Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.1% → 99.8%

Time: 11.5s

Alternatives: 14

Speedup: 66.2×

• could not determine a ground truth

  1. wj = -2.8139132750677795e+61
  2. x = -1.9663195031783656e+257

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.1% 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: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-wj}\\ \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(wj, e^{wj}, -x\right), \frac{t\_0}{-1 - wj}, wj\right)\\ \mathbf{elif}\;wj \leq 1.65 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), 1 - wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - x \cdot \left(\frac{wj}{\mathsf{fma}\left(x, wj, x\right)} - \frac{t\_0}{wj + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (exp (- wj))))
   (if (<= wj -4.5e-6)
     (fma (fma wj (exp wj) (- x)) (/ t_0 (- -1.0 wj)) wj)
     (if (<= wj 1.65e-7)
       (fma
        wj
        (fma wj (fma x (fma wj -2.6666666666666665 2.5) (- 1.0 wj)) (* x -2.0))
        x)
       (- wj (* x (- (/ wj (fma x wj x)) (/ t_0 (+ wj 1.0)))))))))

C

double code(double wj, double x) {
	double t_0 = exp(-wj);
	double tmp;
	if (wj <= -4.5e-6) {
		tmp = fma(fma(wj, exp(wj), -x), (t_0 / (-1.0 - wj)), wj);
	} else if (wj <= 1.65e-7) {
		tmp = fma(wj, fma(wj, fma(x, fma(wj, -2.6666666666666665, 2.5), (1.0 - wj)), (x * -2.0)), x);
	} else {
		tmp = wj - (x * ((wj / fma(x, wj, x)) - (t_0 / (wj + 1.0))));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = exp(Float64(-wj))
	tmp = 0.0
	if (wj <= -4.5e-6)
		tmp = fma(fma(wj, exp(wj), Float64(-x)), Float64(t_0 / Float64(-1.0 - wj)), wj);
	elseif (wj <= 1.65e-7)
		tmp = fma(wj, fma(wj, fma(x, fma(wj, -2.6666666666666665, 2.5), Float64(1.0 - wj)), Float64(x * -2.0)), x);
	else
		tmp = Float64(wj - Float64(x * Float64(Float64(wj / fma(x, wj, x)) - Float64(t_0 / Float64(wj + 1.0)))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[Exp[(-wj)], $MachinePrecision]}, If[LessEqual[wj, -4.5e-6], N[(N[(wj * N[Exp[wj], $MachinePrecision] + (-x)), $MachinePrecision] * N[(t$95$0 / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision], If[LessEqual[wj, 1.65e-7], N[(wj * N[(wj * N[(x * N[(wj * -2.6666666666666665 + 2.5), $MachinePrecision] + N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj - N[(x * N[(N[(wj / N[(x * wj + x), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -4.50000000000000011e-6

   1. Initial program 68.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. div-inv N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 97.4%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 97.5%

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

   if -4.50000000000000011e-6 < wj < 1.6500000000000001e-7

   1. Initial program 76.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \left(1 + x \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right) - \color{blue}{wj}, x \cdot -2\right), x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

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

   if 1.6500000000000001e-7 < wj

   1. Initial program 56.9%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. unsub-neg N/A

         \[\leadsto wj - x \cdot \left(0 - \color{blue}{\left(\frac{1}{e^{wj} + wj \cdot e^{wj}} + \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)}\right)\right)\right)}\right) \]

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(wj, e^{wj}, -x\right), \frac{e^{-wj}}{-1 - wj}, wj\right)\\ \mathbf{elif}\;wj \leq 1.65 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), 1 - wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - x \cdot \left(\frac{wj}{\mathsf{fma}\left(x, wj, x\right)} - \frac{e^{-wj}}{wj + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq -2 \cdot 10^{+50}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), 1 - wj\right), x \cdot -2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) -2e+50)
     (/ x (* (exp wj) (+ wj 1.0)))
     (fma
      wj
      (fma wj (fma x (fma wj -2.6666666666666665 2.5) (- 1.0 wj)) (* x -2.0))
      x))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= -2e+50) {
		tmp = x / (exp(wj) * (wj + 1.0));
	} else {
		tmp = fma(wj, fma(wj, fma(x, fma(wj, -2.6666666666666665, 2.5), (1.0 - wj)), (x * -2.0)), x);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= -2e+50)
		tmp = Float64(x / Float64(exp(wj) * Float64(wj + 1.0)));
	else
		tmp = fma(wj, fma(wj, fma(x, fma(wj, -2.6666666666666665, 2.5), Float64(1.0 - wj)), Float64(x * -2.0)), x);
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e+50], N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj * N[(wj * N[(x * N[(wj * -2.6666666666666665 + 2.5), $MachinePrecision] + N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < -2.0000000000000002e50

   1. Initial program 98.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -2.0000000000000002e50 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 68.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 97.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \left(1 + x \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right) - \color{blue}{wj}, x \cdot -2\right), x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 97.5%

      \[\leadsto \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right)}, 1 - wj\right), x \cdot -2\right), x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq -2 \cdot 10^{+50}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), 1 - wj\right), x \cdot -2\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{-wj}}{wj + 1}\\ \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 - \frac{wj}{\mathsf{fma}\left(wj, x, x\right)}, x, \frac{wj \cdot x}{x}\right)\\ \mathbf{elif}\;wj \leq 1.65 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), 1 - wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - x \cdot \left(\frac{wj}{\mathsf{fma}\left(x, wj, x\right)} - t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ (exp (- wj)) (+ wj 1.0))))
   (if (<= wj -4.5e-6)
     (fma (- t_0 (/ wj (fma wj x x))) x (/ (* wj x) x))
     (if (<= wj 1.65e-7)
       (fma
        wj
        (fma wj (fma x (fma wj -2.6666666666666665 2.5) (- 1.0 wj)) (* x -2.0))
        x)
       (- wj (* x (- (/ wj (fma x wj x)) t_0)))))))

C

double code(double wj, double x) {
	double t_0 = exp(-wj) / (wj + 1.0);
	double tmp;
	if (wj <= -4.5e-6) {
		tmp = fma((t_0 - (wj / fma(wj, x, x))), x, ((wj * x) / x));
	} else if (wj <= 1.65e-7) {
		tmp = fma(wj, fma(wj, fma(x, fma(wj, -2.6666666666666665, 2.5), (1.0 - wj)), (x * -2.0)), x);
	} else {
		tmp = wj - (x * ((wj / fma(x, wj, x)) - t_0));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(exp(Float64(-wj)) / Float64(wj + 1.0))
	tmp = 0.0
	if (wj <= -4.5e-6)
		tmp = fma(Float64(t_0 - Float64(wj / fma(wj, x, x))), x, Float64(Float64(wj * x) / x));
	elseif (wj <= 1.65e-7)
		tmp = fma(wj, fma(wj, fma(x, fma(wj, -2.6666666666666665, 2.5), Float64(1.0 - wj)), Float64(x * -2.0)), x);
	else
		tmp = Float64(wj - Float64(x * Float64(Float64(wj / fma(x, wj, x)) - t_0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -4.5e-6], N[(N[(t$95$0 - N[(wj / N[(wj * x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + N[(N[(wj * x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 1.65e-7], N[(wj * N[(wj * N[(x * N[(wj * -2.6666666666666665 + 2.5), $MachinePrecision] + N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj - N[(x * N[(N[(wj / N[(x * wj + x), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -4.50000000000000011e-6

   1. Initial program 68.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. div-inv N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 97.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

   7. Applied rewrites 96.9%

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

   9. Applied rewrites 97.5%

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

   if -4.50000000000000011e-6 < wj < 1.6500000000000001e-7

   1. Initial program 76.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \left(1 + x \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right) - \color{blue}{wj}, x \cdot -2\right), x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

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

   if 1.6500000000000001e-7 < wj

   1. Initial program 56.9%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. unsub-neg N/A

         \[\leadsto wj - x \cdot \left(0 - \color{blue}{\left(\frac{1}{e^{wj} + wj \cdot e^{wj}} + \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)}\right)\right)\right)}\right) \]

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-wj}}{wj + 1} - \frac{wj}{\mathsf{fma}\left(wj, x, x\right)}, x, \frac{wj \cdot x}{x}\right)\\ \mathbf{elif}\;wj \leq 1.65 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), 1 - wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - x \cdot \left(\frac{wj}{\mathsf{fma}\left(x, wj, x\right)} - \frac{e^{-wj}}{wj + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;wj - \mathsf{fma}\left(x, \frac{-1}{e^{wj} \cdot \left(wj + 1\right)}, \frac{wj}{wj + 1}\right)\\ \mathbf{elif}\;wj \leq 1.65 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), 1 - wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - x \cdot \left(\frac{wj}{\mathsf{fma}\left(x, wj, x\right)} - \frac{e^{-wj}}{wj + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -4.5e-6)
   (- wj (fma x (/ -1.0 (* (exp wj) (+ wj 1.0))) (/ wj (+ wj 1.0))))
   (if (<= wj 1.65e-7)
     (fma
      wj
      (fma wj (fma x (fma wj -2.6666666666666665 2.5) (- 1.0 wj)) (* x -2.0))
      x)
     (- wj (* x (- (/ wj (fma x wj x)) (/ (exp (- wj)) (+ wj 1.0))))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -4.5e-6) {
		tmp = wj - fma(x, (-1.0 / (exp(wj) * (wj + 1.0))), (wj / (wj + 1.0)));
	} else if (wj <= 1.65e-7) {
		tmp = fma(wj, fma(wj, fma(x, fma(wj, -2.6666666666666665, 2.5), (1.0 - wj)), (x * -2.0)), x);
	} else {
		tmp = wj - (x * ((wj / fma(x, wj, x)) - (exp(-wj) / (wj + 1.0))));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -4.5e-6)
		tmp = Float64(wj - fma(x, Float64(-1.0 / Float64(exp(wj) * Float64(wj + 1.0))), Float64(wj / Float64(wj + 1.0))));
	elseif (wj <= 1.65e-7)
		tmp = fma(wj, fma(wj, fma(x, fma(wj, -2.6666666666666665, 2.5), Float64(1.0 - wj)), Float64(x * -2.0)), x);
	else
		tmp = Float64(wj - Float64(x * Float64(Float64(wj / fma(x, wj, x)) - Float64(exp(Float64(-wj)) / Float64(wj + 1.0)))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -4.5e-6], N[(wj - N[(x * N[(-1.0 / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 1.65e-7], N[(wj * N[(wj * N[(x * N[(wj * -2.6666666666666665 + 2.5), $MachinePrecision] + N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj - N[(x * N[(N[(wj / N[(x * wj + x), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -4.50000000000000011e-6

   1. Initial program 68.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. distribute-lft-neg-in N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. distribute-frac-neg2 N/A

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

      11. div-inv N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 97.5%

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

   if -4.50000000000000011e-6 < wj < 1.6500000000000001e-7

   1. Initial program 76.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \left(1 + x \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right) - \color{blue}{wj}, x \cdot -2\right), x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

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

   if 1.6500000000000001e-7 < wj

   1. Initial program 56.9%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. unsub-neg N/A

         \[\leadsto wj - x \cdot \left(0 - \color{blue}{\left(\frac{1}{e^{wj} + wj \cdot e^{wj}} + \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)}\right)\right)\right)}\right) \]

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;wj - \mathsf{fma}\left(x, \frac{-1}{e^{wj} \cdot \left(wj + 1\right)}, \frac{wj}{wj + 1}\right)\\ \mathbf{elif}\;wj \leq 1.65 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), 1 - wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - x \cdot \left(\frac{wj}{\mathsf{fma}\left(x, wj, x\right)} - \frac{e^{-wj}}{wj + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot \left(wj + 1\right)\\ \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;wj - \mathsf{fma}\left(x, \frac{-1}{t\_0}, \frac{wj}{wj + 1}\right)\\ \mathbf{elif}\;wj \leq 1.65 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), 1 - wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(\frac{x}{t\_0} + \frac{wj}{-1 - wj}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) (+ wj 1.0))))
   (if (<= wj -4.5e-6)
     (- wj (fma x (/ -1.0 t_0) (/ wj (+ wj 1.0))))
     (if (<= wj 1.65e-7)
       (fma
        wj
        (fma wj (fma x (fma wj -2.6666666666666665 2.5) (- 1.0 wj)) (* x -2.0))
        x)
       (+ wj (+ (/ x t_0) (/ wj (- -1.0 wj))))))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * (wj + 1.0);
	double tmp;
	if (wj <= -4.5e-6) {
		tmp = wj - fma(x, (-1.0 / t_0), (wj / (wj + 1.0)));
	} else if (wj <= 1.65e-7) {
		tmp = fma(wj, fma(wj, fma(x, fma(wj, -2.6666666666666665, 2.5), (1.0 - wj)), (x * -2.0)), x);
	} else {
		tmp = wj + ((x / t_0) + (wj / (-1.0 - wj)));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(exp(wj) * Float64(wj + 1.0))
	tmp = 0.0
	if (wj <= -4.5e-6)
		tmp = Float64(wj - fma(x, Float64(-1.0 / t_0), Float64(wj / Float64(wj + 1.0))));
	elseif (wj <= 1.65e-7)
		tmp = fma(wj, fma(wj, fma(x, fma(wj, -2.6666666666666665, 2.5), Float64(1.0 - wj)), Float64(x * -2.0)), x);
	else
		tmp = Float64(wj + Float64(Float64(x / t_0) + Float64(wj / Float64(-1.0 - wj))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -4.5e-6], N[(wj - N[(x * N[(-1.0 / t$95$0), $MachinePrecision] + N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 1.65e-7], N[(wj * N[(wj * N[(x * N[(wj * -2.6666666666666665 + 2.5), $MachinePrecision] + N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj + N[(N[(x / t$95$0), $MachinePrecision] + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -4.50000000000000011e-6

   1. Initial program 68.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. distribute-lft-neg-in N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. distribute-frac-neg2 N/A

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

      11. div-inv N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 97.5%

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

   if -4.50000000000000011e-6 < wj < 1.6500000000000001e-7

   1. Initial program 76.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \left(1 + x \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right) - \color{blue}{wj}, x \cdot -2\right), x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

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

   if 1.6500000000000001e-7 < wj

   1. Initial program 56.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. times-frac N/A

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

      10. *-inverses N/A

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

      11. associate-*l/ N/A

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

      12. *-rgt-identity N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 99.8

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. distribute-rgt1-in N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;wj - \mathsf{fma}\left(x, \frac{-1}{e^{wj} \cdot \left(wj + 1\right)}, \frac{wj}{wj + 1}\right)\\ \mathbf{elif}\;wj \leq 1.65 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), 1 - wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \frac{wj}{-1 - wj}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj + \left(\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \frac{wj}{-1 - wj}\right)\\ \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;wj \leq 1.65 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), 1 - wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (+ wj (+ (/ x (* (exp wj) (+ wj 1.0))) (/ wj (- -1.0 wj))))))
   (if (<= wj -4.5e-6)
     t_0
     (if (<= wj 1.65e-7)
       (fma
        wj
        (fma wj (fma x (fma wj -2.6666666666666665 2.5) (- 1.0 wj)) (* x -2.0))
        x)
       t_0))))

C

double code(double wj, double x) {
	double t_0 = wj + ((x / (exp(wj) * (wj + 1.0))) + (wj / (-1.0 - wj)));
	double tmp;
	if (wj <= -4.5e-6) {
		tmp = t_0;
	} else if (wj <= 1.65e-7) {
		tmp = fma(wj, fma(wj, fma(x, fma(wj, -2.6666666666666665, 2.5), (1.0 - wj)), (x * -2.0)), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj + Float64(Float64(x / Float64(exp(wj) * Float64(wj + 1.0))) + Float64(wj / Float64(-1.0 - wj))))
	tmp = 0.0
	if (wj <= -4.5e-6)
		tmp = t_0;
	elseif (wj <= 1.65e-7)
		tmp = fma(wj, fma(wj, fma(x, fma(wj, -2.6666666666666665, 2.5), Float64(1.0 - wj)), Float64(x * -2.0)), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj + N[(N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -4.5e-6], t$95$0, If[LessEqual[wj, 1.65e-7], N[(wj * N[(wj * N[(x * N[(wj * -2.6666666666666665 + 2.5), $MachinePrecision] + N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if wj < -4.50000000000000011e-6 or 1.6500000000000001e-7 < wj

   1. Initial program 62.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-rgt1-in N/A

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

      9. times-frac N/A

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

      10. *-inverses N/A

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

      11. associate-*l/ N/A

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

      12. *-rgt-identity N/A

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

      13. lower-/.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 84.2

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

      16. lift-+.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. distribute-rgt1-in N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. lower-+.f64 98.5

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

   4. Applied rewrites 98.5%

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

   if -4.50000000000000011e-6 < wj < 1.6500000000000001e-7

   1. Initial program 76.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \left(1 + x \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right) - \color{blue}{wj}, x \cdot -2\right), x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right)}, 1 - wj\right), x \cdot -2\right), x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;wj + \left(\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \frac{wj}{-1 - wj}\right)\\ \mathbf{elif}\;wj \leq 1.65 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), 1 - wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \left(\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \frac{wj}{-1 - wj}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.2% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around 0

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

5. Applied rewrites 95.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \left(1 + x \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right) - \color{blue}{wj}, x \cdot -2\right), x\right) \]
7. Step-by-step derivation

8. Applied rewrites 95.9%

   \[\leadsto \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right)}, 1 - wj\right), x \cdot -2\right), x\right) \]
9. Add Preprocessing

Alternative 8: 95.7% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(wj, wj, \mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, 2.5, -2\right), x\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(wj, wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x, x\right)} \]

5. Applied rewrites 95.2%

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

6. Taylor expanded in x around 0

   \[\leadsto x \cdot \left(1 + wj \cdot \left(\frac{5}{2} \cdot wj - 2\right)\right) + \color{blue}{{wj}^{2}} \]
7. Step-by-step derivation

8. Applied rewrites 95.2%

   \[\leadsto \mathsf{fma}\left(wj, \color{blue}{wj}, \mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, 2.5, -2\right), x\right)\right) \]
9. Add Preprocessing

Alternative 9: 95.7% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, 2.5, -2\right), wj\right), x\right) \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (fma wj (fma x (fma wj 2.5 -2.0) wj) x))

C

double code(double wj, double x) {
	return fma(wj, fma(x, fma(wj, 2.5, -2.0), wj), x);
}

Julia

function code(wj, x)
	return fma(wj, fma(x, fma(wj, 2.5, -2.0), wj), x)
end

Wolfram

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

Derivation

1. Initial program 75.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around 0

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

5. Applied rewrites 95.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)} \]

6. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. distribute-rgt-in N/A

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

   5. distribute-rgt-out N/A

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

   6. metadata-eval N/A

      \[\leadsto \left(\left(wj \cdot \left(1 - x \cdot \color{blue}{\frac{-5}{2}}\right)\right) \cdot wj + \left(-2 \cdot x\right) \cdot wj\right) + x \]

   7. *-commutative N/A

      \[\leadsto \left(\left(wj \cdot \left(1 - \color{blue}{\frac{-5}{2} \cdot x}\right)\right) \cdot wj + \left(-2 \cdot x\right) \cdot wj\right) + x \]

   8. cancel-sign-sub-inv N/A

      \[\leadsto \left(\left(wj \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{-5}{2}\right)\right) \cdot x\right)}\right) \cdot wj + \left(-2 \cdot x\right) \cdot wj\right) + x \]

   9. metadata-eval N/A

      \[\leadsto \left(\left(wj \cdot \left(1 + \color{blue}{\frac{5}{2}} \cdot x\right)\right) \cdot wj + \left(-2 \cdot x\right) \cdot wj\right) + x \]

   10. distribute-rgt-in N/A

       \[\leadsto \color{blue}{wj \cdot \left(wj \cdot \left(1 + \frac{5}{2} \cdot x\right) + -2 \cdot x\right)} + x \]

   11. +-commutative N/A

       \[\leadsto wj \cdot \color{blue}{\left(-2 \cdot x + wj \cdot \left(1 + \frac{5}{2} \cdot x\right)\right)} + x \]

8. Applied rewrites 95.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, 2.5, -2\right), wj\right), x\right)} \]
9. Add Preprocessing

Alternative 10: 95.3% accurate, 22.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(wj, wj \cdot \left(1 - wj\right), x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around 0

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

5. Applied rewrites 95.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 95.1%

   \[\leadsto \mathsf{fma}\left(wj, wj \cdot \color{blue}{\left(1 - wj\right)}, x\right) \]
9. Add Preprocessing

Alternative 11: 95.5% accurate, 25.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(wj, \mathsf{fma}\left(x, -2, wj\right), x\right) \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (fma wj (fma x -2.0 wj) x))

C

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

Julia

function code(wj, x)
	return fma(wj, fma(x, -2.0, wj), x)
end

Wolfram

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

Derivation

1. Initial program 75.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around 0

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

5. Applied rewrites 95.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)} \]

6. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. distribute-rgt-in N/A

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

   5. distribute-rgt-out N/A

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

   6. metadata-eval N/A

      \[\leadsto \left(\left(wj \cdot \left(1 - x \cdot \color{blue}{\frac{-5}{2}}\right)\right) \cdot wj + \left(-2 \cdot x\right) \cdot wj\right) + x \]

   7. *-commutative N/A

      \[\leadsto \left(\left(wj \cdot \left(1 - \color{blue}{\frac{-5}{2} \cdot x}\right)\right) \cdot wj + \left(-2 \cdot x\right) \cdot wj\right) + x \]

   8. cancel-sign-sub-inv N/A

      \[\leadsto \left(\left(wj \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{-5}{2}\right)\right) \cdot x\right)}\right) \cdot wj + \left(-2 \cdot x\right) \cdot wj\right) + x \]

   9. metadata-eval N/A

      \[\leadsto \left(\left(wj \cdot \left(1 + \color{blue}{\frac{5}{2}} \cdot x\right)\right) \cdot wj + \left(-2 \cdot x\right) \cdot wj\right) + x \]

   10. distribute-rgt-in N/A

       \[\leadsto \color{blue}{wj \cdot \left(wj \cdot \left(1 + \frac{5}{2} \cdot x\right) + -2 \cdot x\right)} + x \]

   11. +-commutative N/A

       \[\leadsto wj \cdot \color{blue}{\left(-2 \cdot x + wj \cdot \left(1 + \frac{5}{2} \cdot x\right)\right)} + x \]

8. Applied rewrites 95.2%

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

9. Taylor expanded in wj around 0

   \[\leadsto \mathsf{fma}\left(wj, \mathsf{fma}\left(x, -2, wj\right), x\right) \]
10. Step-by-step derivation

11. Applied rewrites 94.9%

    \[\leadsto \mathsf{fma}\left(wj, \mathsf{fma}\left(x, -2, wj\right), x\right) \]
12. Add Preprocessing

Alternative 12: 83.9% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 85.0

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

5. Applied rewrites 85.0%

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

Alternative 13: 83.4% accurate, 66.2× speedup

Math

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

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 Float64(-Float64(-x))
end

MATLAB

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

Wolfram

code[wj_, x_] := (-(-x))

Derivation

1. Initial program 75.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around 0

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

5. Applied rewrites 95.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \left(1 + x \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right) - \color{blue}{wj}, x \cdot -2\right), x\right) \]
7. Step-by-step derivation

8. Applied rewrites 95.9%

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

9. Taylor expanded in x around -inf

   \[\leadsto -1 \cdot \color{blue}{\left(x \cdot \left(\left(-1 \cdot \frac{{wj}^{2} \cdot \left(1 - wj\right)}{x} + wj \cdot \left(2 + -1 \cdot \left(wj \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)\right)\right) - 1\right)\right)} \]
10. Step-by-step derivation

11. Applied rewrites 95.8%

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

12. Taylor expanded in wj around 0

    \[\leadsto \mathsf{neg}\left(-1 \cdot x\right) \]
13. Step-by-step derivation

14. Applied rewrites 84.7%

    \[\leadsto -\left(-x\right) \]
15. Add Preprocessing

Alternative 14: 4.2% accurate, 82.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj - 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around inf

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

5. Applied rewrites 4.2%

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

Developer Target 1: 78.1% accurate, 1.4× 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 2024233 
(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))))))