Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.5% → 99.7%

Time: 7.6s

Alternatives: 9

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -1835827130780.4658
  2. x = 4.1434900734093464e+57

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{e^{wj}} - wj\\ t_1 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq -3.65 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{t_0}{wj + 1}\\ \mathbf{elif}\;wj \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot t_1\right) - x \cdot -3\right) - x \cdot 0.6666666666666666\right) + \left(\left(1 - t_1\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{wj + 1}, wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (- (/ x (exp wj)) wj)) (t_1 (+ (* x -4.0) (* x 1.5))))
   (if (<= wj -3.65e-6)
     (+ wj (/ t_0 (+ wj 1.0)))
     (if (<= wj 3.8e-6)
       (+
        (*
         (pow wj 3.0)
         (- (- (- -1.0 (* -2.0 t_1)) (* x -3.0)) (* x 0.6666666666666666)))
        (+ (* (- 1.0 t_1) (pow wj 2.0)) (+ x (* -2.0 (* wj x)))))
       (fma t_0 (/ 1.0 (+ wj 1.0)) wj)))))

C

double code(double wj, double x) {
	double t_0 = (x / exp(wj)) - wj;
	double t_1 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= -3.65e-6) {
		tmp = wj + (t_0 / (wj + 1.0));
	} else if (wj <= 3.8e-6) {
		tmp = (pow(wj, 3.0) * (((-1.0 - (-2.0 * t_1)) - (x * -3.0)) - (x * 0.6666666666666666))) + (((1.0 - t_1) * pow(wj, 2.0)) + (x + (-2.0 * (wj * x))));
	} else {
		tmp = fma(t_0, (1.0 / (wj + 1.0)), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(Float64(x / exp(wj)) - wj)
	t_1 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (wj <= -3.65e-6)
		tmp = Float64(wj + Float64(t_0 / Float64(wj + 1.0)));
	elseif (wj <= 3.8e-6)
		tmp = Float64(Float64((wj ^ 3.0) * Float64(Float64(Float64(-1.0 - Float64(-2.0 * t_1)) - Float64(x * -3.0)) - Float64(x * 0.6666666666666666))) + Float64(Float64(Float64(1.0 - t_1) * (wj ^ 2.0)) + Float64(x + Float64(-2.0 * Float64(wj * x)))));
	else
		tmp = fma(t_0, Float64(1.0 / Float64(wj + 1.0)), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -3.65e-6], N[(wj + N[(t$95$0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 3.8e-6], N[(N[(N[Power[wj, 3.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(x * -3.0), $MachinePrecision]), $MachinePrecision] - N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 - t$95$1), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -3.65000000000000021e-6

   1. Initial program 70.8%

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

      1. sub-neg 70.8%

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

      2. div-sub 70.8%

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

      3. sub-neg 70.8%

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

      4. +-commutative 70.8%

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

      5. distribute-neg-in 70.8%

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

      6. remove-double-neg 70.8%

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

      7. sub-neg 70.8%

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

      8. div-sub 70.8%

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

      9. distribute-rgt1-in 99.1%

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

      10. associate-/l/ 99.1%

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

   3. Simplified 99.1%

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

   if -3.65000000000000021e-6 < wj < 3.8e-6

   1. Initial program 78.5%

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

      1. sub-neg 78.5%

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

      2. div-sub 78.5%

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

      3. sub-neg 78.5%

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

      4. +-commutative 78.5%

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

      5. distribute-neg-in 78.5%

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

      6. remove-double-neg 78.5%

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

      7. sub-neg 78.5%

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

      8. div-sub 78.5%

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

      9. distribute-rgt1-in 78.5%

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

      10. associate-/l/ 78.6%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in wj around 0 100.0%

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

   if 3.8e-6 < wj

   1. Initial program 39.7%

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

      1. sub-neg 39.7%

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

      2. div-sub 39.7%

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

      3. sub-neg 39.7%

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

      4. +-commutative 39.7%

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

      5. distribute-neg-in 39.7%

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

      6. remove-double-neg 39.7%

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

      7. sub-neg 39.7%

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

      8. div-sub 39.7%

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

      9. distribute-rgt1-in 39.7%

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

      10. associate-/l/ 39.7%

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

   3. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. div-inv 99.7%

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

      3. fma-def 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.65 \cdot 10^{-6}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot -3\right) - x \cdot 0.6666666666666666\right) + \left(\left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{e^{wj}} - wj\\ \mathbf{if}\;wj \leq -4.5 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{t_0}{wj + 1}\\ \mathbf{elif}\;wj \leq 7 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, \mathsf{fma}\left(-2, wj \cdot x, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{wj + 1}, wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (- (/ x (exp wj)) wj)))
   (if (<= wj -4.5e-9)
     (+ wj (/ t_0 (+ wj 1.0)))
     (if (<= wj 7e-9)
       (fma wj wj (fma -2.0 (* wj x) x))
       (fma t_0 (/ 1.0 (+ wj 1.0)) wj)))))

C

double code(double wj, double x) {
	double t_0 = (x / exp(wj)) - wj;
	double tmp;
	if (wj <= -4.5e-9) {
		tmp = wj + (t_0 / (wj + 1.0));
	} else if (wj <= 7e-9) {
		tmp = fma(wj, wj, fma(-2.0, (wj * x), x));
	} else {
		tmp = fma(t_0, (1.0 / (wj + 1.0)), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(Float64(x / exp(wj)) - wj)
	tmp = 0.0
	if (wj <= -4.5e-9)
		tmp = Float64(wj + Float64(t_0 / Float64(wj + 1.0)));
	elseif (wj <= 7e-9)
		tmp = fma(wj, wj, fma(-2.0, Float64(wj * x), x));
	else
		tmp = fma(t_0, Float64(1.0 / Float64(wj + 1.0)), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision]}, If[LessEqual[wj, -4.5e-9], N[(wj + N[(t$95$0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 7e-9], N[(wj * wj + N[(-2.0 * N[(wj * x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -4.49999999999999976e-9

   1. Initial program 74.4%

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

      1. sub-neg 74.4%

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

      2. div-sub 74.4%

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

      3. sub-neg 74.4%

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

      4. +-commutative 74.4%

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

      5. distribute-neg-in 74.4%

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

      6. remove-double-neg 74.4%

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

      7. sub-neg 74.4%

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

      8. div-sub 74.4%

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

      9. distribute-rgt1-in 99.2%

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

      10. associate-/l/ 99.2%

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

   3. Simplified 99.2%

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

   if -4.49999999999999976e-9 < wj < 6.9999999999999998e-9

   1. Initial program 78.4%

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

      1. sub-neg 78.4%

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

      2. div-sub 78.4%

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

      3. sub-neg 78.4%

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

      4. +-commutative 78.4%

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

      5. distribute-neg-in 78.4%

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

      6. remove-double-neg 78.4%

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

      7. sub-neg 78.4%

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

      8. div-sub 78.4%

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

      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 78.4%

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

   4. Taylor expanded in wj around 0 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. unpow2 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in wj around 0 99.7%

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

      1. unpow2 99.7%

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

      2. fma-def 99.7%

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

      3. fma-udef 99.7%

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

      4. *-commutative 99.7%

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

   10. Simplified 99.7%

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

   if 6.9999999999999998e-9 < wj

   1. Initial program 49.5%

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

      1. sub-neg 49.5%

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

      2. div-sub 49.5%

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

      3. sub-neg 49.5%

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

      4. +-commutative 49.5%

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

      5. distribute-neg-in 49.5%

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

      6. remove-double-neg 49.5%

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

      7. sub-neg 49.5%

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

      8. div-sub 49.5%

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

      9. distribute-rgt1-in 49.5%

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

      10. associate-/l/ 49.5%

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

   3. Simplified 99.5%

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

      1. +-commutative 99.5%

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

      2. div-inv 99.5%

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

      3. fma-def 99.7%

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

   5. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.5 \cdot 10^{-9}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{elif}\;wj \leq 7 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, \mathsf{fma}\left(-2, wj \cdot x, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4.4 \cdot 10^{-9} \lor \neg \left(wj \leq 7.6 \cdot 10^{-9}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, \mathsf{fma}\left(-2, wj \cdot x, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -4.4e-9) (not (<= wj 7.6e-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 <= -4.4e-9) || !(wj <= 7.6e-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 <= -4.4e-9) || !(wj <= 7.6e-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[Or[LessEqual[wj, -4.4e-9], N[Not[LessEqual[wj, 7.6e-9]], $MachinePrecision]], 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 < -4.3999999999999997e-9 or 7.60000000000000023e-9 < wj

   1. Initial program 63.7%

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

      1. sub-neg 63.7%

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

      2. div-sub 63.7%

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

      3. sub-neg 63.7%

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

      4. +-commutative 63.7%

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

      5. distribute-neg-in 63.7%

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

      6. remove-double-neg 63.7%

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

      7. sub-neg 63.7%

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

      8. div-sub 63.7%

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

      9. distribute-rgt1-in 77.9%

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

      10. associate-/l/ 77.9%

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

   3. Simplified 99.3%

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

   if -4.3999999999999997e-9 < wj < 7.60000000000000023e-9

   1. Initial program 78.4%

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

      1. sub-neg 78.4%

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

      2. div-sub 78.4%

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

      3. sub-neg 78.4%

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

      4. +-commutative 78.4%

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

      5. distribute-neg-in 78.4%

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

      6. remove-double-neg 78.4%

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

      7. sub-neg 78.4%

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

      8. div-sub 78.4%

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

      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 78.4%

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

   4. Taylor expanded in wj around 0 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. unpow2 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in wj around 0 99.7%

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

      1. unpow2 99.7%

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

      2. fma-def 99.7%

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

      3. fma-udef 99.7%

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

      4. *-commutative 99.7%

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

   10. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.4 \cdot 10^{-9} \lor \neg \left(wj \leq 7.6 \cdot 10^{-9}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, \mathsf{fma}\left(-2, wj \cdot x, x\right)\right)\\ \end{array} \]

Alternative 4: 99.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -1.52 \cdot 10^{-6} \lor \neg \left(wj \leq 5.8 \cdot 10^{-8}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -1.52e-6) (not (<= wj 5.8e-8)))
   (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0)))
   (+
    (* (- 1.0 (+ (* x -4.0) (* x 1.5))) (pow wj 2.0))
    (+ x (* -2.0 (* wj x))))))

C

double code(double wj, double x) {
	double tmp;
	if ((wj <= -1.52e-6) || !(wj <= 5.8e-8)) {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = ((1.0 - ((x * -4.0) + (x * 1.5))) * pow(wj, 2.0)) + (x + (-2.0 * (wj * x)));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((wj <= (-1.52d-6)) .or. (.not. (wj <= 5.8d-8))) then
        tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0d0))
    else
        tmp = ((1.0d0 - ((x * (-4.0d0)) + (x * 1.5d0))) * (wj ** 2.0d0)) + (x + ((-2.0d0) * (wj * x)))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((wj <= -1.52e-6) || !(wj <= 5.8e-8)) {
		tmp = wj + (((x / Math.exp(wj)) - wj) / (wj + 1.0));
	} else {
		tmp = ((1.0 - ((x * -4.0) + (x * 1.5))) * Math.pow(wj, 2.0)) + (x + (-2.0 * (wj * x)));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (wj <= -1.52e-6) or not (wj <= 5.8e-8):
		tmp = wj + (((x / math.exp(wj)) - wj) / (wj + 1.0))
	else:
		tmp = ((1.0 - ((x * -4.0) + (x * 1.5))) * math.pow(wj, 2.0)) + (x + (-2.0 * (wj * x)))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((wj <= -1.52e-6) || !(wj <= 5.8e-8))
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(Float64(x * -4.0) + Float64(x * 1.5))) * (wj ^ 2.0)) + Float64(x + Float64(-2.0 * Float64(wj * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((wj <= -1.52e-6) || ~((wj <= 5.8e-8)))
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	else
		tmp = ((1.0 - ((x * -4.0) + (x * 1.5))) * (wj ^ 2.0)) + (x + (-2.0 * (wj * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[wj, -1.52e-6], N[Not[LessEqual[wj, 5.8e-8]], $MachinePrecision]], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -1.52000000000000006e-6 or 5.8000000000000003e-8 < wj

   1. Initial program 57.8%

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

      1. sub-neg 57.8%

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

      2. div-sub 57.8%

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

      3. sub-neg 57.8%

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

      4. +-commutative 57.8%

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

      5. distribute-neg-in 57.8%

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

      6. remove-double-neg 57.8%

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

      7. sub-neg 57.8%

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

      8. div-sub 57.8%

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

      9. distribute-rgt1-in 74.3%

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

      10. associate-/l/ 74.3%

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

   3. Simplified 99.3%

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

   if -1.52000000000000006e-6 < wj < 5.8000000000000003e-8

   1. Initial program 78.5%

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

      1. sub-neg 78.5%

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

      2. div-sub 78.5%

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

      3. sub-neg 78.5%

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

      4. +-commutative 78.5%

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

      5. distribute-neg-in 78.5%

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

      6. remove-double-neg 78.5%

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

      7. sub-neg 78.5%

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

      8. div-sub 78.5%

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

      9. distribute-rgt1-in 78.5%

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

      10. associate-/l/ 78.6%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in wj around 0 99.7%

      \[\leadsto \color{blue}{\left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) \cdot {wj}^{2} + \left(-2 \cdot \left(wj \cdot x\right) + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -1.52 \cdot 10^{-6} \lor \neg \left(wj \leq 5.8 \cdot 10^{-8}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2} + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\\ \end{array} \]

Alternative 5: 99.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4.4 \cdot 10^{-9} \lor \neg \left(wj \leq 7.6 \cdot 10^{-9}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -4.4e-9) (not (<= wj 7.6e-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 <= -4.4e-9) || !(wj <= 7.6e-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 <= (-4.4d-9)) .or. (.not. (wj <= 7.6d-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 <= -4.4e-9) || !(wj <= 7.6e-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 <= -4.4e-9) or not (wj <= 7.6e-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 <= -4.4e-9) || !(wj <= 7.6e-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 <= -4.4e-9) || ~((wj <= 7.6e-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[Or[LessEqual[wj, -4.4e-9], N[Not[LessEqual[wj, 7.6e-9]], $MachinePrecision]], 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 < -4.3999999999999997e-9 or 7.60000000000000023e-9 < wj

   1. Initial program 63.7%

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

      1. sub-neg 63.7%

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

      2. div-sub 63.7%

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

      3. sub-neg 63.7%

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

      4. +-commutative 63.7%

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

      5. distribute-neg-in 63.7%

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

      6. remove-double-neg 63.7%

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

      7. sub-neg 63.7%

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

      8. div-sub 63.7%

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

      9. distribute-rgt1-in 77.9%

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

      10. associate-/l/ 77.9%

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

   3. Simplified 99.3%

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

   if -4.3999999999999997e-9 < wj < 7.60000000000000023e-9

   1. Initial program 78.4%

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

      1. sub-neg 78.4%

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

      2. div-sub 78.4%

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

      3. sub-neg 78.4%

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

      4. +-commutative 78.4%

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

      5. distribute-neg-in 78.4%

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

      6. remove-double-neg 78.4%

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

      7. sub-neg 78.4%

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

      8. div-sub 78.4%

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

      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 78.4%

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

   4. Taylor expanded in wj around 0 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. unpow2 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.4 \cdot 10^{-9} \lor \neg \left(wj \leq 7.6 \cdot 10^{-9}\right):\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2 \cdot \left(wj \cdot x\right)\right) + wj \cdot wj\\ \end{array} \]

Alternative 6: 95.6% 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 77.6%

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

   1. sub-neg 77.6%

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

   2. div-sub 77.6%

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

   3. sub-neg 77.6%

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

   4. +-commutative 77.6%

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

   5. distribute-neg-in 77.6%

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

   6. remove-double-neg 77.6%

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

   7. sub-neg 77.6%

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

   8. div-sub 77.6%

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

   9. distribute-rgt1-in 78.3%

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

   10. associate-/l/ 78.4%

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

3. Simplified 79.5%

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

4. Taylor expanded in wj around 0 95.5%

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

5. Taylor expanded in x around 0 95.4%

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

   1. unpow2 95.4%

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

7. Simplified 95.4%

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

8. Final simplification 95.4%

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

Alternative 7: 95.1% accurate, 62.6× speedup

Math

\[\begin{array}{l} \\ x + wj \cdot wj \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

   1. sub-neg 77.6%

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

   2. div-sub 77.6%

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

   3. sub-neg 77.6%

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

   4. +-commutative 77.6%

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

   5. distribute-neg-in 77.6%

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

   6. remove-double-neg 77.6%

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

   7. sub-neg 77.6%

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

   8. div-sub 77.6%

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

   9. distribute-rgt1-in 78.3%

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

   10. associate-/l/ 78.4%

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

3. Simplified 79.5%

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

4. Taylor expanded in wj around 0 95.5%

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

5. Taylor expanded in x around 0 95.4%

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

   1. unpow2 95.4%

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

7. Simplified 95.4%

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

8. Taylor expanded in wj around 0 94.8%

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

9. Final simplification 94.8%

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

Alternative 8: 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 77.6%

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

   1. sub-neg 77.6%

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

   2. div-sub 77.6%

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

   3. sub-neg 77.6%

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

   4. +-commutative 77.6%

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

   5. distribute-neg-in 77.6%

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

   6. remove-double-neg 77.6%

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

   7. sub-neg 77.6%

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

   8. div-sub 77.6%

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

   9. distribute-rgt1-in 78.3%

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

   10. associate-/l/ 78.4%

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

3. Simplified 79.5%

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

4. Taylor expanded in wj around inf 4.6%

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

5. Final simplification 4.6%

   \[\leadsto wj \]

Alternative 9: 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 77.6%

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

   1. sub-neg 77.6%

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

   2. div-sub 77.6%

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

   3. sub-neg 77.6%

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

   4. +-commutative 77.6%

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

   5. distribute-neg-in 77.6%

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

   6. remove-double-neg 77.6%

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

   7. sub-neg 77.6%

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

   8. div-sub 77.6%

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

   9. distribute-rgt1-in 78.3%

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

   10. associate-/l/ 78.4%

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

3. Simplified 79.5%

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

4. Taylor expanded in wj around 0 84.3%

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

5. Final simplification 84.3%

   \[\leadsto x \]

Developer target: 78.5% 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 2023196 
(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))))))