000000000003e-30

   1. Initial program 83.8%

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

      1. sub-neg 83.8%

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

      2. div-sub 83.8%

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

      3. sub-neg 83.8%

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

      4. +-commutative 83.8%

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

      5. distribute-neg-in 83.8%

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

      6. remove-double-neg 83.8%

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

      7. sub-neg 83.8%

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

      8. div-sub 83.8%

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

      9. distribute-rgt1-in 84.7%

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

      10. associate-/l/ 84.8%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in wj around 0 91.9%

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

   if 3.5000000000000003e-30 < wj

   1. Initial program 45.0%

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

      1. distribute-rgt1-in 44.9%

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

   3. Simplified 44.9%

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

   4. Taylor expanded in wj around 0 26.1%

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{1 + 2 \cdot wj}} \]
   5. Step-by-step derivation

      1. count-2 26.1%

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

   6. Simplified 26.1%

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

   7. Taylor expanded in x around 0 12.3%

      \[\leadsto \color{blue}{wj - \frac{e^{wj} \cdot wj}{1 + 2 \cdot wj}} \]
   8. Step-by-step derivation

      1. associate-/l* 12.7%

         \[\leadsto wj - \color{blue}{\frac{e^{wj}}{\frac{1 + 2 \cdot wj}{wj}}} \]

      2. count-2 12.7%

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

   9. Simplified 12.7%

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

   10. Taylor expanded in wj around 0 48.7%

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

       1. unpow2 48.7%

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

   12. Simplified 48.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 3.5 \cdot 10^{-30}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 11: 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 80.8%

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

   1. sub-neg 80.8%

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

   2. div-sub 80.8%

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

   3. sub-neg 80.8%

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

   4. +-commutative 80.8%

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

   5. distribute-neg-in 80.8%

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

   6. remove-double-neg 80.8%

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

   7. sub-neg 80.8%

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

   8. div-sub 80.8%

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

   9. distribute-rgt1-in 81.6%

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

   10. associate-/l/ 81.7%

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

3. Simplified 82.8%

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

4. Taylor expanded in wj around inf 4.6%

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

5. Final simplification 4.6%

   \[\leadsto wj \]

Alternative 12: 84.9% 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 80.8%

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

   1. sub-neg 80.8%

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

   2. div-sub 80.8%

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

   3. sub-neg 80.8%

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

   4. +-commutative 80.8%

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

   5. distribute-neg-in 80.8%

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

   6. remove-double-neg 80.8%

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

   7. sub-neg 80.8%

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

   8. div-sub 80.8%

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

   9. distribute-rgt1-in 81.6%

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

   10. associate-/l/ 81.7%

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

3. Simplified 82.8%

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

4. Taylor expanded in wj around 0 86.0%

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

5. Final simplification 86.0%

   \[\leadsto x \]

Developer target: 79.7% 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 2023230 
(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))))))