Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.3% → 98.4%

Time: 10.4s

Alternatives: 12

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -1.2926239702709963e+274
  2. x = 3.381808814092577e-112

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: 78.3% 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: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t_0}{e^{wj} + t_0} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot -3\right) - x \cdot 0.6666666666666666\right) + \left(wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 2e-13)
     (+
      (*
       (pow wj 3.0)
       (-
        (- (- -1.0 (* -2.0 (+ (* x -4.0) (* x 1.5)))) (* x -3.0))
        (* x 0.6666666666666666)))
      (+ (* wj wj) (+ x (* -2.0 (* wj x)))))
     (fma (- (/ x (exp wj)) wj) (/ 1.0 (+ wj 1.0)) wj))))

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-13) {
		tmp = (pow(wj, 3.0) * (((-1.0 - (-2.0 * ((x * -4.0) + (x * 1.5)))) - (x * -3.0)) - (x * 0.6666666666666666))) + ((wj * wj) + (x + (-2.0 * (wj * x))));
	} else {
		tmp = fma(((x / exp(wj)) - wj), (1.0 / (wj + 1.0)), wj);
	}
	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-13)
		tmp = Float64(Float64((wj ^ 3.0) * Float64(Float64(Float64(-1.0 - Float64(-2.0 * Float64(Float64(x * -4.0) + Float64(x * 1.5)))) - Float64(x * -3.0)) - Float64(x * 0.6666666666666666))) + Float64(Float64(wj * wj) + Float64(x + Float64(-2.0 * Float64(wj * x)))));
	else
		tmp = fma(Float64(Float64(x / exp(wj)) - wj), Float64(1.0 / Float64(wj + 1.0)), wj);
	end
	return 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-13], N[(N[(N[Power[wj, 3.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[(-2.0 * N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * -3.0), $MachinePrecision]), $MachinePrecision] - N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(wj * wj), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + wj), $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))))) < 2.0000000000000001e-13

   1. Initial program 69.4%

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

      1. sub-neg 69.4%

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

      2. div-sub 69.4%

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

      3. sub-neg 69.4%

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

      4. +-commutative 69.4%

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

      5. distribute-neg-in 69.4%

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

      6. remove-double-neg 69.4%

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

      7. sub-neg 69.4%

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

      8. div-sub 69.4%

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

      9. distribute-rgt1-in 69.4%

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

      10. associate-/l/ 69.4%

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

   3. Simplified 69.4%

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

   4. Taylor expanded in wj around 0 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. unpow2 99.9%

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

   7. Simplified 99.9%

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

   if 2.0000000000000001e-13 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 96.8%

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

      1. sub-neg 96.8%

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

      2. div-sub 96.8%

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

      3. sub-neg 96.8%

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

      4. +-commutative 96.8%

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

      5. distribute-neg-in 96.8%

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

      6. remove-double-neg 96.8%

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

      7. sub-neg 96.8%

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

      8. div-sub 96.8%

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

      9. distribute-rgt1-in 96.9%

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

      10. associate-/l/ 96.9%

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

   3. Simplified 99.6%

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

      1. +-commutative 99.6%

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

      2. div-inv 99.6%

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

      3. fma-def 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;{wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot -3\right) - x \cdot 0.6666666666666666\right) + \left(wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{e^{wj}} - wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \]

Alternative 2: 96.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ {wj}^{3} \cdot \left(\left(\left(-1 - -2 \cdot \left(x \cdot -4 + x \cdot 1.5\right)\right) - x \cdot -3\right) - x \cdot 0.6666666666666666\right) + \left(wj \cdot wj + \left(x + -2 \cdot \left(wj \cdot x\right)\right)\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (+
  (*
   (pow wj 3.0)
   (-
    (- (- -1.0 (* -2.0 (+ (* x -4.0) (* x 1.5)))) (* x -3.0))
    (* x 0.6666666666666666)))
  (+ (* wj wj) (+ x (* -2.0 (* wj x))))))

C

double code(double wj, double x) {
	return (pow(wj, 3.0) * (((-1.0 - (-2.0 * ((x * -4.0) + (x * 1.5)))) - (x * -3.0)) - (x * 0.6666666666666666))) + ((wj * wj) + (x + (-2.0 * (wj * x))));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = ((wj ** 3.0d0) * ((((-1.0d0) - ((-2.0d0) * ((x * (-4.0d0)) + (x * 1.5d0)))) - (x * (-3.0d0))) - (x * 0.6666666666666666d0))) + ((wj * wj) + (x + ((-2.0d0) * (wj * x))))
end function

Java

public static double code(double wj, double x) {
	return (Math.pow(wj, 3.0) * (((-1.0 - (-2.0 * ((x * -4.0) + (x * 1.5)))) - (x * -3.0)) - (x * 0.6666666666666666))) + ((wj * wj) + (x + (-2.0 * (wj * x))));
}

Python

def code(wj, x):
	return (math.pow(wj, 3.0) * (((-1.0 - (-2.0 * ((x * -4.0) + (x * 1.5)))) - (x * -3.0)) - (x * 0.6666666666666666))) + ((wj * wj) + (x + (-2.0 * (wj * x))))

Julia

function code(wj, x)
	return Float64(Float64((wj ^ 3.0) * Float64(Float64(Float64(-1.0 - Float64(-2.0 * Float64(Float64(x * -4.0) + Float64(x * 1.5)))) - Float64(x * -3.0)) - Float64(x * 0.6666666666666666))) + Float64(Float64(wj * wj) + Float64(x + Float64(-2.0 * Float64(wj * x)))))
end

MATLAB

function tmp = code(wj, x)
	tmp = ((wj ^ 3.0) * (((-1.0 - (-2.0 * ((x * -4.0) + (x * 1.5)))) - (x * -3.0)) - (x * 0.6666666666666666))) + ((wj * wj) + (x + (-2.0 * (wj * x))));
end

Wolfram

code[wj_, x_] := N[(N[(N[Power[wj, 3.0], $MachinePrecision] * N[(N[(N[(-1.0 - N[(-2.0 * N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * -3.0), $MachinePrecision]), $MachinePrecision] - N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(wj * wj), $MachinePrecision] + N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.0%

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

   1. sub-neg 77.0%

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

   2. div-sub 77.0%

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

   3. sub-neg 77.0%

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

   4. +-commutative 77.0%

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

   5. distribute-neg-in 77.0%

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

   6. remove-double-neg 77.0%

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

   7. sub-neg 77.0%

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

   8. div-sub 77.0%

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

   9. distribute-rgt1-in 77.0%

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

   10. associate-/l/ 77.0%

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

3. Simplified 77.8%

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

4. Taylor expanded in wj around 0 97.1%

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

5. Taylor expanded in x around 0 97.0%

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

   1. unpow2 97.0%

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

7. Simplified 97.0%

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

8. Final simplification 97.0%

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

Alternative 3: 96.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \left(x + -2 \cdot \left(wj \cdot x\right)\right) + \left(1 - \left(x \cdot -4 + x \cdot 1.5\right)\right) \cdot {wj}^{2} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (+ (+ x (* -2.0 (* wj x))) (* (- 1.0 (+ (* x -4.0) (* x 1.5))) (pow wj 2.0))))

C

double code(double wj, double x) {
	return (x + (-2.0 * (wj * x))) + ((1.0 - ((x * -4.0) + (x * 1.5))) * pow(wj, 2.0));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = (x + ((-2.0d0) * (wj * x))) + ((1.0d0 - ((x * (-4.0d0)) + (x * 1.5d0))) * (wj ** 2.0d0))
end function

Java

public static double code(double wj, double x) {
	return (x + (-2.0 * (wj * x))) + ((1.0 - ((x * -4.0) + (x * 1.5))) * Math.pow(wj, 2.0));
}

Python

def code(wj, x):
	return (x + (-2.0 * (wj * x))) + ((1.0 - ((x * -4.0) + (x * 1.5))) * math.pow(wj, 2.0))

Julia

function code(wj, x)
	return Float64(Float64(x + Float64(-2.0 * Float64(wj * x))) + Float64(Float64(1.0 - Float64(Float64(x * -4.0) + Float64(x * 1.5))) * (wj ^ 2.0)))
end

MATLAB

function tmp = code(wj, x)
	tmp = (x + (-2.0 * (wj * x))) + ((1.0 - ((x * -4.0) + (x * 1.5))) * (wj ^ 2.0));
end

Wolfram

code[wj_, x_] := N[(N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[wj, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.0%

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

   1. sub-neg 77.0%

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

   2. div-sub 77.0%

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

   3. sub-neg 77.0%

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

   4. +-commutative 77.0%

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

   5. distribute-neg-in 77.0%

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

   6. remove-double-neg 77.0%

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

   7. sub-neg 77.0%

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

   8. div-sub 77.0%

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

   9. distribute-rgt1-in 77.0%

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

   10. associate-/l/ 77.0%

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

3. Simplified 77.8%

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

4. Taylor expanded in wj around 0 96.8%

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

5. Final simplification 96.8%

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

Alternative 4: 84.3% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -3.4 \cdot 10^{-30}:\\ \;\;\;\;wj \cdot \left(wj \cdot \left(1 - wj\right)\right)\\ \mathbf{elif}\;wj \leq 2.6 \cdot 10^{-55}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 8.9 \cdot 10^{-45}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj + x \cdot \left(wj + -1\right)}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -3.4e-30)
   (* wj (* wj (- 1.0 wj)))
   (if (<= wj 2.6e-55)
     x
     (if (<= wj 8.9e-45)
       (* wj wj)
       (- wj (/ (+ wj (* x (+ wj -1.0))) (+ wj 1.0)))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -3.4e-30) {
		tmp = wj * (wj * (1.0 - wj));
	} else if (wj <= 2.6e-55) {
		tmp = x;
	} else if (wj <= 8.9e-45) {
		tmp = wj * wj;
	} else {
		tmp = wj - ((wj + (x * (wj + -1.0))) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-3.4d-30)) then
        tmp = wj * (wj * (1.0d0 - wj))
    else if (wj <= 2.6d-55) then
        tmp = x
    else if (wj <= 8.9d-45) then
        tmp = wj * wj
    else
        tmp = wj - ((wj + (x * (wj + (-1.0d0)))) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -3.4e-30) {
		tmp = wj * (wj * (1.0 - wj));
	} else if (wj <= 2.6e-55) {
		tmp = x;
	} else if (wj <= 8.9e-45) {
		tmp = wj * wj;
	} else {
		tmp = wj - ((wj + (x * (wj + -1.0))) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -3.4e-30:
		tmp = wj * (wj * (1.0 - wj))
	elif wj <= 2.6e-55:
		tmp = x
	elif wj <= 8.9e-45:
		tmp = wj * wj
	else:
		tmp = wj - ((wj + (x * (wj + -1.0))) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -3.4e-30)
		tmp = Float64(wj * Float64(wj * Float64(1.0 - wj)));
	elseif (wj <= 2.6e-55)
		tmp = x;
	elseif (wj <= 8.9e-45)
		tmp = Float64(wj * wj);
	else
		tmp = Float64(wj - Float64(Float64(wj + Float64(x * Float64(wj + -1.0))) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -3.4e-30)
		tmp = wj * (wj * (1.0 - wj));
	elseif (wj <= 2.6e-55)
		tmp = x;
	elseif (wj <= 8.9e-45)
		tmp = wj * wj;
	else
		tmp = wj - ((wj + (x * (wj + -1.0))) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -3.4e-30], N[(wj * N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 2.6e-55], x, If[LessEqual[wj, 8.9e-45], N[(wj * wj), $MachinePrecision], N[(wj - N[(N[(wj + N[(x * N[(wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if wj < -3.4000000000000003e-30

   1. Initial program 53.3%

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

      1. sub-neg 53.3%

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

      2. div-sub 53.3%

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

      3. sub-neg 53.3%

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

      4. +-commutative 53.3%

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

      5. distribute-neg-in 53.3%

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

      6. remove-double-neg 53.3%

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

      7. sub-neg 53.3%

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

      8. div-sub 53.3%

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

      9. distribute-rgt1-in 53.4%

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

      10. associate-/l/ 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in x around 0 20.8%

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

      1. +-commutative 20.8%

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

   6. Simplified 20.8%

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

   7. Taylor expanded in wj around 0 17.8%

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

      1. associate-+r+ 17.8%

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

      2. +-commutative 17.8%

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

      3. cube-mult 17.8%

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

      4. unpow2 17.8%

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

      5. distribute-rgt-out 17.8%

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

      6. unpow2 17.8%

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

      7. +-commutative 17.8%

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

   9. Simplified 17.8%

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

       1. associate--r+ 62.2%

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

       2. +-inverses 62.2%

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

       3. metadata-eval 62.2%

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

       4. associate-*l* 62.3%

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

       5. cancel-sign-sub-inv 62.3%

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

       6. metadata-eval 62.3%

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

   11. Applied egg-rr 62.3%

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

   if -3.4000000000000003e-30 < wj < 2.5999999999999999e-55

   1. Initial program 81.3%

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

      1. sub-neg 81.3%

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

      2. div-sub 81.3%

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

      3. sub-neg 81.3%

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

      4. +-commutative 81.3%

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

      5. distribute-neg-in 81.3%

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

      6. remove-double-neg 81.3%

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

      7. sub-neg 81.3%

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

      8. div-sub 81.3%

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

      9. distribute-rgt1-in 81.3%

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

      10. associate-/l/ 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in wj around 0 92.9%

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

   if 2.5999999999999999e-55 < wj < 8.90000000000000034e-45

   1. Initial program 17.7%

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

      1. sub-neg 17.7%

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

      2. div-sub 17.7%

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

      3. sub-neg 17.7%

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

      4. +-commutative 17.7%

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

      5. distribute-neg-in 17.7%

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

      6. remove-double-neg 17.7%

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

      7. sub-neg 17.7%

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

      8. div-sub 17.7%

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

      9. distribute-rgt1-in 17.7%

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

      10. associate-/l/ 17.7%

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

   3. Simplified 17.7%

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

   4. Taylor expanded in x around 0 3.8%

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

      1. +-commutative 3.8%

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

   6. Simplified 3.8%

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

   7. Taylor expanded in wj around 0 86.0%

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

      1. unpow2 86.0%

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

   9. Simplified 86.0%

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

   if 8.90000000000000034e-45 < wj

   1. Initial program 65.2%

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

      1. sub-neg 65.2%

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

      2. div-sub 65.2%

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

      3. sub-neg 65.2%

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

      4. +-commutative 65.2%

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

      5. distribute-neg-in 65.2%

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

      6. remove-double-neg 65.2%

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

      7. sub-neg 65.2%

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

      8. div-sub 65.2%

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

      9. distribute-rgt1-in 65.4%

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

      10. associate-/l/ 65.6%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in wj around 0 69.3%

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

      1. associate-*r* 69.3%

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

      2. neg-mul-1 69.3%

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

      3. distribute-lft1-in 69.3%

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

      4. +-commutative 69.3%

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

      5. sub-neg 69.3%

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

   6. Simplified 69.3%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.4 \cdot 10^{-30}:\\ \;\;\;\;wj \cdot \left(wj \cdot \left(1 - wj\right)\right)\\ \mathbf{elif}\;wj \leq 2.6 \cdot 10^{-55}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 8.9 \cdot 10^{-45}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj + x \cdot \left(wj + -1\right)}{wj + 1}\\ \end{array} \]

Alternative 5: 95.9% accurate, 16.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. sub-neg 77.0%

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

   2. div-sub 77.0%

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

   3. sub-neg 77.0%

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

   4. +-commutative 77.0%

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

   5. distribute-neg-in 77.0%

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

   6. remove-double-neg 77.0%

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

   7. sub-neg 77.0%

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

   8. div-sub 77.0%

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

   9. distribute-rgt1-in 77.0%

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

   10. associate-/l/ 77.0%

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

3. Simplified 77.8%

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

4. Taylor expanded in wj around 0 76.1%

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

   1. associate-*r* 76.1%

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

   2. neg-mul-1 76.1%

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

   3. distribute-lft1-in 76.1%

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

   4. +-commutative 76.1%

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

   5. sub-neg 76.1%

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

6. Simplified 76.1%

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

7. Taylor expanded in wj around 0 96.7%

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

8. Taylor expanded in x around 0 96.7%

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

   1. unpow2 96.7%

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

   2. *-commutative 96.7%

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

   3. unpow2 96.7%

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

10. Simplified 96.7%

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

11. Final simplification 96.7%

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

Alternative 6: 95.8% accurate, 28.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. sub-neg 77.0%

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

   2. div-sub 77.0%

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

   3. sub-neg 77.0%

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

   4. +-commutative 77.0%

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

   5. distribute-neg-in 77.0%

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

   6. remove-double-neg 77.0%

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

   7. sub-neg 77.0%

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

   8. div-sub 77.0%

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

   9. distribute-rgt1-in 77.0%

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

   10. associate-/l/ 77.0%

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

3. Simplified 77.8%

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

4. Taylor expanded in wj around 0 76.1%

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

   1. associate-*r* 76.1%

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

   2. neg-mul-1 76.1%

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

   3. distribute-lft1-in 76.1%

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

   4. +-commutative 76.1%

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

   5. sub-neg 76.1%

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

6. Simplified 76.1%

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

7. Taylor expanded in wj around 0 96.7%

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

8. Taylor expanded in x around 0 96.7%

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

   1. unpow2 96.7%

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

   2. *-commutative 96.7%

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

   3. unpow2 96.7%

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

10. Simplified 96.7%

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

11. Taylor expanded in x around 0 96.7%

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

    1. unpow2 96.7%

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

13. Simplified 96.7%

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

14. Final simplification 96.7%

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

Alternative 7: 82.8% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -1.45 \cdot 10^{-30}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq 2.6 \cdot 10^{-55}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 8.8 \cdot 10^{-45}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -1.45e-30)
   (* wj wj)
   (if (<= wj 2.6e-55) x (if (<= wj 8.8e-45) (* wj wj) x))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -1.45e-30) {
		tmp = wj * wj;
	} else if (wj <= 2.6e-55) {
		tmp = x;
	} else if (wj <= 8.8e-45) {
		tmp = wj * wj;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= (-1.45d-30)) then
        tmp = wj * wj
    else if (wj <= 2.6d-55) then
        tmp = x
    else if (wj <= 8.8d-45) then
        tmp = wj * wj
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -1.45e-30) {
		tmp = wj * wj;
	} else if (wj <= 2.6e-55) {
		tmp = x;
	} else if (wj <= 8.8e-45) {
		tmp = wj * wj;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -1.45e-30:
		tmp = wj * wj
	elif wj <= 2.6e-55:
		tmp = x
	elif wj <= 8.8e-45:
		tmp = wj * wj
	else:
		tmp = x
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -1.45e-30)
		tmp = Float64(wj * wj);
	elseif (wj <= 2.6e-55)
		tmp = x;
	elseif (wj <= 8.8e-45)
		tmp = Float64(wj * wj);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -1.45e-30)
		tmp = wj * wj;
	elseif (wj <= 2.6e-55)
		tmp = x;
	elseif (wj <= 8.8e-45)
		tmp = wj * wj;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -1.45e-30], N[(wj * wj), $MachinePrecision], If[LessEqual[wj, 2.6e-55], x, If[LessEqual[wj, 8.8e-45], N[(wj * wj), $MachinePrecision], x]]]

Derivation

1. Split input into 2 regimes

2. if wj < -1.44999999999999995e-30 or 2.5999999999999999e-55 < wj < 8.79999999999999974e-45

   1. Initial program 40.2%

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

      1. sub-neg 40.2%

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

      2. div-sub 40.2%

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

      3. sub-neg 40.2%

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

      4. +-commutative 40.2%

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

      5. distribute-neg-in 40.2%

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

      6. remove-double-neg 40.2%

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

      7. sub-neg 40.2%

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

      8. div-sub 40.2%

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

      9. distribute-rgt1-in 40.3%

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

      10. associate-/l/ 40.2%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in x around 0 14.6%

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

      1. +-commutative 14.6%

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

   6. Simplified 14.6%

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

   7. Taylor expanded in wj around 0 67.1%

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

      1. unpow2 67.1%

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

   9. Simplified 67.1%

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

   if -1.44999999999999995e-30 < wj < 2.5999999999999999e-55 or 8.79999999999999974e-45 < wj

   1. Initial program 80.0%

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

      1. sub-neg 80.0%

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

      2. div-sub 80.0%

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

      3. sub-neg 80.0%

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

      4. +-commutative 80.0%

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

      5. distribute-neg-in 80.0%

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

      6. remove-double-neg 80.0%

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

      7. sub-neg 80.0%

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

      8. div-sub 80.0%

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

      9. distribute-rgt1-in 80.0%

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

      10. associate-/l/ 80.0%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in wj around 0 89.8%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -1.45 \cdot 10^{-30}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq 2.6 \cdot 10^{-55}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 8.8 \cdot 10^{-45}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 82.9% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -1.45 \cdot 10^{-30}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq 2.6 \cdot 10^{-55}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 8.8 \cdot 10^{-45}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -1.45e-30)
   (* wj wj)
   (if (<= wj 2.6e-55) x (if (<= wj 8.8e-45) (* wj wj) (+ wj x)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -1.45e-30) {
		tmp = wj * wj;
	} else if (wj <= 2.6e-55) {
		tmp = x;
	} else if (wj <= 8.8e-45) {
		tmp = wj * wj;
	} else {
		tmp = 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 <= (-1.45d-30)) then
        tmp = wj * wj
    else if (wj <= 2.6d-55) then
        tmp = x
    else if (wj <= 8.8d-45) then
        tmp = wj * wj
    else
        tmp = wj + x
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -1.45e-30) {
		tmp = wj * wj;
	} else if (wj <= 2.6e-55) {
		tmp = x;
	} else if (wj <= 8.8e-45) {
		tmp = wj * wj;
	} else {
		tmp = wj + x;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -1.45e-30:
		tmp = wj * wj
	elif wj <= 2.6e-55:
		tmp = x
	elif wj <= 8.8e-45:
		tmp = wj * wj
	else:
		tmp = wj + x
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -1.45e-30)
		tmp = Float64(wj * wj);
	elseif (wj <= 2.6e-55)
		tmp = x;
	elseif (wj <= 8.8e-45)
		tmp = Float64(wj * wj);
	else
		tmp = Float64(wj + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -1.45e-30)
		tmp = wj * wj;
	elseif (wj <= 2.6e-55)
		tmp = x;
	elseif (wj <= 8.8e-45)
		tmp = wj * wj;
	else
		tmp = wj + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -1.45e-30], N[(wj * wj), $MachinePrecision], If[LessEqual[wj, 2.6e-55], x, If[LessEqual[wj, 8.8e-45], N[(wj * wj), $MachinePrecision], N[(wj + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -1.44999999999999995e-30 or 2.5999999999999999e-55 < wj < 8.79999999999999974e-45

   1. Initial program 40.2%

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

      1. sub-neg 40.2%

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

      2. div-sub 40.2%

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

      3. sub-neg 40.2%

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

      4. +-commutative 40.2%

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

      5. distribute-neg-in 40.2%

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

      6. remove-double-neg 40.2%

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

      7. sub-neg 40.2%

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

      8. div-sub 40.2%

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

      9. distribute-rgt1-in 40.3%

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

      10. associate-/l/ 40.2%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in x around 0 14.6%

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

      1. +-commutative 14.6%

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

   6. Simplified 14.6%

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

   7. Taylor expanded in wj around 0 67.1%

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

      1. unpow2 67.1%

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

   9. Simplified 67.1%

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

   if -1.44999999999999995e-30 < wj < 2.5999999999999999e-55

   1. Initial program 81.3%

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

      1. sub-neg 81.3%

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

      2. div-sub 81.3%

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

      3. sub-neg 81.3%

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

      4. +-commutative 81.3%

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

      5. distribute-neg-in 81.3%

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

      6. remove-double-neg 81.3%

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

      7. sub-neg 81.3%

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

      8. div-sub 81.3%

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

      9. distribute-rgt1-in 81.3%

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

      10. associate-/l/ 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in wj around 0 92.9%

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

   if 8.79999999999999974e-45 < wj

   1. Initial program 65.2%

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

      1. distribute-rgt1-in 65.4%

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

   3. Simplified 65.4%

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

   4. Taylor expanded in wj around 0 59.0%

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

      1. neg-mul-1 59.0%

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

   6. Simplified 59.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -1.45 \cdot 10^{-30}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{elif}\;wj \leq 2.6 \cdot 10^{-55}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 8.8 \cdot 10^{-45}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + x\\ \end{array} \]

Alternative 9: 83.1% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -3.2 \cdot 10^{-30}:\\ \;\;\;\;wj \cdot \left(wj - wj \cdot wj\right)\\ \mathbf{elif}\;wj \leq 2.6 \cdot 10^{-55}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 8.8 \cdot 10^{-45}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -3.2e-30)
   (* wj (- wj (* wj wj)))
   (if (<= wj 2.6e-55) x (if (<= wj 8.8e-45) (* wj wj) (+ wj x)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -3.2e-30) {
		tmp = wj * (wj - (wj * wj));
	} else if (wj <= 2.6e-55) {
		tmp = x;
	} else if (wj <= 8.8e-45) {
		tmp = wj * wj;
	} else {
		tmp = 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 <= (-3.2d-30)) then
        tmp = wj * (wj - (wj * wj))
    else if (wj <= 2.6d-55) then
        tmp = x
    else if (wj <= 8.8d-45) then
        tmp = wj * wj
    else
        tmp = wj + x
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -3.2e-30) {
		tmp = wj * (wj - (wj * wj));
	} else if (wj <= 2.6e-55) {
		tmp = x;
	} else if (wj <= 8.8e-45) {
		tmp = wj * wj;
	} else {
		tmp = wj + x;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -3.2e-30:
		tmp = wj * (wj - (wj * wj))
	elif wj <= 2.6e-55:
		tmp = x
	elif wj <= 8.8e-45:
		tmp = wj * wj
	else:
		tmp = wj + x
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -3.2e-30)
		tmp = Float64(wj * Float64(wj - Float64(wj * wj)));
	elseif (wj <= 2.6e-55)
		tmp = x;
	elseif (wj <= 8.8e-45)
		tmp = Float64(wj * wj);
	else
		tmp = Float64(wj + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -3.2e-30)
		tmp = wj * (wj - (wj * wj));
	elseif (wj <= 2.6e-55)
		tmp = x;
	elseif (wj <= 8.8e-45)
		tmp = wj * wj;
	else
		tmp = wj + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -3.2e-30], N[(wj * N[(wj - N[(wj * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 2.6e-55], x, If[LessEqual[wj, 8.8e-45], N[(wj * wj), $MachinePrecision], N[(wj + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if wj < -3.2e-30

   1. Initial program 53.3%

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

      1. sub-neg 53.3%

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

      2. div-sub 53.3%

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

      3. sub-neg 53.3%

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

      4. +-commutative 53.3%

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

      5. distribute-neg-in 53.3%

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

      6. remove-double-neg 53.3%

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

      7. sub-neg 53.3%

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

      8. div-sub 53.3%

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

      9. distribute-rgt1-in 53.4%

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

      10. associate-/l/ 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in x around 0 20.8%

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

      1. +-commutative 20.8%

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

   6. Simplified 20.8%

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

   7. Taylor expanded in wj around 0 62.2%

      \[\leadsto \color{blue}{{wj}^{2} + -1 \cdot {wj}^{3}} \]
   8. Step-by-step derivation

      1. mul-1-neg 62.2%

         \[\leadsto {wj}^{2} + \color{blue}{\left(-{wj}^{3}\right)} \]

      2. unsub-neg 62.2%

         \[\leadsto \color{blue}{{wj}^{2} - {wj}^{3}} \]

      3. unpow2 62.2%

         \[\leadsto \color{blue}{wj \cdot wj} - {wj}^{3} \]

   9. Simplified 62.2%

      \[\leadsto \color{blue}{wj \cdot wj - {wj}^{3}} \]
   10. Step-by-step derivation

       1. cube-mult 62.2%

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

       2. distribute-lft-out-- 62.2%

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

   11. Applied egg-rr 62.2%

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

   if -3.2e-30 < wj < 2.5999999999999999e-55

   1. Initial program 81.3%

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

      1. sub-neg 81.3%

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

      2. div-sub 81.3%

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

      3. sub-neg 81.3%

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

      4. +-commutative 81.3%

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

      5. distribute-neg-in 81.3%

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

      6. remove-double-neg 81.3%

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

      7. sub-neg 81.3%

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

      8. div-sub 81.3%

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

      9. distribute-rgt1-in 81.3%

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

      10. associate-/l/ 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in wj around 0 92.9%

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

   if 2.5999999999999999e-55 < wj < 8.79999999999999974e-45

   1. Initial program 17.7%

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

      1. sub-neg 17.7%

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

      2. div-sub 17.7%

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

      3. sub-neg 17.7%

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

      4. +-commutative 17.7%

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

      5. distribute-neg-in 17.7%

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

      6. remove-double-neg 17.7%

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

      7. sub-neg 17.7%

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

      8. div-sub 17.7%

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

      9. distribute-rgt1-in 17.7%

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

      10. associate-/l/ 17.7%

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

   3. Simplified 17.7%

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

   4. Taylor expanded in x around 0 3.8%

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

      1. +-commutative 3.8%

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

   6. Simplified 3.8%

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

   7. Taylor expanded in wj around 0 86.0%

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

      1. unpow2 86.0%

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

   9. Simplified 86.0%

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

   if 8.79999999999999974e-45 < wj

   1. Initial program 65.2%

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

      1. distribute-rgt1-in 65.4%

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

   3. Simplified 65.4%

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

   4. Taylor expanded in wj around 0 59.0%

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

      1. neg-mul-1 59.0%

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

   6. Simplified 59.0%

      \[\leadsto wj - \color{blue}{\left(-x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.2 \cdot 10^{-30}:\\ \;\;\;\;wj \cdot \left(wj - wj \cdot wj\right)\\ \mathbf{elif}\;wj \leq 2.6 \cdot 10^{-55}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 8.8 \cdot 10^{-45}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + x\\ \end{array} \]

Alternative 10: 83.1% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -3.1 \cdot 10^{-30}:\\ \;\;\;\;wj \cdot \left(wj \cdot \left(1 - wj\right)\right)\\ \mathbf{elif}\;wj \leq 2.6 \cdot 10^{-55}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 8.8 \cdot 10^{-45}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -3.1e-30)
   (* wj (* wj (- 1.0 wj)))
   (if (<= wj 2.6e-55) x (if (<= wj 8.8e-45) (* wj wj) (+ wj x)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -3.1e-30) {
		tmp = wj * (wj * (1.0 - wj));
	} else if (wj <= 2.6e-55) {
		tmp = x;
	} else if (wj <= 8.8e-45) {
		tmp = wj * wj;
	} else {
		tmp = 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 <= (-3.1d-30)) then
        tmp = wj * (wj * (1.0d0 - wj))
    else if (wj <= 2.6d-55) then
        tmp = x
    else if (wj <= 8.8d-45) then
        tmp = wj * wj
    else
        tmp = wj + x
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= -3.1e-30) {
		tmp = wj * (wj * (1.0 - wj));
	} else if (wj <= 2.6e-55) {
		tmp = x;
	} else if (wj <= 8.8e-45) {
		tmp = wj * wj;
	} else {
		tmp = wj + x;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= -3.1e-30:
		tmp = wj * (wj * (1.0 - wj))
	elif wj <= 2.6e-55:
		tmp = x
	elif wj <= 8.8e-45:
		tmp = wj * wj
	else:
		tmp = wj + x
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -3.1e-30)
		tmp = Float64(wj * Float64(wj * Float64(1.0 - wj)));
	elseif (wj <= 2.6e-55)
		tmp = x;
	elseif (wj <= 8.8e-45)
		tmp = Float64(wj * wj);
	else
		tmp = Float64(wj + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= -3.1e-30)
		tmp = wj * (wj * (1.0 - wj));
	elseif (wj <= 2.6e-55)
		tmp = x;
	elseif (wj <= 8.8e-45)
		tmp = wj * wj;
	else
		tmp = wj + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -3.1e-30], N[(wj * N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 2.6e-55], x, If[LessEqual[wj, 8.8e-45], N[(wj * wj), $MachinePrecision], N[(wj + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if wj < -3.09999999999999991e-30

   1. Initial program 53.3%

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

      1. sub-neg 53.3%

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

      2. div-sub 53.3%

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

      3. sub-neg 53.3%

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

      4. +-commutative 53.3%

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

      5. distribute-neg-in 53.3%

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

      6. remove-double-neg 53.3%

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

      7. sub-neg 53.3%

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

      8. div-sub 53.3%

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

      9. distribute-rgt1-in 53.4%

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

      10. associate-/l/ 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in x around 0 20.8%

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

      1. +-commutative 20.8%

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

   6. Simplified 20.8%

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

   7. Taylor expanded in wj around 0 17.8%

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

      1. associate-+r+ 17.8%

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

      2. +-commutative 17.8%

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

      3. cube-mult 17.8%

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

      4. unpow2 17.8%

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

      5. distribute-rgt-out 17.8%

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

      6. unpow2 17.8%

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

      7. +-commutative 17.8%

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

   9. Simplified 17.8%

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

       1. associate--r+ 62.2%

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

       2. +-inverses 62.2%

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

       3. metadata-eval 62.2%

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

       4. associate-*l* 62.3%

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

       5. cancel-sign-sub-inv 62.3%

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

       6. metadata-eval 62.3%

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

   11. Applied egg-rr 62.3%

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

   if -3.09999999999999991e-30 < wj < 2.5999999999999999e-55

   1. Initial program 81.3%

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

      1. sub-neg 81.3%

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

      2. div-sub 81.3%

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

      3. sub-neg 81.3%

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

      4. +-commutative 81.3%

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

      5. distribute-neg-in 81.3%

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

      6. remove-double-neg 81.3%

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

      7. sub-neg 81.3%

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

      8. div-sub 81.3%

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

      9. distribute-rgt1-in 81.3%

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

      10. associate-/l/ 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in wj around 0 92.9%

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

   if 2.5999999999999999e-55 < wj < 8.79999999999999974e-45

   1. Initial program 17.7%

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

      1. sub-neg 17.7%

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

      2. div-sub 17.7%

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

      3. sub-neg 17.7%

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

      4. +-commutative 17.7%

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

      5. distribute-neg-in 17.7%

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

      6. remove-double-neg 17.7%

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

      7. sub-neg 17.7%

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

      8. div-sub 17.7%

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

      9. distribute-rgt1-in 17.7%

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

      10. associate-/l/ 17.7%

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

   3. Simplified 17.7%

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

   4. Taylor expanded in x around 0 3.8%

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

      1. +-commutative 3.8%

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

   6. Simplified 3.8%

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

   7. Taylor expanded in wj around 0 86.0%

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

      1. unpow2 86.0%

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

   9. Simplified 86.0%

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

   if 8.79999999999999974e-45 < wj

   1. Initial program 65.2%

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

      1. distribute-rgt1-in 65.4%

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

   3. Simplified 65.4%

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

   4. Taylor expanded in wj around 0 59.0%

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

      1. neg-mul-1 59.0%

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

   6. Simplified 59.0%

      \[\leadsto wj - \color{blue}{\left(-x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.1 \cdot 10^{-30}:\\ \;\;\;\;wj \cdot \left(wj \cdot \left(1 - wj\right)\right)\\ \mathbf{elif}\;wj \leq 2.6 \cdot 10^{-55}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj \leq 8.8 \cdot 10^{-45}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + x\\ \end{array} \]

Alternative 11: 4.3% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 77.0%

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

   1. sub-neg 77.0%

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

   2. div-sub 77.0%

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

   3. sub-neg 77.0%

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

   4. +-commutative 77.0%

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

   5. distribute-neg-in 77.0%

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

   6. remove-double-neg 77.0%

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

   7. sub-neg 77.0%

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

   8. div-sub 77.0%

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

   9. distribute-rgt1-in 77.0%

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

   10. associate-/l/ 77.0%

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

3. Simplified 77.8%

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

4. Taylor expanded in wj around inf 4.4%

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

5. Final simplification 4.4%

   \[\leadsto wj \]

Alternative 12: 84.3% 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 77.0%

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

   1. sub-neg 77.0%

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

   2. div-sub 77.0%

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

   3. sub-neg 77.0%

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

   4. +-commutative 77.0%

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

   5. distribute-neg-in 77.0%

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

   6. remove-double-neg 77.0%

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

   7. sub-neg 77.0%

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

   8. div-sub 77.0%

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

   9. distribute-rgt1-in 77.0%

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

   10. associate-/l/ 77.0%

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

3. Simplified 77.8%

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

4. Taylor expanded in wj around 0 83.9%

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

5. Final simplification 83.9%

   \[\leadsto x \]

Developer target: 79.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023182 
(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))))))