Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.0% → 99.6%

Time: 15.0s

Alternatives: 15

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -5.9128230382261035e+169
  2. x = 3.157283369308592e-266

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj - \frac{x}{e^{wj}}\\ t_1 := \frac{t\_0}{wj + 1}\\ t_2 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \left(\frac{wj}{x} + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\right)\\ \mathbf{elif}\;wj \leq 3.55 \cdot 10^{-6}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(t\_2 + \left(-1 + wj \cdot \left(1 + \left(x \cdot -3 + \left(-2 \cdot t\_2 + x \cdot 0.6666666666666666\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj - t\_1\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, t\_0, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (- wj (/ x (exp wj))))
        (t_1 (/ t_0 (+ wj 1.0)))
        (t_2 (+ (* x -4.0) (* x 1.5))))
   (if (<= wj -3.5e-6)
     (*
      x
      (+ (/ (exp (- wj)) (+ wj 1.0)) (+ (/ wj x) (/ wj (* x (- -1.0 wj))))))
     (if (<= wj 3.55e-6)
       (-
        x
        (*
         wj
         (+
          (* x 2.0)
          (*
           wj
           (+
            t_2
            (+
             -1.0
             (*
              wj
              (+
               1.0
               (+ (* x -3.0) (+ (* -2.0 t_2) (* x 0.6666666666666666)))))))))))
       (+ (- wj t_1) (fma (/ -1.0 (+ wj 1.0)) t_0 t_1))))))

C

double code(double wj, double x) {
	double t_0 = wj - (x / exp(wj));
	double t_1 = t_0 / (wj + 1.0);
	double t_2 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= -3.5e-6) {
		tmp = x * ((exp(-wj) / (wj + 1.0)) + ((wj / x) + (wj / (x * (-1.0 - wj)))));
	} else if (wj <= 3.55e-6) {
		tmp = x - (wj * ((x * 2.0) + (wj * (t_2 + (-1.0 + (wj * (1.0 + ((x * -3.0) + ((-2.0 * t_2) + (x * 0.6666666666666666))))))))));
	} else {
		tmp = (wj - t_1) + fma((-1.0 / (wj + 1.0)), t_0, t_1);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj - Float64(x / exp(wj)))
	t_1 = Float64(t_0 / Float64(wj + 1.0))
	t_2 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (wj <= -3.5e-6)
		tmp = Float64(x * Float64(Float64(exp(Float64(-wj)) / Float64(wj + 1.0)) + Float64(Float64(wj / x) + Float64(wj / Float64(x * Float64(-1.0 - wj))))));
	elseif (wj <= 3.55e-6)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(t_2 + Float64(-1.0 + Float64(wj * Float64(1.0 + Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_2) + Float64(x * 0.6666666666666666)))))))))));
	else
		tmp = Float64(Float64(wj - t_1) + fma(Float64(-1.0 / Float64(wj + 1.0)), t_0, t_1));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -3.5e-6], N[(x * N[(N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(wj / x), $MachinePrecision] + N[(wj / N[(x * N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 3.55e-6], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(t$95$2 + N[(-1.0 + N[(wj * N[(1.0 + N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * t$95$2), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(wj - t$95$1), $MachinePrecision] + N[(N[(-1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -3.49999999999999995e-6

   1. Initial program 85.3%

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

      1. distribute-rgt1-in 99.8%

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

      2. associate-/l/ 99.6%

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

      3. div-sub 85.3%

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

      4. associate-/l* 85.3%

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

      5. *-inverses 99.6%

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

      6. *-rgt-identity 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. associate--l+ 99.6%

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

      2. associate-/r* 99.8%

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

      3. rec-exp 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   if -3.49999999999999995e-6 < wj < 3.5499999999999999e-6

   1. Initial program 74.0%

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

      1. distribute-rgt1-in 74.0%

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

      2. associate-/l/ 74.0%

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

      3. div-sub 74.0%

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

      4. associate-/l* 74.0%

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

      5. *-inverses 74.0%

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

      6. *-rgt-identity 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in wj around 0 99.9%

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

   if 3.5499999999999999e-6 < wj

   1. Initial program 44.6%

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

      1. distribute-rgt1-in 44.4%

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

      2. associate-/l/ 44.5%

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

      3. div-sub 44.5%

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

      4. associate-/l* 44.5%

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

      5. *-inverses 98.4%

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

      6. *-rgt-identity 98.4%

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

   3. Simplified 98.4%

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

      1. *-un-lft-identity 98.4%

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

      2. div-inv 98.3%

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

      3. prod-diff 98.5%

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

      4. associate-/r/ 98.1%

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

      5. clear-num 98.2%

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

      6. fma-neg 98.2%

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

      7. *-un-lft-identity 98.2%

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

      8. associate-/r/ 98.4%

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

      9. clear-num 98.5%

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

   6. Applied egg-rr 98.5%

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

      1. distribute-neg-frac 98.5%

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

      2. metadata-eval 98.5%

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

   8. Simplified 98.5%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \left(\frac{wj}{x} + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\right)\\ \mathbf{elif}\;wj \leq 3.55 \cdot 10^{-6}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(\left(x \cdot -4 + x \cdot 1.5\right) + \left(-1 + wj \cdot \left(1 + \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right) + \mathsf{fma}\left(\frac{-1}{wj + 1}, wj - \frac{x}{e^{wj}}, \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq -4.3 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \left(\frac{wj}{x} + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\right)\\ \mathbf{elif}\;wj \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(t\_0 + \left(-1 + wj \cdot \left(1 + \left(x \cdot -3 + \left(-2 \cdot t\_0 + x \cdot 0.6666666666666666\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{\mathsf{fma}\left(wj, wj, -1\right)} \cdot \left(1 - wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (<= wj -4.3e-6)
     (*
      x
      (+ (/ (exp (- wj)) (+ wj 1.0)) (+ (/ wj x) (/ wj (* x (- -1.0 wj))))))
     (if (<= wj 6.2e-6)
       (-
        x
        (*
         wj
         (+
          (* x 2.0)
          (*
           wj
           (+
            t_0
            (+
             -1.0
             (*
              wj
              (+
               1.0
               (+ (* x -3.0) (+ (* -2.0 t_0) (* x 0.6666666666666666)))))))))))
       (+ wj (* (/ (- wj (/ x (exp wj))) (fma wj wj -1.0)) (- 1.0 wj)))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= -4.3e-6) {
		tmp = x * ((exp(-wj) / (wj + 1.0)) + ((wj / x) + (wj / (x * (-1.0 - wj)))));
	} else if (wj <= 6.2e-6) {
		tmp = x - (wj * ((x * 2.0) + (wj * (t_0 + (-1.0 + (wj * (1.0 + ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))))))));
	} else {
		tmp = wj + (((wj - (x / exp(wj))) / fma(wj, wj, -1.0)) * (1.0 - wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (wj <= -4.3e-6)
		tmp = Float64(x * Float64(Float64(exp(Float64(-wj)) / Float64(wj + 1.0)) + Float64(Float64(wj / x) + Float64(wj / Float64(x * Float64(-1.0 - wj))))));
	elseif (wj <= 6.2e-6)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(t_0 + Float64(-1.0 + Float64(wj * Float64(1.0 + Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_0) + Float64(x * 0.6666666666666666)))))))))));
	else
		tmp = Float64(wj + Float64(Float64(Float64(wj - Float64(x / exp(wj))) / fma(wj, wj, -1.0)) * Float64(1.0 - wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -4.3e-6], N[(x * N[(N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(wj / x), $MachinePrecision] + N[(wj / N[(x * N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 6.2e-6], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(t$95$0 + N[(-1.0 + N[(wj * N[(1.0 + N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * t$95$0), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -4.30000000000000033e-6

   1. Initial program 85.3%

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

      1. distribute-rgt1-in 99.8%

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

      2. associate-/l/ 99.6%

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

      3. div-sub 85.3%

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

      4. associate-/l* 85.3%

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

      5. *-inverses 99.6%

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

      6. *-rgt-identity 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. associate--l+ 99.6%

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

      2. associate-/r* 99.8%

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

      3. rec-exp 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   if -4.30000000000000033e-6 < wj < 6.1999999999999999e-6

   1. Initial program 74.0%

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

      1. distribute-rgt1-in 74.0%

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

      2. associate-/l/ 74.0%

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

      3. div-sub 74.0%

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

      4. associate-/l* 74.0%

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

      5. *-inverses 74.0%

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

      6. *-rgt-identity 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in wj around 0 99.9%

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

   if 6.1999999999999999e-6 < wj

   1. Initial program 44.6%

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

      1. distribute-rgt1-in 44.4%

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

      2. associate-/l/ 44.5%

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

      3. div-sub 44.5%

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

      4. associate-/l* 44.5%

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

      5. *-inverses 98.4%

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

      6. *-rgt-identity 98.4%

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

   3. Simplified 98.4%

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

      1. flip-+ 98.5%

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

      2. associate-/r/ 98.5%

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

      3. metadata-eval 98.5%

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

      4. fma-neg 98.5%

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

      5. metadata-eval 98.5%

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

      6. sub-neg 98.5%

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

      7. metadata-eval 98.5%

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

   6. Applied egg-rr 98.5%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.3 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \left(\frac{wj}{x} + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\right)\\ \mathbf{elif}\;wj \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(\left(x \cdot -4 + x \cdot 1.5\right) + \left(-1 + wj \cdot \left(1 + \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - \frac{x}{e^{wj}}}{\mathsf{fma}\left(wj, wj, -1\right)} \cdot \left(1 - wj\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq -3.2 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \left(\frac{wj}{x} + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\right)\\ \mathbf{elif}\;wj \leq 3.55 \cdot 10^{-6}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(t\_0 + \left(-1 + wj \cdot \left(1 + \left(x \cdot -3 + \left(-2 \cdot t\_0 + x \cdot 0.6666666666666666\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (<= wj -3.2e-6)
     (*
      x
      (+ (/ (exp (- wj)) (+ wj 1.0)) (+ (/ wj x) (/ wj (* x (- -1.0 wj))))))
     (if (<= wj 3.55e-6)
       (-
        x
        (*
         wj
         (+
          (* x 2.0)
          (*
           wj
           (+
            t_0
            (+
             -1.0
             (*
              wj
              (+
               1.0
               (+ (* x -3.0) (+ (* -2.0 t_0) (* x 0.6666666666666666)))))))))))
       (- wj (/ (- wj (/ x (exp wj))) (+ wj 1.0)))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= -3.2e-6) {
		tmp = x * ((exp(-wj) / (wj + 1.0)) + ((wj / x) + (wj / (x * (-1.0 - wj)))));
	} else if (wj <= 3.55e-6) {
		tmp = x - (wj * ((x * 2.0) + (wj * (t_0 + (-1.0 + (wj * (1.0 + ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))))))));
	} else {
		tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * (-4.0d0)) + (x * 1.5d0)
    if (wj <= (-3.2d-6)) then
        tmp = x * ((exp(-wj) / (wj + 1.0d0)) + ((wj / x) + (wj / (x * ((-1.0d0) - wj)))))
    else if (wj <= 3.55d-6) then
        tmp = x - (wj * ((x * 2.0d0) + (wj * (t_0 + ((-1.0d0) + (wj * (1.0d0 + ((x * (-3.0d0)) + (((-2.0d0) * t_0) + (x * 0.6666666666666666d0))))))))))
    else
        tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= -3.2e-6) {
		tmp = x * ((Math.exp(-wj) / (wj + 1.0)) + ((wj / x) + (wj / (x * (-1.0 - wj)))));
	} else if (wj <= 3.55e-6) {
		tmp = x - (wj * ((x * 2.0) + (wj * (t_0 + (-1.0 + (wj * (1.0 + ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))))))));
	} else {
		tmp = wj - ((wj - (x / Math.exp(wj))) / (wj + 1.0));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if wj <= -3.2e-6:
		tmp = x * ((math.exp(-wj) / (wj + 1.0)) + ((wj / x) + (wj / (x * (-1.0 - wj)))))
	elif wj <= 3.55e-6:
		tmp = x - (wj * ((x * 2.0) + (wj * (t_0 + (-1.0 + (wj * (1.0 + ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))))))))
	else:
		tmp = wj - ((wj - (x / math.exp(wj))) / (wj + 1.0))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (wj <= -3.2e-6)
		tmp = Float64(x * Float64(Float64(exp(Float64(-wj)) / Float64(wj + 1.0)) + Float64(Float64(wj / x) + Float64(wj / Float64(x * Float64(-1.0 - wj))))));
	elseif (wj <= 3.55e-6)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(t_0 + Float64(-1.0 + Float64(wj * Float64(1.0 + Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_0) + Float64(x * 0.6666666666666666)))))))))));
	else
		tmp = Float64(wj - Float64(Float64(wj - Float64(x / exp(wj))) / Float64(wj + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if (wj <= -3.2e-6)
		tmp = x * ((exp(-wj) / (wj + 1.0)) + ((wj / x) + (wj / (x * (-1.0 - wj)))));
	elseif (wj <= 3.55e-6)
		tmp = x - (wj * ((x * 2.0) + (wj * (t_0 + (-1.0 + (wj * (1.0 + ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))))))));
	else
		tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -3.2e-6], N[(x * N[(N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(wj / x), $MachinePrecision] + N[(wj / N[(x * N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 3.55e-6], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(t$95$0 + N[(-1.0 + N[(wj * N[(1.0 + N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * t$95$0), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -3.1999999999999999e-6

   1. Initial program 85.3%

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

      1. distribute-rgt1-in 99.8%

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

      2. associate-/l/ 99.6%

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

      3. div-sub 85.3%

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

      4. associate-/l* 85.3%

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

      5. *-inverses 99.6%

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

      6. *-rgt-identity 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. associate--l+ 99.6%

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

      2. associate-/r* 99.8%

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

      3. rec-exp 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   7. Simplified 99.8%

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

   if -3.1999999999999999e-6 < wj < 3.5499999999999999e-6

   1. Initial program 74.0%

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

      1. distribute-rgt1-in 74.0%

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

      2. associate-/l/ 74.0%

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

      3. div-sub 74.0%

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

      4. associate-/l* 74.0%

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

      5. *-inverses 74.0%

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

      6. *-rgt-identity 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in wj around 0 99.9%

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

   if 3.5499999999999999e-6 < wj

   1. Initial program 44.6%

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

      1. distribute-rgt1-in 44.4%

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

      2. associate-/l/ 44.5%

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

      3. div-sub 44.5%

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

      4. associate-/l* 44.5%

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

      5. *-inverses 98.4%

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

      6. *-rgt-identity 98.4%

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

   3. Simplified 98.4%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.2 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\frac{e^{-wj}}{wj + 1} + \left(\frac{wj}{x} + \frac{wj}{x \cdot \left(-1 - wj\right)}\right)\right)\\ \mathbf{elif}\;wj \leq 3.55 \cdot 10^{-6}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(\left(x \cdot -4 + x \cdot 1.5\right) + \left(-1 + wj \cdot \left(1 + \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq -3.5 \cdot 10^{-6} \lor \neg \left(wj \leq 5.5 \cdot 10^{-6}\right):\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(t\_0 + \left(-1 + wj \cdot \left(1 + \left(x \cdot -3 + \left(-2 \cdot t\_0 + x \cdot 0.6666666666666666\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (or (<= wj -3.5e-6) (not (<= wj 5.5e-6)))
     (- wj (/ (- wj (/ x (exp wj))) (+ wj 1.0)))
     (-
      x
      (*
       wj
       (+
        (* x 2.0)
        (*
         wj
         (+
          t_0
          (+
           -1.0
           (*
            wj
            (+
             1.0
             (+
              (* x -3.0)
              (+ (* -2.0 t_0) (* x 0.6666666666666666))))))))))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if ((wj <= -3.5e-6) || !(wj <= 5.5e-6)) {
		tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0));
	} else {
		tmp = x - (wj * ((x * 2.0) + (wj * (t_0 + (-1.0 + (wj * (1.0 + ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))))))));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * (-4.0d0)) + (x * 1.5d0)
    if ((wj <= (-3.5d-6)) .or. (.not. (wj <= 5.5d-6))) then
        tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0d0))
    else
        tmp = x - (wj * ((x * 2.0d0) + (wj * (t_0 + ((-1.0d0) + (wj * (1.0d0 + ((x * (-3.0d0)) + (((-2.0d0) * t_0) + (x * 0.6666666666666666d0))))))))))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if ((wj <= -3.5e-6) || !(wj <= 5.5e-6)) {
		tmp = wj - ((wj - (x / Math.exp(wj))) / (wj + 1.0));
	} else {
		tmp = x - (wj * ((x * 2.0) + (wj * (t_0 + (-1.0 + (wj * (1.0 + ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))))))));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if (wj <= -3.5e-6) or not (wj <= 5.5e-6):
		tmp = wj - ((wj - (x / math.exp(wj))) / (wj + 1.0))
	else:
		tmp = x - (wj * ((x * 2.0) + (wj * (t_0 + (-1.0 + (wj * (1.0 + ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))))))))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if ((wj <= -3.5e-6) || !(wj <= 5.5e-6))
		tmp = Float64(wj - Float64(Float64(wj - Float64(x / exp(wj))) / Float64(wj + 1.0)));
	else
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(t_0 + Float64(-1.0 + Float64(wj * Float64(1.0 + Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_0) + Float64(x * 0.6666666666666666)))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if ((wj <= -3.5e-6) || ~((wj <= 5.5e-6)))
		tmp = wj - ((wj - (x / exp(wj))) / (wj + 1.0));
	else
		tmp = x - (wj * ((x * 2.0) + (wj * (t_0 + (-1.0 + (wj * (1.0 + ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[wj, -3.5e-6], N[Not[LessEqual[wj, 5.5e-6]], $MachinePrecision]], N[(wj - N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(t$95$0 + N[(-1.0 + N[(wj * N[(1.0 + N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * t$95$0), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if wj < -3.49999999999999995e-6 or 5.4999999999999999e-6 < wj

   1. Initial program 58.8%

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

      1. distribute-rgt1-in 63.8%

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

      2. associate-/l/ 63.8%

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

      3. div-sub 58.8%

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

      4. associate-/l* 58.8%

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

      5. *-inverses 98.8%

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

      6. *-rgt-identity 98.8%

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

   3. Simplified 98.8%

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

   if -3.49999999999999995e-6 < wj < 5.4999999999999999e-6

   1. Initial program 74.0%

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

      1. distribute-rgt1-in 74.0%

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

      2. associate-/l/ 74.0%

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

      3. div-sub 74.0%

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

      4. associate-/l* 74.0%

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

      5. *-inverses 74.0%

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

      6. *-rgt-identity 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in wj around 0 99.9%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -3.5 \cdot 10^{-6} \lor \neg \left(wj \leq 5.5 \cdot 10^{-6}\right):\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \mathbf{else}:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(\left(x \cdot -4 + x \cdot 1.5\right) + \left(-1 + wj \cdot \left(1 + \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot -4 + x \cdot 1.5\\ \mathbf{if}\;wj \leq 0.24:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(t\_0 + \left(-1 + wj \cdot \left(1 + \left(x \cdot -3 + \left(-2 \cdot t\_0 + x \cdot 0.6666666666666666\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ (* x -4.0) (* x 1.5))))
   (if (<= wj 0.24)
     (-
      x
      (*
       wj
       (+
        (* x 2.0)
        (*
         wj
         (+
          t_0
          (+
           -1.0
           (*
            wj
            (+
             1.0
             (+ (* x -3.0) (+ (* -2.0 t_0) (* x 0.6666666666666666)))))))))))
     (+ wj (/ wj (- -1.0 wj))))))

C

double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= 0.24) {
		tmp = x - (wj * ((x * 2.0) + (wj * (t_0 + (-1.0 + (wj * (1.0 + ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))))))));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * (-4.0d0)) + (x * 1.5d0)
    if (wj <= 0.24d0) then
        tmp = x - (wj * ((x * 2.0d0) + (wj * (t_0 + ((-1.0d0) + (wj * (1.0d0 + ((x * (-3.0d0)) + (((-2.0d0) * t_0) + (x * 0.6666666666666666d0))))))))))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = (x * -4.0) + (x * 1.5);
	double tmp;
	if (wj <= 0.24) {
		tmp = x - (wj * ((x * 2.0) + (wj * (t_0 + (-1.0 + (wj * (1.0 + ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))))))));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = (x * -4.0) + (x * 1.5)
	tmp = 0
	if wj <= 0.24:
		tmp = x - (wj * ((x * 2.0) + (wj * (t_0 + (-1.0 + (wj * (1.0 + ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))))))))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(Float64(x * -4.0) + Float64(x * 1.5))
	tmp = 0.0
	if (wj <= 0.24)
		tmp = Float64(x - Float64(wj * Float64(Float64(x * 2.0) + Float64(wj * Float64(t_0 + Float64(-1.0 + Float64(wj * Float64(1.0 + Float64(Float64(x * -3.0) + Float64(Float64(-2.0 * t_0) + Float64(x * 0.6666666666666666)))))))))));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = (x * -4.0) + (x * 1.5);
	tmp = 0.0;
	if (wj <= 0.24)
		tmp = x - (wj * ((x * 2.0) + (wj * (t_0 + (-1.0 + (wj * (1.0 + ((x * -3.0) + ((-2.0 * t_0) + (x * 0.6666666666666666))))))))));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(x * -4.0), $MachinePrecision] + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, 0.24], N[(x - N[(wj * N[(N[(x * 2.0), $MachinePrecision] + N[(wj * N[(t$95$0 + N[(-1.0 + N[(wj * N[(1.0 + N[(N[(x * -3.0), $MachinePrecision] + N[(N[(-2.0 * t$95$0), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.23999999999999999

   1. Initial program 74.7%

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

      1. distribute-rgt1-in 75.1%

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

      2. associate-/l/ 75.1%

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

      3. div-sub 74.7%

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

      4. associate-/l* 74.7%

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

      5. *-inverses 75.1%

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

      6. *-rgt-identity 75.1%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in wj around 0 96.9%

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

   if 0.23999999999999999 < wj

   1. Initial program 22.2%

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

      1. distribute-rgt1-in 22.2%

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

      2. associate-/l/ 22.2%

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

      3. div-sub 22.2%

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

      4. associate-/l* 22.2%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.24:\\ \;\;\;\;x - wj \cdot \left(x \cdot 2 + wj \cdot \left(\left(x \cdot -4 + x \cdot 1.5\right) + \left(-1 + wj \cdot \left(1 + \left(x \cdot -3 + \left(-2 \cdot \left(x \cdot -4 + x \cdot 1.5\right) + x \cdot 0.6666666666666666\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.5% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.24:\\ \;\;\;\;x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + wj \cdot \left(\frac{-1}{x} - 2.6666666666666665\right)\right)\right)\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.24)
   (+
    x
    (*
     wj
     (-
      (*
       x
       (* wj (+ 2.5 (+ (/ 1.0 x) (* wj (- (/ -1.0 x) 2.6666666666666665))))))
      (* x 2.0))))
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.24) {
		tmp = x + (wj * ((x * (wj * (2.5 + ((1.0 / x) + (wj * ((-1.0 / x) - 2.6666666666666665)))))) - (x * 2.0)));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 0.24d0) then
        tmp = x + (wj * ((x * (wj * (2.5d0 + ((1.0d0 / x) + (wj * (((-1.0d0) / x) - 2.6666666666666665d0)))))) - (x * 2.0d0)))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 0.24) {
		tmp = x + (wj * ((x * (wj * (2.5 + ((1.0 / x) + (wj * ((-1.0 / x) - 2.6666666666666665)))))) - (x * 2.0)));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.24:
		tmp = x + (wj * ((x * (wj * (2.5 + ((1.0 / x) + (wj * ((-1.0 / x) - 2.6666666666666665)))))) - (x * 2.0)))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.24)
		tmp = Float64(x + Float64(wj * Float64(Float64(x * Float64(wj * Float64(2.5 + Float64(Float64(1.0 / x) + Float64(wj * Float64(Float64(-1.0 / x) - 2.6666666666666665)))))) - Float64(x * 2.0))));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.24)
		tmp = x + (wj * ((x * (wj * (2.5 + ((1.0 / x) + (wj * ((-1.0 / x) - 2.6666666666666665)))))) - (x * 2.0)));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.24], N[(x + N[(wj * N[(N[(x * N[(wj * N[(2.5 + N[(N[(1.0 / x), $MachinePrecision] + N[(wj * N[(N[(-1.0 / x), $MachinePrecision] - 2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.23999999999999999

   1. Initial program 74.7%

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

      1. distribute-rgt1-in 75.1%

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

      2. associate-/l/ 75.1%

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

      3. div-sub 74.7%

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

      4. associate-/l* 74.7%

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

      5. *-inverses 75.1%

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

      6. *-rgt-identity 75.1%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in wj around 0 96.9%

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

   6. Taylor expanded in x around inf 96.9%

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

      1. fma-define 96.9%

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

      2. *-commutative 96.9%

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

      3. associate-/l* 96.8%

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

      4. mul-1-neg 96.8%

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

      5. sub-neg 96.8%

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

   8. Simplified 96.8%

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

   9. Taylor expanded in wj around 0 96.8%

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

       1. +-commutative 96.8%

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

       2. mul-1-neg 96.8%

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

       3. unsub-neg 96.8%

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

       4. +-commutative 96.8%

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

   11. Simplified 96.8%

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

   if 0.23999999999999999 < wj

   1. Initial program 22.2%

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

      1. distribute-rgt1-in 22.2%

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

      2. associate-/l/ 22.2%

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

      3. div-sub 22.2%

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

      4. associate-/l* 22.2%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.24:\\ \;\;\;\;x + wj \cdot \left(x \cdot \left(wj \cdot \left(2.5 + \left(\frac{1}{x} + wj \cdot \left(\frac{-1}{x} - 2.6666666666666665\right)\right)\right)\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.3% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.24:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.24)
   (+ x (* wj (- (* wj (- 1.0 wj)) (* x 2.0))))
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.24) {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 0.24d0) then
        tmp = x + (wj * ((wj * (1.0d0 - wj)) - (x * 2.0d0)))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 0.24) {
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.24:
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.24)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(1.0 - wj)) - Float64(x * 2.0))));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.24)
		tmp = x + (wj * ((wj * (1.0 - wj)) - (x * 2.0)));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.24], N[(x + N[(wj * N[(N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] - N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.23999999999999999

   1. Initial program 74.7%

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

      1. distribute-rgt1-in 75.1%

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

      2. associate-/l/ 75.1%

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

      3. div-sub 74.7%

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

      4. associate-/l* 74.7%

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

      5. *-inverses 75.1%

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

      6. *-rgt-identity 75.1%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in wj around 0 96.9%

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

   6. Taylor expanded in x around 0 96.4%

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

      1. mul-1-neg 96.4%

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

      2. sub-neg 96.4%

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

   8. Simplified 96.4%

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

   if 0.23999999999999999 < wj

   1. Initial program 22.2%

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

      1. distribute-rgt1-in 22.2%

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

      2. associate-/l/ 22.2%

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

      3. div-sub 22.2%

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

      4. associate-/l* 22.2%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.24:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 - wj\right) - x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.8% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.24:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.24) (+ x (* wj (+ wj (* x -2.0)))) (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.24) {
		tmp = x + (wj * (wj + (x * -2.0)));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 0.24d0) then
        tmp = x + (wj * (wj + (x * (-2.0d0))))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 0.24) {
		tmp = x + (wj * (wj + (x * -2.0)));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.24:
		tmp = x + (wj * (wj + (x * -2.0)))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.24)
		tmp = Float64(x + Float64(wj * Float64(wj + Float64(x * -2.0))));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.24)
		tmp = x + (wj * (wj + (x * -2.0)));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.24], N[(x + N[(wj * N[(wj + N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.23999999999999999

   1. Initial program 74.7%

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

      1. distribute-rgt1-in 75.1%

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

      2. associate-/l/ 75.1%

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

      3. div-sub 74.7%

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

      4. associate-/l* 74.7%

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

      5. *-inverses 75.1%

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

      6. *-rgt-identity 75.1%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in wj around 0 96.9%

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

   6. Taylor expanded in x around 0 96.4%

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

      1. mul-1-neg 96.4%

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

      2. sub-neg 96.4%

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

   8. Simplified 96.4%

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

   9. Taylor expanded in wj around 0 96.0%

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

       1. *-commutative 96.0%

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

   11. Simplified 96.0%

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

   if 0.23999999999999999 < wj

   1. Initial program 22.2%

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

      1. distribute-rgt1-in 22.2%

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

      2. associate-/l/ 22.2%

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

      3. div-sub 22.2%

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

      4. associate-/l* 22.2%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.24:\\ \;\;\;\;x + wj \cdot \left(wj + x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.9% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.4e-7) (/ x (+ 1.0 (* wj 2.0))) (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.4e-7) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 1.4d-7) then
        tmp = x / (1.0d0 + (wj * 2.0d0))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.4e-7) {
		tmp = x / (1.0 + (wj * 2.0));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.4e-7:
		tmp = x / (1.0 + (wj * 2.0))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.4e-7)
		tmp = Float64(x / Float64(1.0 + Float64(wj * 2.0)));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.4e-7)
		tmp = x / (1.0 + (wj * 2.0));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.4e-7], N[(x / N[(1.0 + N[(wj * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.4000000000000001e-7

   1. Initial program 74.4%

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

      1. distribute-rgt1-in 74.8%

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

      2. associate-/l/ 74.8%

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

      3. div-sub 74.4%

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

      4. associate-/l* 74.4%

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

      5. *-inverses 74.8%

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

      6. *-rgt-identity 74.8%

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

   3. Simplified 74.8%

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

   5. Taylor expanded in x around inf 84.4%

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

      1. +-commutative 84.4%

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

      2. rem-exp-log 83.1%

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

      3. +-commutative 83.1%

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

      4. log1p-undefine 83.1%

         \[\leadsto \frac{x}{e^{wj} \cdot e^{\color{blue}{\mathsf{log1p}\left(wj\right)}}} \]

      5. exp-sum 83.1%

         \[\leadsto \frac{x}{\color{blue}{e^{wj + \mathsf{log1p}\left(wj\right)}}} \]

   7. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{x}{e^{wj + \mathsf{log1p}\left(wj\right)}}} \]

   8. Taylor expanded in wj around 0 82.8%

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

      1. *-commutative 82.8%

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

   10. Simplified 82.8%

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

   if 1.4000000000000001e-7 < wj

   1. Initial program 46.1%

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

      1. distribute-rgt1-in 46.0%

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

      2. associate-/l/ 46.1%

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

      3. div-sub 46.1%

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

      4. associate-/l* 46.1%

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

      5. *-inverses 96.1%

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

      6. *-rgt-identity 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in x around 0 75.6%

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

      1. +-commutative 75.6%

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

   7. Simplified 75.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{1 + wj \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.9% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 1.4e-7) (+ x (* -2.0 (* wj x))) (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.4e-7) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 1.4d-7) then
        tmp = x + ((-2.0d0) * (wj * x))
    else
        tmp = wj + (wj / ((-1.0d0) - wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.4e-7) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.4e-7:
		tmp = x + (-2.0 * (wj * x))
	else:
		tmp = wj + (wj / (-1.0 - wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.4e-7)
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.4e-7)
		tmp = x + (-2.0 * (wj * x));
	else
		tmp = wj + (wj / (-1.0 - wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.4e-7], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.4000000000000001e-7

   1. Initial program 74.4%

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

      1. distribute-rgt1-in 74.8%

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

      2. associate-/l/ 74.8%

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

      3. div-sub 74.4%

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

      4. associate-/l* 74.4%

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

      5. *-inverses 74.8%

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

      6. *-rgt-identity 74.8%

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

   3. Simplified 74.8%

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

   5. Taylor expanded in wj around 0 82.6%

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

      1. *-commutative 82.6%

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

   7. Simplified 82.6%

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

   if 1.4000000000000001e-7 < wj

   1. Initial program 46.1%

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

      1. distribute-rgt1-in 46.0%

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

      2. associate-/l/ 46.1%

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

      3. div-sub 46.1%

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

      4. associate-/l* 46.1%

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

      5. *-inverses 96.1%

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

      6. *-rgt-identity 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in x around 0 75.6%

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

      1. +-commutative 75.6%

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

   7. Simplified 75.6%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.4% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.95:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.95) (+ x (* -2.0 (* wj x))) (+ wj -1.0)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.95) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 0.95d0) then
        tmp = x + ((-2.0d0) * (wj * x))
    else
        tmp = wj + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 0.95) {
		tmp = x + (-2.0 * (wj * x));
	} else {
		tmp = wj + -1.0;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 0.95:
		tmp = x + (-2.0 * (wj * x))
	else:
		tmp = wj + -1.0
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.95)
		tmp = Float64(x + Float64(-2.0 * Float64(wj * x)));
	else
		tmp = Float64(wj + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 0.95)
		tmp = x + (-2.0 * (wj * x));
	else
		tmp = wj + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.95], N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.94999999999999996

   1. Initial program 74.7%

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

      1. distribute-rgt1-in 75.1%

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

      2. associate-/l/ 75.1%

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

      3. div-sub 74.7%

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

      4. associate-/l* 74.7%

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

      5. *-inverses 75.1%

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

      6. *-rgt-identity 75.1%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in wj around 0 81.4%

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

      1. *-commutative 81.4%

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

   7. Simplified 81.4%

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

   if 0.94999999999999996 < wj

   1. Initial program 22.2%

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

      1. distribute-rgt1-in 22.2%

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

      2. associate-/l/ 22.2%

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

      3. div-sub 22.2%

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

      4. associate-/l* 22.2%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in wj around inf 75.2%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.95:\\ \;\;\;\;x + -2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.8% accurate, 39.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 1.8:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (if (<= wj 1.8) x (+ wj -1.0)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 1.8) {
		tmp = x;
	} else {
		tmp = 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 <= 1.8d0) then
        tmp = x
    else
        tmp = wj + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 1.8) {
		tmp = x;
	} else {
		tmp = wj + -1.0;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 1.8:
		tmp = x
	else:
		tmp = wj + -1.0
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 1.8)
		tmp = x;
	else
		tmp = Float64(wj + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 1.8)
		tmp = x;
	else
		tmp = wj + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 1.8], x, N[(wj + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.80000000000000004

   1. Initial program 74.7%

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

      1. distribute-rgt1-in 75.1%

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

      2. associate-/l/ 75.1%

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

      3. div-sub 74.7%

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

      4. associate-/l* 74.7%

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

      5. *-inverses 75.1%

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

      6. *-rgt-identity 75.1%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in wj around 0 80.6%

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

   if 1.80000000000000004 < wj

   1. Initial program 22.2%

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

      1. distribute-rgt1-in 22.2%

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

      2. associate-/l/ 22.2%

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

      3. div-sub 22.2%

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

      4. associate-/l* 22.2%

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

      5. *-inverses 100.0%

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

      6. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in wj around inf 75.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 1.8:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.0% 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 72.9%

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

   1. distribute-rgt1-in 73.2%

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

   2. associate-/l/ 73.2%

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

   3. div-sub 72.9%

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

   4. associate-/l* 72.9%

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

   5. *-inverses 76.0%

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

   6. *-rgt-identity 76.0%

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

3. Simplified 76.0%

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

5. Taylor expanded in wj around 0 77.9%

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

Alternative 14: 4.4% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 72.9%

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

   1. distribute-rgt1-in 73.2%

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

   2. associate-/l/ 73.2%

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

   3. div-sub 72.9%

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

   4. associate-/l* 72.9%

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

   5. *-inverses 76.0%

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

   6. *-rgt-identity 76.0%

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

3. Simplified 76.0%

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

5. Taylor expanded in wj around inf 5.4%

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

Alternative 15: 3.3% accurate, 313.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return -1.0

Julia

function code(wj, x)
	return -1.0
end

MATLAB

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

Wolfram

code[wj_, x_] := -1.0

Derivation

1. Initial program 72.9%

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

   1. distribute-rgt1-in 73.2%

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

   2. associate-/l/ 73.2%

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

   3. div-sub 72.9%

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

   4. associate-/l* 72.9%

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

   5. *-inverses 76.0%

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

   6. *-rgt-identity 76.0%

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

3. Simplified 76.0%

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

5. Taylor expanded in wj around inf 5.9%

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

6. Taylor expanded in wj around 0 3.2%

   \[\leadsto \color{blue}{-1} \]
7. Add Preprocessing

Developer Target 1: 79.1% 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 2024123 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

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

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