Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.0% → 98.7%

Time: 9.7s

Alternatives: 11

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -6.1539687024492326e+209
  2. x = 3.8377947049925936e-208

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: 78.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ t_1 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t_1}{e^{wj} + t_1} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot t_0\right) - x \cdot -3\right) - x \cdot 0.6666666666666666\right) + \left(\left(1 - t_0\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))) (t_1 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_1) (+ (exp wj) t_1))) 5e-17)
     (+
      (*
       (pow wj 3.0)
       (- (- (- -1.0 (* -2.0 t_0)) (* x -3.0)) (* x 0.6666666666666666)))
      (+ (* (- 1.0 t_0) (pow wj 2.0)) (+ x (* -2.0 (* wj x)))))
     (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double t_1 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_1) / (exp(wj) + t_1))) <= 5e-17) {
		tmp = (pow(wj, 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (x * 0.6666666666666666))) + (((1.0 - t_0) * pow(wj, 2.0)) + (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 = (x * (-4.0d0)) + (x * 1.5d0)
    t_1 = wj * exp(wj)
    if ((wj + ((x - t_1) / (exp(wj) + t_1))) <= 5d-17) then
        tmp = ((wj ** 3.0d0) * ((((-1.0d0) - ((-2.0d0) * t_0)) - (x * (-3.0d0))) - (x * 0.6666666666666666d0))) + (((1.0d0 - t_0) * (wj ** 2.0d0)) + (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 = (x * -4.0) + (x * 1.5);
	double t_1 = wj * Math.exp(wj);
	double tmp;
	if ((wj + ((x - t_1) / (Math.exp(wj) + t_1))) <= 5e-17) {
		tmp = (Math.pow(wj, 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (x * 0.6666666666666666))) + (((1.0 - t_0) * Math.pow(wj, 2.0)) + (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 = (x * -4.0) + (x * 1.5)
	t_1 = wj * math.exp(wj)
	tmp = 0
	if (wj + ((x - t_1) / (math.exp(wj) + t_1))) <= 5e-17:
		tmp = (math.pow(wj, 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (x * 0.6666666666666666))) + (((1.0 - t_0) * math.pow(wj, 2.0)) + (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(Float64(x * -4.0) + Float64(x * 1.5))
	t_1 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_1) / Float64(exp(wj) + t_1))) <= 5e-17)
		tmp = Float64(Float64((wj ^ 3.0) * Float64(Float64(Float64(-1.0 - Float64(-2.0 * t_0)) - Float64(x * -3.0)) - Float64(x * 0.6666666666666666))) + Float64(Float64(Float64(1.0 - t_0) * (wj ^ 2.0)) + Float64(x + Float64(-2.0 * Float64(wj * x)))));
	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)
	t_0 = (x * -4.0) + (x * 1.5);
	t_1 = wj * exp(wj);
	tmp = 0.0;
	if ((wj + ((x - t_1) / (exp(wj) + t_1))) <= 5e-17)
		tmp = ((wj ^ 3.0) * (((-1.0 - (-2.0 * t_0)) - (x * -3.0)) - (x * 0.6666666666666666))) + (((1.0 - t_0) * (wj ^ 2.0)) + (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[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$1), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-17], N[(N[(N[Power[wj, 3.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x * -3.0), $MachinePrecision]), $MachinePrecision] - N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 - t$95$0), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $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 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 4.9999999999999999e-17

   1. Initial program 66.6%

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

      1. sub-neg 66.6%

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

      2. div-sub 66.6%

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

      3. sub-neg 66.6%

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

      4. +-commutative 66.6%

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

      5. distribute-neg-in 66.6%

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

      6. remove-double-neg 66.6%

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

      7. sub-neg 66.6%

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

      8. div-sub 66.6%

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

      9. distribute-rgt1-in 67.1%

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

      10. associate-/l/ 67.2%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in wj around 0 98.8%

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

   if 4.9999999999999999e-17 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 94.4%

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

      1. sub-neg 94.4%

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

      2. div-sub 94.4%

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

      3. sub-neg 94.4%

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

      4. +-commutative 94.4%

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

      5. distribute-neg-in 94.4%

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

      6. remove-double-neg 94.4%

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

      7. sub-neg 94.4%

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

      8. div-sub 94.4%

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

      9. distribute-rgt1-in 96.9%

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

      10. associate-/l/ 96.9%

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

   3. Simplified 99.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot -3\right) - x \cdot 0.6666666666666666\right) + \left(\left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]

Alternative 2: 97.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -5.85 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, \mathsf{fma}\left(-2, wj \cdot x, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -5.85e-9)
   (+ wj (/ (- x (* wj (exp wj))) (* (exp wj) (+ wj 1.0))))
   (fma wj wj (fma -2.0 (* wj x) x))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -5.85e-9) {
		tmp = wj + ((x - (wj * exp(wj))) / (exp(wj) * (wj + 1.0)));
	} else {
		tmp = fma(wj, wj, fma(-2.0, (wj * x), x));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -5.85e-9)
		tmp = Float64(wj + Float64(Float64(x - Float64(wj * exp(wj))) / Float64(exp(wj) * Float64(wj + 1.0))));
	else
		tmp = fma(wj, wj, fma(-2.0, Float64(wj * x), x));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -5.85e-9], N[(wj + N[(N[(x - N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj * wj + N[(-2.0 * N[(wj * x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -5.85000000000000027e-9

   1. Initial program 66.8%

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

      1. distribute-rgt1-in 96.8%

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

   3. Simplified 96.8%

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

   if -5.85000000000000027e-9 < wj

   1. Initial program 75.7%

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

      1. distribute-rgt1-in 75.7%

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

   3. Simplified 75.7%

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

   4. Taylor expanded in wj around 0 75.3%

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

      1. count-2 75.3%

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

   6. Simplified 75.3%

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

   7. Taylor expanded in wj around 0 97.8%

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

   8. Taylor expanded in x around 0 98.2%

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

      1. unpow2 98.2%

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

   10. Simplified 98.2%

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

   11. Taylor expanded in wj around 0 98.2%

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

       1. unpow2 98.2%

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

       2. fma-def 98.2%

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

       3. fma-udef 98.2%

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

       4. *-commutative 98.2%

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

   13. Simplified 98.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -5.85 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, \mathsf{fma}\left(-2, wj \cdot x, x\right)\right)\\ \end{array} \]

Alternative 3: 97.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -5.85 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, \mathsf{fma}\left(-2, wj \cdot x, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -5.85e-9)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (fma wj wj (fma -2.0 (* wj x) x))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -5.85e-9) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = fma(wj, wj, fma(-2.0, (wj * x), x));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -5.85e-9)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = fma(wj, wj, fma(-2.0, Float64(wj * x), x));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -5.85e-9], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj * wj + N[(-2.0 * N[(wj * x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -5.85000000000000027e-9

   1. Initial program 66.8%

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

      1. sub-neg 66.8%

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

      2. div-sub 66.8%

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

      3. sub-neg 66.8%

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

      4. +-commutative 66.8%

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

      5. distribute-neg-in 66.8%

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

      6. remove-double-neg 66.8%

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

      7. sub-neg 66.8%

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

      8. div-sub 66.8%

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

      9. distribute-rgt1-in 96.8%

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

      10. associate-/l/ 96.5%

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

   3. Simplified 96.5%

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

   if -5.85000000000000027e-9 < wj

   1. Initial program 75.7%

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

      1. distribute-rgt1-in 75.7%

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

   3. Simplified 75.7%

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

   4. Taylor expanded in wj around 0 75.3%

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

      1. count-2 75.3%

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

   6. Simplified 75.3%

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

   7. Taylor expanded in wj around 0 97.8%

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

   8. Taylor expanded in x around 0 98.2%

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

      1. unpow2 98.2%

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

   10. Simplified 98.2%

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

   11. Taylor expanded in wj around 0 98.2%

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

       1. unpow2 98.2%

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

       2. fma-def 98.2%

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

       3. fma-udef 98.2%

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

       4. *-commutative 98.2%

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

   13. Simplified 98.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -5.85 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, \mathsf{fma}\left(-2, wj \cdot x, x\right)\right)\\ \end{array} \]

Alternative 4: 97.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -5.85 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -5.85e-9)
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+ (+ x (* -2.0 (* wj x))) (* wj wj))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -5.85e-9) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-5.85d-9)) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = (x + ((-2.0d0) * (wj * x))) + (wj * wj)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -5.85e-9) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -5.85e-9:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj)
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -5.85e-9)
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(Float64(x + Float64(-2.0 * Float64(wj * x))) + Float64(wj * wj));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -5.85e-9)
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = (x + (-2.0 * (wj * x))) + (wj * wj);
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -5.85e-9], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj * wj), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -5.85000000000000027e-9

   1. Initial program 66.8%

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

      1. sub-neg 66.8%

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

      2. div-sub 66.8%

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

      3. sub-neg 66.8%

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

      4. +-commutative 66.8%

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

      5. distribute-neg-in 66.8%

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

      6. remove-double-neg 66.8%

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

      7. sub-neg 66.8%

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

      8. div-sub 66.8%

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

      9. distribute-rgt1-in 96.8%

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

      10. associate-/l/ 96.5%

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

   3. Simplified 96.5%

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

   if -5.85000000000000027e-9 < wj

   1. Initial program 75.7%

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

      1. distribute-rgt1-in 75.7%

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

   3. Simplified 75.7%

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

   4. Taylor expanded in wj around 0 75.3%

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

      1. count-2 75.3%

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

   6. Simplified 75.3%

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

   7. Taylor expanded in wj around 0 97.8%

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

   8. Taylor expanded in x around 0 98.2%

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

      1. unpow2 98.2%

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

   10. Simplified 98.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -5.85 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \end{array} \]

Alternative 5: 84.6% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 4.2 \cdot 10^{-24} \lor \neg \left(wj \leq 1.65 \cdot 10^{-13}\right):\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj 4.2e-24) (not (<= wj 1.65e-13)))
   (+ x (* -2.0 (* wj x)))
   (* wj wj)))

C

double code(double wj, double x) {
	double tmp;
	if ((wj <= 4.2e-24) || !(wj <= 1.65e-13)) {
		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 <= 4.2d-24) .or. (.not. (wj <= 1.65d-13))) 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 <= 4.2e-24) || !(wj <= 1.65e-13)) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (wj <= 4.2e-24) or not (wj <= 1.65e-13):
		tmp = x + (-2.0 * (wj * x))
	else:
		tmp = wj * wj
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((wj <= 4.2e-24) || !(wj <= 1.65e-13))
		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 <= 4.2e-24) || ~((wj <= 1.65e-13)))
		tmp = x + (-2.0 * (wj * x));
	else
		tmp = wj * wj;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[wj, 4.2e-24], N[Not[LessEqual[wj, 1.65e-13]], $MachinePrecision]], 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 < 4.1999999999999999e-24 or 1.65e-13 < wj

   1. Initial program 77.2%

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

      1. sub-neg 77.2%

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

      2. div-sub 77.2%

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

      3. sub-neg 77.2%

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

      4. +-commutative 77.2%

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

      5. distribute-neg-in 77.2%

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

      6. remove-double-neg 77.2%

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

      7. sub-neg 77.2%

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

      8. div-sub 77.2%

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

      9. distribute-rgt1-in 78.4%

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

      10. associate-/l/ 78.4%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in wj around 0 83.6%

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

   if 4.1999999999999999e-24 < wj < 1.65e-13

   1. Initial program 31.2%

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

      1. distribute-rgt1-in 31.2%

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

   3. Simplified 31.2%

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

   4. Taylor expanded in wj around 0 33.8%

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

      1. count-2 33.8%

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

   6. Simplified 33.8%

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

   7. Taylor expanded in wj around 0 96.0%

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

   8. Taylor expanded in x around 0 96.0%

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

      1. unpow2 96.0%

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

   10. Simplified 96.0%

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

   11. Taylor expanded in wj around inf 74.3%

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

       1. unpow2 74.3%

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

   13. Simplified 74.3%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 4.2 \cdot 10^{-24} \lor \neg \left(wj \leq 1.65 \cdot 10^{-13}\right):\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 6: 85.9% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 3.2 \cdot 10^{-24}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{elif}\;wj \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 3.2e-24)
   (+ x (* -2.0 (* wj x)))
   (if (<= wj 3.6e-13) (* wj wj) (- wj (/ wj (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 3.2e-24) {
		tmp = x + (-2.0 * (wj * x));
	} else if (wj <= 3.6e-13) {
		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 <= 3.2d-24) then
        tmp = x + ((-2.0d0) * (wj * x))
    else if (wj <= 3.6d-13) 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 <= 3.2e-24) {
		tmp = x + (-2.0 * (wj * x));
	} else if (wj <= 3.6e-13) {
		tmp = wj * wj;
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 3.2e-24:
		tmp = x + (-2.0 * (wj * x))
	elif wj <= 3.6e-13:
		tmp = wj * wj
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 3.2e-24)
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	elseif (wj <= 3.6e-13)
		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 <= 3.2e-24)
		tmp = x + (-2.0 * (wj * x));
	elseif (wj <= 3.6e-13)
		tmp = wj * wj;
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 3.2e-24], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 3.6e-13], 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 < 3.20000000000000012e-24

   1. Initial program 77.5%

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

      1. sub-neg 77.5%

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

      2. div-sub 77.5%

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

      3. sub-neg 77.5%

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

      4. +-commutative 77.5%

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

      5. distribute-neg-in 77.5%

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

      6. remove-double-neg 77.5%

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

      7. sub-neg 77.5%

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

      8. div-sub 77.5%

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

      9. distribute-rgt1-in 78.8%

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

      10. associate-/l/ 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in wj around 0 84.8%

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

   if 3.20000000000000012e-24 < wj < 3.5999999999999998e-13

   1. Initial program 31.2%

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

      1. distribute-rgt1-in 31.2%

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

   3. Simplified 31.2%

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

   4. Taylor expanded in wj around 0 33.8%

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

      1. count-2 33.8%

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

   6. Simplified 33.8%

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

   7. Taylor expanded in wj around 0 96.0%

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

   8. Taylor expanded in x around 0 96.0%

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

      1. unpow2 96.0%

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

   10. Simplified 96.0%

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

   11. Taylor expanded in wj around inf 74.3%

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

       1. unpow2 74.3%

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

   13. Simplified 74.3%

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

   if 3.5999999999999998e-13 < wj

   1. Initial program 63.4%

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

      1. sub-neg 63.4%

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

      2. div-sub 63.4%

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

      3. sub-neg 63.4%

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

      4. +-commutative 63.4%

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

      5. distribute-neg-in 63.4%

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

      6. remove-double-neg 63.4%

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

      7. sub-neg 63.4%

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

      8. div-sub 63.4%

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

      9. distribute-rgt1-in 63.4%

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

      10. associate-/l/ 63.4%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in x around 0 64.6%

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

      1. +-commutative 64.6%

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

   6. Simplified 64.6%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 3.2 \cdot 10^{-24}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{elif}\;wj \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 7: 85.9% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 5.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{1 + \left(wj + wj\right)}\\ \mathbf{elif}\;wj \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 5.2e-24)
   (/ x (+ 1.0 (+ wj wj)))
   (if (<= wj 3.6e-13) (* wj wj) (- wj (/ wj (+ wj 1.0))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 5.2e-24) {
		tmp = x / (1.0 + (wj + wj));
	} else if (wj <= 3.6e-13) {
		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 <= 5.2d-24) then
        tmp = x / (1.0d0 + (wj + wj))
    else if (wj <= 3.6d-13) 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 <= 5.2e-24) {
		tmp = x / (1.0 + (wj + wj));
	} else if (wj <= 3.6e-13) {
		tmp = wj * wj;
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 5.2e-24:
		tmp = x / (1.0 + (wj + wj))
	elif wj <= 3.6e-13:
		tmp = wj * wj
	else:
		tmp = wj - (wj / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 5.2e-24)
		tmp = Float64(x / Float64(1.0 + Float64(wj + wj)));
	elseif (wj <= 3.6e-13)
		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 <= 5.2e-24)
		tmp = x / (1.0 + (wj + wj));
	elseif (wj <= 3.6e-13)
		tmp = wj * wj;
	else
		tmp = wj - (wj / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 5.2e-24], N[(x / N[(1.0 + N[(wj + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 3.6e-13], 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 < 5.2e-24

   1. Initial program 77.5%

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

      1. distribute-rgt1-in 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in wj around 0 76.4%

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

      1. count-2 76.4%

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

   6. Simplified 76.4%

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

   7. Taylor expanded in x around inf 84.9%

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

      1. count-2 84.9%

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

   9. Simplified 84.9%

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

   if 5.2e-24 < wj < 3.5999999999999998e-13

   1. Initial program 31.2%

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

      1. distribute-rgt1-in 31.2%

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

   3. Simplified 31.2%

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

   4. Taylor expanded in wj around 0 33.8%

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

      1. count-2 33.8%

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

   6. Simplified 33.8%

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

   7. Taylor expanded in wj around 0 96.0%

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

   8. Taylor expanded in x around 0 96.0%

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

      1. unpow2 96.0%

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

   10. Simplified 96.0%

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

   11. Taylor expanded in wj around inf 74.3%

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

       1. unpow2 74.3%

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

   13. Simplified 74.3%

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

   if 3.5999999999999998e-13 < wj

   1. Initial program 63.4%

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

      1. sub-neg 63.4%

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

      2. div-sub 63.4%

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

      3. sub-neg 63.4%

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

      4. +-commutative 63.4%

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

      5. distribute-neg-in 63.4%

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

      6. remove-double-neg 63.4%

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

      7. sub-neg 63.4%

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

      8. div-sub 63.4%

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

      9. distribute-rgt1-in 63.4%

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

      10. associate-/l/ 63.4%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in x around 0 64.6%

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

      1. +-commutative 64.6%

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

   6. Simplified 64.6%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 5.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{x}{1 + \left(wj + wj\right)}\\ \mathbf{elif}\;wj \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \]

Alternative 8: 95.7% accurate, 28.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

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

   1. distribute-rgt1-in 76.6%

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

3. Simplified 76.6%

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

4. Taylor expanded in wj around 0 73.9%

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

   1. count-2 73.9%

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

6. Simplified 73.9%

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

7. Taylor expanded in wj around 0 95.4%

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

8. Taylor expanded in x around 0 95.8%

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

   1. unpow2 95.8%

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

10. Simplified 95.8%

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

11. Final simplification 95.8%

    \[\leadsto \left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj \]

Alternative 9: 84.2% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 4.5 \cdot 10^{-24}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (if (<= wj 4.5e-24) x (* wj wj)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 4.5e-24) {
		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 <= 4.5d-24) 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 <= 4.5e-24) {
		tmp = x;
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 4.5e-24:
		tmp = x
	else:
		tmp = wj * wj
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 4.5e-24)
		tmp = x;
	else
		tmp = Float64(wj * wj);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 4.5e-24)
		tmp = x;
	else
		tmp = wj * wj;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 4.5e-24], x, N[(wj * wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 4.4999999999999997e-24

   1. Initial program 77.5%

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

      1. sub-neg 77.5%

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

      2. div-sub 77.5%

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

      3. sub-neg 77.5%

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

      4. +-commutative 77.5%

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

      5. distribute-neg-in 77.5%

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

      6. remove-double-neg 77.5%

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

      7. sub-neg 77.5%

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

      8. div-sub 77.5%

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

      9. distribute-rgt1-in 78.8%

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

      10. associate-/l/ 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in wj around 0 84.2%

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

   if 4.4999999999999997e-24 < wj

   1. Initial program 43.3%

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

      1. distribute-rgt1-in 43.3%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in wj around 0 36.1%

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

      1. count-2 36.1%

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

   6. Simplified 36.1%

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

   7. Taylor expanded in wj around 0 77.0%

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

   8. Taylor expanded in x around 0 76.8%

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

      1. unpow2 76.8%

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

   10. Simplified 76.8%

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

   11. Taylor expanded in wj around inf 51.7%

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

       1. unpow2 51.7%

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

   13. Simplified 51.7%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 4.5 \cdot 10^{-24}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]

Alternative 10: 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 75.4%

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

   1. sub-neg 75.4%

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

   2. div-sub 75.4%

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

   3. sub-neg 75.4%

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

   4. +-commutative 75.4%

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

   5. distribute-neg-in 75.4%

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

   6. remove-double-neg 75.4%

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

   7. sub-neg 75.4%

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

   8. div-sub 75.4%

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

   9. distribute-rgt1-in 76.6%

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

   10. associate-/l/ 76.6%

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

3. Simplified 77.4%

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

4. Taylor expanded in wj around inf 4.4%

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

5. Final simplification 4.4%

   \[\leadsto wj \]

Alternative 11: 84.1% 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 75.4%

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

   1. sub-neg 75.4%

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

   2. div-sub 75.4%

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

   3. sub-neg 75.4%

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

   4. +-commutative 75.4%

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

   5. distribute-neg-in 75.4%

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

   6. remove-double-neg 75.4%

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

   7. sub-neg 75.4%

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

   8. div-sub 75.4%

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

   9. distribute-rgt1-in 76.6%

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

   10. associate-/l/ 76.6%

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

3. Simplified 77.4%

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

4. Taylor expanded in wj around 0 80.5%

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

5. Final simplification 80.5%

   \[\leadsto x \]

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 2023200 
(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))))))