Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.2% → 98.7%

Time: 12.4s

Alternatives: 13

Speedup: 313.0×

• could not determine a ground truth

  1. wj = 8.876651272435733e+82
  2. x = 2.017608814823458e+175

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ wj - \frac{t\_0 - x}{e^{wj} + t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ wj - \frac{t\_0 - x}{e^{wj} + t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(\left(wj + -1\right) - x \cdot \left(wj \cdot -2.6666666666666665 + 2.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 2e-14)
     (-
      x
      (*
       wj
       (+
        (* x 2.0)
        (* wj (- (+ wj -1.0) (* x (+ (* wj -2.6666666666666665) 2.5)))))))
     (+ wj (/ (- (* x (exp (- wj))) wj) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 2e-14) {
		tmp = x - (wj * ((x * 2.0) + (wj * ((wj + -1.0) - (x * ((wj * -2.6666666666666665) + 2.5))))));
	} 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) :: tmp
    t_0 = wj * exp(wj)
    if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 2d-14) then
        tmp = x - (wj * ((x * 2.0d0) + (wj * ((wj + (-1.0d0)) - (x * ((wj * (-2.6666666666666665d0)) + 2.5d0))))))
    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 tmp;
	if ((wj + ((x - t_0) / (Math.exp(wj) + t_0))) <= 2e-14) {
		tmp = x - (wj * ((x * 2.0) + (wj * ((wj + -1.0) - (x * ((wj * -2.6666666666666665) + 2.5))))));
	} else {
		tmp = wj + (((x * Math.exp(-wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	tmp = 0
	if (wj + ((x - t_0) / (math.exp(wj) + t_0))) <= 2e-14:
		tmp = x - (wj * ((x * 2.0) + (wj * ((wj + -1.0) - (x * ((wj * -2.6666666666666665) + 2.5))))))
	else:
		tmp = wj + (((x * math.exp(-wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 2e-14)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(Float64(wj + -1.0) - Float64(x * Float64(Float64(wj * -2.6666666666666665) + 2.5)))))));
	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);
	tmp = 0.0;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 2e-14)
		tmp = x - (wj * ((x * 2.0) + (wj * ((wj + -1.0) - (x * ((wj * -2.6666666666666665) + 2.5))))));
	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]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-14], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(N[(wj + -1.0), $MachinePrecision] - N[(x * N[(N[(wj * -2.6666666666666665), $MachinePrecision] + 2.5), $MachinePrecision]), $MachinePrecision]), $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))))) < 2e-14

   1. Initial program 73.5%

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

      1. distribute-rgt1-in 73.5%

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

      2. associate-/l/ 73.5%

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

      3. div-sub 73.5%

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

      4. associate-/l* 73.5%

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

      5. *-inverses 73.5%

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

      6. *-rgt-identity 73.5%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in wj around 0 98.9%

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

   6. Taylor expanded in x around 0 98.9%

      \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 98.9%

         \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]

      2. mul-1-neg 98.9%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + \color{blue}{\left(-wj\right)}\right)\right) - 2 \cdot x\right) \]

      3. sub-neg 98.9%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \color{blue}{\left(1 - wj\right)}\right) - 2 \cdot x\right) \]

      4. +-commutative 98.9%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \color{blue}{\left(-2.6666666666666665 \cdot wj + 2.5\right)} + \left(1 - wj\right)\right) - 2 \cdot x\right) \]

      5. *-commutative 98.9%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(\color{blue}{wj \cdot -2.6666666666666665} + 2.5\right) + \left(1 - wj\right)\right) - 2 \cdot x\right) \]

   8. Simplified 98.9%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(wj \cdot -2.6666666666666665 + 2.5\right) + \left(1 - wj\right)\right)} - 2 \cdot x\right) \]

   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 96.3%

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

      1. distribute-rgt1-in 96.2%

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

      2. associate-/l/ 96.4%

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

      3. div-sub 96.4%

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

      4. associate-/l* 96.4%

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

      5. *-inverses 99.6%

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

      6. *-rgt-identity 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 99.6%

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

      3. rec-exp 99.7%

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

   6. Applied egg-rr 99.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(\left(wj + -1\right) - x \cdot \left(wj \cdot -2.6666666666666665 + 2.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{x \cdot e^{-wj} - wj}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 5.6 \cdot 10^{-6}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(\left(wj + -1\right) - x \cdot \left(wj \cdot -2.6666666666666665 + 2.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 5.6e-6)
   (-
    x
    (*
     wj
     (+
      (* x 2.0)
      (* wj (- (+ wj -1.0) (* x (+ (* wj -2.6666666666666665) 2.5)))))))
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 5.6e-6) {
		tmp = x - (wj * ((x * 2.0) + (wj * ((wj + -1.0) - (x * ((wj * -2.6666666666666665) + 2.5))))));
	} 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 <= 5.6d-6) then
        tmp = x - (wj * ((x * 2.0d0) + (wj * ((wj + (-1.0d0)) - (x * ((wj * (-2.6666666666666665d0)) + 2.5d0))))))
    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 <= 5.6e-6) {
		tmp = x - (wj * ((x * 2.0) + (wj * ((wj + -1.0) - (x * ((wj * -2.6666666666666665) + 2.5))))));
	} else {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 5.6e-6:
		tmp = x - (wj * ((x * 2.0) + (wj * ((wj + -1.0) - (x * ((wj * -2.6666666666666665) + 2.5))))))
	else:
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 5.6e-6)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(Float64(wj + -1.0) - Float64(x * Float64(Float64(wj * -2.6666666666666665) + 2.5)))))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 5.6e-6)
		tmp = x - (wj * ((x * 2.0) + (wj * ((wj + -1.0) - (x * ((wj * -2.6666666666666665) + 2.5))))));
	else
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 5.6e-6], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(N[(wj + -1.0), $MachinePrecision] - N[(x * N[(N[(wj * -2.6666666666666665), $MachinePrecision] + 2.5), $MachinePrecision]), $MachinePrecision]), $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 wj < 5.59999999999999975e-6

   1. Initial program 79.2%

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

      1. distribute-rgt1-in 79.2%

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

      2. associate-/l/ 79.2%

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

      3. div-sub 79.2%

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

      4. associate-/l* 79.2%

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

      5. *-inverses 79.2%

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

      6. *-rgt-identity 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in wj around 0 98.4%

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

   6. Taylor expanded in x around 0 98.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
   7. Step-by-step derivation

      1. distribute-lft-out 98.4%

         \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]

      2. mul-1-neg 98.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + \color{blue}{\left(-wj\right)}\right)\right) - 2 \cdot x\right) \]

      3. sub-neg 98.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \color{blue}{\left(1 - wj\right)}\right) - 2 \cdot x\right) \]

      4. +-commutative 98.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \color{blue}{\left(-2.6666666666666665 \cdot wj + 2.5\right)} + \left(1 - wj\right)\right) - 2 \cdot x\right) \]

      5. *-commutative 98.4%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(\color{blue}{wj \cdot -2.6666666666666665} + 2.5\right) + \left(1 - wj\right)\right) - 2 \cdot x\right) \]

   8. Simplified 98.4%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(wj \cdot -2.6666666666666665 + 2.5\right) + \left(1 - wj\right)\right)} - 2 \cdot x\right) \]

   if 5.59999999999999975e-6 < wj

   1. Initial program 68.4%

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

      1. distribute-rgt1-in 68.2%

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

      2. associate-/l/ 68.7%

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

      3. div-sub 68.7%

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

      4. associate-/l* 68.7%

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

      5. *-inverses 97.2%

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

      6. *-rgt-identity 97.2%

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

   3. Simplified 97.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 5.6 \cdot 10^{-6}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(\left(wj + -1\right) - x \cdot \left(wj \cdot -2.6666666666666665 + 2.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ x - wj \cdot \left(x \cdot 2 + wj \cdot \left(\left(wj + -1\right) - x \cdot \left(wj \cdot -2.6666666666666665 + 2.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (-
  x
  (*
   wj
   (+
    (* x 2.0)
    (* wj (- (+ wj -1.0) (* x (+ (* wj -2.6666666666666665) 2.5))))))))

C

double code(double wj, double x) {
	return x - (wj * ((x * 2.0) + (wj * ((wj + -1.0) - (x * ((wj * -2.6666666666666665) + 2.5))))));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x - (wj * ((x * 2.0d0) + (wj * ((wj + (-1.0d0)) - (x * ((wj * (-2.6666666666666665d0)) + 2.5d0))))))
end function

Java

public static double code(double wj, double x) {
	return x - (wj * ((x * 2.0) + (wj * ((wj + -1.0) - (x * ((wj * -2.6666666666666665) + 2.5))))));
}

Python

def code(wj, x):
	return x - (wj * ((x * 2.0) + (wj * ((wj + -1.0) - (x * ((wj * -2.6666666666666665) + 2.5))))))

Julia

function code(wj, x)
	return Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(Float64(wj + -1.0) - Float64(x * Float64(Float64(wj * -2.6666666666666665) + 2.5)))))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x - (wj * ((x * 2.0) + (wj * ((wj + -1.0) - (x * ((wj * -2.6666666666666665) + 2.5))))));
end

Wolfram

code[wj_, x_] := N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(N[(wj + -1.0), $MachinePrecision] - N[(x * N[(N[(wj * -2.6666666666666665), $MachinePrecision] + 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

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

   1. distribute-rgt1-in 78.9%

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

   2. associate-/l/ 78.9%

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

   3. div-sub 78.9%

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

   4. associate-/l* 78.9%

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

   5. *-inverses 79.7%

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

   6. *-rgt-identity 79.7%

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

3. Simplified 79.7%

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

5. Taylor expanded in wj around 0 96.6%

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

6. Taylor expanded in x around 0 96.6%

   \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
7. Step-by-step derivation

   1. distribute-lft-out 96.6%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]

   2. mul-1-neg 96.6%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + \color{blue}{\left(-wj\right)}\right)\right) - 2 \cdot x\right) \]

   3. sub-neg 96.6%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \color{blue}{\left(1 - wj\right)}\right) - 2 \cdot x\right) \]

   4. +-commutative 96.6%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \color{blue}{\left(-2.6666666666666665 \cdot wj + 2.5\right)} + \left(1 - wj\right)\right) - 2 \cdot x\right) \]

   5. *-commutative 96.6%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(\color{blue}{wj \cdot -2.6666666666666665} + 2.5\right) + \left(1 - wj\right)\right) - 2 \cdot x\right) \]

8. Simplified 96.6%

   \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(wj \cdot -2.6666666666666665 + 2.5\right) + \left(1 - wj\right)\right)} - 2 \cdot x\right) \]

9. Final simplification 96.6%

   \[\leadsto x - wj \cdot \left(x \cdot 2 + wj \cdot \left(\left(wj + -1\right) - x \cdot \left(wj \cdot -2.6666666666666665 + 2.5\right)\right)\right) \]
10. Add Preprocessing

Alternative 4: 96.2% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. distribute-rgt1-in 78.9%

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

   2. associate-/l/ 78.9%

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

   3. div-sub 78.9%

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

   4. associate-/l* 78.9%

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

   5. *-inverses 79.7%

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

   6. *-rgt-identity 79.7%

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

3. Simplified 79.7%

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

5. Taylor expanded in wj around 0 96.6%

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

6. Taylor expanded in x around 0 96.6%

   \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
7. Step-by-step derivation

   1. distribute-lft-out 96.6%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]

   2. mul-1-neg 96.6%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + \color{blue}{\left(-wj\right)}\right)\right) - 2 \cdot x\right) \]

   3. sub-neg 96.6%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \color{blue}{\left(1 - wj\right)}\right) - 2 \cdot x\right) \]

   4. +-commutative 96.6%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \color{blue}{\left(-2.6666666666666665 \cdot wj + 2.5\right)} + \left(1 - wj\right)\right) - 2 \cdot x\right) \]

   5. *-commutative 96.6%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(\color{blue}{wj \cdot -2.6666666666666665} + 2.5\right) + \left(1 - wj\right)\right) - 2 \cdot x\right) \]

8. Simplified 96.6%

   \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(wj \cdot -2.6666666666666665 + 2.5\right) + \left(1 - wj\right)\right)} - 2 \cdot x\right) \]

9. Taylor expanded in wj around 0 96.5%

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

    1. *-commutative 96.5%

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

11. Simplified 96.5%

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

12. Final simplification 96.5%

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

Alternative 5: 86.2% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{1 + wj \cdot \left(2 + wj \cdot 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 3.5e-5)
   (/ x (+ 1.0 (* wj (+ 2.0 (* wj 1.5)))))
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 3.5e-5) {
		tmp = x / (1.0 + (wj * (2.0 + (wj * 1.5))));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 3.5d-5) then
        tmp = x / (1.0d0 + (wj * (2.0d0 + (wj * 1.5d0))))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 3.5e-5) {
		tmp = x / (1.0 + (wj * (2.0 + (wj * 1.5))));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 3.5e-5:
		tmp = x / (1.0 + (wj * (2.0 + (wj * 1.5))))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 3.5e-5)
		tmp = Float64(x / Float64(1.0 + Float64(wj * Float64(2.0 + Float64(wj * 1.5)))));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 3.5e-5)
		tmp = x / (1.0 + (wj * (2.0 + (wj * 1.5))));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 3.5e-5], N[(x / N[(1.0 + N[(wj * N[(2.0 + N[(wj * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 3.4999999999999997e-5

   1. Initial program 79.3%

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

      1. distribute-rgt1-in 79.3%

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

      2. associate-/l/ 79.3%

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

      3. div-sub 79.3%

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

      4. associate-/l* 79.3%

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

      5. *-inverses 79.3%

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

      6. *-rgt-identity 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in x around inf 88.3%

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

      1. +-commutative 88.3%

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

   7. Simplified 88.3%

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

   8. Taylor expanded in wj around 0 87.8%

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

      1. *-commutative 87.8%

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

   10. Simplified 87.8%

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

   if 3.4999999999999997e-5 < 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. distribute-rgt1-in 63.2%

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

      2. associate-/l/ 63.4%

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

      3. div-sub 63.4%

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

      4. associate-/l* 63.4%

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

      5. *-inverses 96.8%

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

      6. *-rgt-identity 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in x around 0 81.2%

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

      1. +-commutative 81.2%

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

   7. Simplified 81.2%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{1 + wj \cdot \left(2 + wj \cdot 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.1% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{-1}{1 - \frac{-1}{wj}}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.15e-7)
   (/ x (+ 1.0 (* wj 2.0)))
   (+ wj (/ -1.0 (- 1.0 (/ -1.0 wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.15e-7) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj + (-1.0 / (1.0 - (-1.0 / wj)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 1.15d-7) then
        tmp = x / (1.0d0 + (wj * 2.0d0))
    else
        tmp = wj + ((-1.0d0) / (1.0d0 - ((-1.0d0) / wj)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.15e-7) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj + (-1.0 / (1.0 - (-1.0 / wj)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.15e-7:
		tmp = x / (1.0 + (wj * 2.0))
	else:
		tmp = wj + (-1.0 / (1.0 - (-1.0 / wj)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.15e-7)
		tmp = Float64(x / Float64(1.0 + Float64(wj * 2.0)));
	else
		tmp = Float64(wj + Float64(-1.0 / Float64(1.0 - Float64(-1.0 / wj))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.15e-7)
		tmp = x / (1.0 + (wj * 2.0));
	else
		tmp = wj + (-1.0 / (1.0 - (-1.0 / wj)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.15e-7], N[(x / N[(1.0 + N[(wj * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(-1.0 / N[(1.0 - N[(-1.0 / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.14999999999999997e-7

   1. Initial program 79.3%

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

      1. distribute-rgt1-in 79.2%

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

      2. associate-/l/ 79.3%

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

      3. div-sub 79.3%

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

      4. associate-/l* 79.3%

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

      5. *-inverses 79.3%

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

      6. *-rgt-identity 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in x around inf 88.6%

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

      1. +-commutative 88.6%

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

   7. Simplified 88.6%

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

   8. Taylor expanded in wj around 0 88.0%

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

      1. *-commutative 88.0%

         \[\leadsto \frac{x}{1 + \color{blue}{wj \cdot 2}} \]

   10. Simplified 88.0%

       \[\leadsto \frac{x}{\color{blue}{1 + wj \cdot 2}} \]

   if 1.14999999999999997e-7 < wj

   1. Initial program 68.0%

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

      1. distribute-rgt1-in 67.8%

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

      2. associate-/l/ 68.2%

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

      3. div-sub 68.2%

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

      4. associate-/l* 68.2%

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

      5. *-inverses 93.2%

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

      6. *-rgt-identity 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in x around 0 69.6%

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

      1. +-commutative 69.6%

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

   7. Simplified 69.6%

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

      1. clear-num 69.6%

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

      2. inv-pow 69.6%

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

   9. Applied egg-rr 69.6%

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

       1. unpow-1 69.6%

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

   11. Simplified 69.6%

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

   12. Taylor expanded in wj around inf 69.9%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{-1}{1 - \frac{-1}{wj}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.1% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.26 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj + wj \cdot \frac{1}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.26e-7)
   (/ x (+ 1.0 (* wj 2.0)))
   (+ wj (* wj (/ 1.0 (- -1.0 wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.26e-7) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj + (wj * (1.0 / (-1.0 - wj)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 1.26d-7) then
        tmp = x / (1.0d0 + (wj * 2.0d0))
    else
        tmp = wj + (wj * (1.0d0 / ((-1.0d0) - wj)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.26e-7) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj + (wj * (1.0 / (-1.0 - wj)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.26e-7:
		tmp = x / (1.0 + (wj * 2.0))
	else:
		tmp = wj + (wj * (1.0 / (-1.0 - wj)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.26e-7)
		tmp = Float64(x / Float64(1.0 + Float64(wj * 2.0)));
	else
		tmp = Float64(wj + Float64(wj * Float64(1.0 / Float64(-1.0 - wj))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.26e-7)
		tmp = x / (1.0 + (wj * 2.0));
	else
		tmp = wj + (wj * (1.0 / (-1.0 - wj)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.26e-7], N[(x / N[(1.0 + N[(wj * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj * N[(1.0 / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.2599999999999999e-7

   1. Initial program 79.3%

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

      1. distribute-rgt1-in 79.2%

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

      2. associate-/l/ 79.3%

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

      3. div-sub 79.3%

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

      4. associate-/l* 79.3%

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

      5. *-inverses 79.3%

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

      6. *-rgt-identity 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in x around inf 88.6%

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

      1. +-commutative 88.6%

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

   7. Simplified 88.6%

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

   8. Taylor expanded in wj around 0 88.0%

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

      1. *-commutative 88.0%

         \[\leadsto \frac{x}{1 + \color{blue}{wj \cdot 2}} \]

   10. Simplified 88.0%

       \[\leadsto \frac{x}{\color{blue}{1 + wj \cdot 2}} \]

   if 1.2599999999999999e-7 < wj

   1. Initial program 68.0%

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

      1. distribute-rgt1-in 67.8%

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

      2. associate-/l/ 68.2%

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

      3. div-sub 68.2%

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

      4. associate-/l* 68.2%

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

      5. *-inverses 93.2%

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

      6. *-rgt-identity 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in x around 0 69.6%

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

      1. +-commutative 69.6%

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

   7. Simplified 69.6%

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

      1. div-inv 69.9%

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

   9. Applied egg-rr 69.9%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.26 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj + wj \cdot \frac{1}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.0% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. distribute-rgt1-in 78.9%

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

   2. associate-/l/ 78.9%

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

   3. div-sub 78.9%

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

   4. associate-/l* 78.9%

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

   5. *-inverses 79.7%

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

   6. *-rgt-identity 79.7%

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

3. Simplified 79.7%

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

5. Taylor expanded in wj around 0 96.6%

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

6. Taylor expanded in x around 0 96.6%

   \[\leadsto x + wj \cdot \left(\color{blue}{\left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right) + wj \cdot \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]
7. Step-by-step derivation

   1. distribute-lft-out 96.6%

      \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + -1 \cdot wj\right)\right)} - 2 \cdot x\right) \]

   2. mul-1-neg 96.6%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \left(1 + \color{blue}{\left(-wj\right)}\right)\right) - 2 \cdot x\right) \]

   3. sub-neg 96.6%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right) + \color{blue}{\left(1 - wj\right)}\right) - 2 \cdot x\right) \]

   4. +-commutative 96.6%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \color{blue}{\left(-2.6666666666666665 \cdot wj + 2.5\right)} + \left(1 - wj\right)\right) - 2 \cdot x\right) \]

   5. *-commutative 96.6%

      \[\leadsto x + wj \cdot \left(wj \cdot \left(x \cdot \left(\color{blue}{wj \cdot -2.6666666666666665} + 2.5\right) + \left(1 - wj\right)\right) - 2 \cdot x\right) \]

8. Simplified 96.6%

   \[\leadsto x + wj \cdot \left(\color{blue}{wj \cdot \left(x \cdot \left(wj \cdot -2.6666666666666665 + 2.5\right) + \left(1 - wj\right)\right)} - 2 \cdot x\right) \]

9. Taylor expanded in wj around 0 96.5%

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

    1. *-commutative 96.5%

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

11. Simplified 96.5%

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

12. Taylor expanded in x around 0 96.2%

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

13. Final simplification 96.2%

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

Alternative 9: 86.1% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 10^{-5}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1e-5) (/ x (+ 1.0 (* wj 2.0))) (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1e-5) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 1d-5) then
        tmp = x / (1.0d0 + (wj * 2.0d0))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1e-5) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1e-5:
		tmp = x / (1.0 + (wj * 2.0))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1e-5)
		tmp = Float64(x / Float64(1.0 + Float64(wj * 2.0)));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1e-5)
		tmp = x / (1.0 + (wj * 2.0));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1e-5], N[(x / N[(1.0 + N[(wj * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.00000000000000008e-5

   1. Initial program 79.3%

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

      1. distribute-rgt1-in 79.3%

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

      2. associate-/l/ 79.3%

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

      3. div-sub 79.3%

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

      4. associate-/l* 79.3%

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

      5. *-inverses 79.3%

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

      6. *-rgt-identity 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in x around inf 88.3%

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

      1. +-commutative 88.3%

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

   7. Simplified 88.3%

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

   8. Taylor expanded in wj around 0 87.6%

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

      1. *-commutative 87.6%

         \[\leadsto \frac{x}{1 + \color{blue}{wj \cdot 2}} \]

   10. Simplified 87.6%

       \[\leadsto \frac{x}{\color{blue}{1 + wj \cdot 2}} \]

   if 1.00000000000000008e-5 < 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. distribute-rgt1-in 63.2%

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

      2. associate-/l/ 63.4%

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

      3. div-sub 63.4%

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

      4. associate-/l* 63.4%

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

      5. *-inverses 96.8%

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

      6. *-rgt-identity 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in x around 0 81.2%

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

      1. +-commutative 81.2%

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

   7. Simplified 81.2%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 10^{-5}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.0% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.6 \cdot 10^{-5}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.6e-5) (+ x (* -2.0 (* wj x))) (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.6e-5) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 2.6d-5) then
        tmp = x + ((-2.0d0) * (wj * x))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 2.6e-5) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 2.6e-5:
		tmp = x + (-2.0 * (wj * x))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.6e-5)
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 2.6e-5)
		tmp = x + (-2.0 * (wj * x));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.6e-5], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.59999999999999984e-5

   1. Initial program 79.3%

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

      1. distribute-rgt1-in 79.3%

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

      2. associate-/l/ 79.3%

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

      3. div-sub 79.3%

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

      4. associate-/l* 79.3%

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

      5. *-inverses 79.3%

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

      6. *-rgt-identity 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in wj around 0 87.6%

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

      1. *-commutative 87.6%

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

   7. Simplified 87.6%

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

   if 2.59999999999999984e-5 < 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. distribute-rgt1-in 63.2%

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

      2. associate-/l/ 63.4%

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

      3. div-sub 63.4%

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

      4. associate-/l* 63.4%

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

      5. *-inverses 96.8%

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

      6. *-rgt-identity 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in x around 0 81.2%

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

      1. +-commutative 81.2%

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

   7. Simplified 81.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.6 \cdot 10^{-5}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.6% accurate, 44.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. distribute-rgt1-in 78.9%

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

   2. associate-/l/ 78.9%

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

   3. div-sub 78.9%

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

   4. associate-/l* 78.9%

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

   5. *-inverses 79.7%

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

   6. *-rgt-identity 79.7%

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

3. Simplified 79.7%

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

5. Taylor expanded in wj around 0 85.6%

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

   1. *-commutative 85.6%

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

7. Simplified 85.6%

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

8. Final simplification 85.6%

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

Alternative 12: 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 78.9%

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

   1. distribute-rgt1-in 78.9%

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

   2. associate-/l/ 78.9%

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

   3. div-sub 78.9%

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

   4. associate-/l* 78.9%

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

   5. *-inverses 79.7%

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

   6. *-rgt-identity 79.7%

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

3. Simplified 79.7%

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

5. Taylor expanded in wj around 0 85.2%

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

Alternative 13: 4.4% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 78.9%

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

   1. distribute-rgt1-in 78.9%

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

   2. associate-/l/ 78.9%

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

   3. div-sub 78.9%

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

   4. associate-/l* 78.9%

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

   5. *-inverses 79.7%

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

   6. *-rgt-identity 79.7%

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

3. Simplified 79.7%

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

5. Taylor expanded in wj around inf 4.6%

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

Developer Target 1: 79.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024172 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((ew (exp wj))) (- wj (- (/ wj (+ wj 1)) (/ x (+ ew (* wj ew)))))))

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