Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.1% → 98.9%

Time: 8.7s

Alternatives: 10

Speedup: 27.6×

• could not determine a ground truth

  1. wj = 3.853926095879488e+282
  2. x = 3.9896636618795535e-220

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: 78.1% 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: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot wj\\ \mathbf{if}\;wj - \frac{t\_0 - x}{t\_0 + e^{wj}} \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-wj} \cdot x - wj}{\mathsf{fma}\left(wj \cdot wj, wj, 1\right)}, \mathsf{fma}\left(wj, wj - 1, 1\right), wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) wj)))
   (if (<= (- wj (/ (- t_0 x) (+ t_0 (exp wj)))) 1e-13)
     (fma
      (fma
       (fma 2.5 x (- 1.0 (* (fma 0.6666666666666666 x (fma 2.0 x 1.0)) wj)))
       wj
       (* -2.0 x))
      wj
      x)
     (fma
      (/ (- (* (exp (- wj)) x) wj) (fma (* wj wj) wj 1.0))
      (fma wj (- wj 1.0) 1.0)
      wj))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * wj;
	double tmp;
	if ((wj - ((t_0 - x) / (t_0 + exp(wj)))) <= 1e-13) {
		tmp = fma(fma(fma(2.5, x, (1.0 - (fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = fma((((exp(-wj) * x) - wj) / fma((wj * wj), wj, 1.0)), fma(wj, (wj - 1.0), 1.0), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(exp(wj) * wj)
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(t_0 + exp(wj)))) <= 1e-13)
		tmp = fma(fma(fma(2.5, x, Float64(1.0 - Float64(fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = fma(Float64(Float64(Float64(exp(Float64(-wj)) * x) - wj) / fma(Float64(wj * wj), wj, 1.0)), fma(wj, Float64(wj - 1.0), 1.0), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[Exp[wj], $MachinePrecision] * wj), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(t$95$0 + N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-13], N[(N[(N[(2.5 * x + N[(1.0 - N[(N[(0.6666666666666666 * x + N[(2.0 * x + 1.0), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(N[(N[(N[(N[Exp[(-wj)], $MachinePrecision] * x), $MachinePrecision] - wj), $MachinePrecision] / N[(N[(wj * wj), $MachinePrecision] * wj + 1.0), $MachinePrecision]), $MachinePrecision] * N[(wj * N[(wj - 1.0), $MachinePrecision] + 1.0), $MachinePrecision] + wj), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 1e-13

   1. Initial program 72.5%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.8%

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

   if 1e-13 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 99.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. flip3-+ N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 99.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}} \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-wj} \cdot x - wj}{\mathsf{fma}\left(wj \cdot wj, wj, 1\right)}, \mathsf{fma}\left(wj, wj - 1, 1\right), wj\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot wj\\ t_1 := wj - \frac{t\_0 - x}{t\_0 + e^{wj}}\\ t_2 := wj - \left(-x\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-250}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) wj))
        (t_1 (- wj (/ (- t_0 x) (+ t_0 (exp wj)))))
        (t_2 (- wj (- x))))
   (if (<= t_1 -1e-250) t_2 (if (<= t_1 0.0) (* wj wj) t_2))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * wj;
	double t_1 = wj - ((t_0 - x) / (t_0 + exp(wj)));
	double t_2 = wj - -x;
	double tmp;
	if (t_1 <= -1e-250) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = wj * wj;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = exp(wj) * wj
    t_1 = wj - ((t_0 - x) / (t_0 + exp(wj)))
    t_2 = wj - -x
    if (t_1 <= (-1d-250)) then
        tmp = t_2
    else if (t_1 <= 0.0d0) then
        tmp = wj * wj
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = Math.exp(wj) * wj;
	double t_1 = wj - ((t_0 - x) / (t_0 + Math.exp(wj)));
	double t_2 = wj - -x;
	double tmp;
	if (t_1 <= -1e-250) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = wj * wj;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = math.exp(wj) * wj
	t_1 = wj - ((t_0 - x) / (t_0 + math.exp(wj)))
	t_2 = wj - -x
	tmp = 0
	if t_1 <= -1e-250:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = wj * wj
	else:
		tmp = t_2
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(exp(wj) * wj)
	t_1 = Float64(wj - Float64(Float64(t_0 - x) / Float64(t_0 + exp(wj))))
	t_2 = Float64(wj - Float64(-x))
	tmp = 0.0
	if (t_1 <= -1e-250)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(wj * wj);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = exp(wj) * wj;
	t_1 = wj - ((t_0 - x) / (t_0 + exp(wj)));
	t_2 = wj - -x;
	tmp = 0.0;
	if (t_1 <= -1e-250)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = wj * wj;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[Exp[wj], $MachinePrecision] * wj), $MachinePrecision]}, Block[{t$95$1 = N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(t$95$0 + N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(wj - (-x)), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-250], t$95$2, If[LessEqual[t$95$1, 0.0], N[(wj * wj), $MachinePrecision], t$95$2]]]]]

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))))) < -1.0000000000000001e-250 or 0.0 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 98.2%

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

   3. Taylor expanded in wj around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.0

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

   5. Applied rewrites 90.0%

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

   if -1.0000000000000001e-250 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 0.0

   1. Initial program 6.1%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.0%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}} \leq -1 \cdot 10^{-250}:\\ \;\;\;\;wj - \left(-x\right)\\ \mathbf{elif}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}} \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot wj\\ \mathbf{if}\;wj - \frac{t\_0 - x}{t\_0 + e^{wj}} \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-wj} \cdot x - wj}{1 - wj \cdot wj}, 1 - wj, wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) wj)))
   (if (<= (- wj (/ (- t_0 x) (+ t_0 (exp wj)))) 1e-13)
     (fma
      (fma
       (fma 2.5 x (- 1.0 (* (fma 0.6666666666666666 x (fma 2.0 x 1.0)) wj)))
       wj
       (* -2.0 x))
      wj
      x)
     (fma (/ (- (* (exp (- wj)) x) wj) (- 1.0 (* wj wj))) (- 1.0 wj) wj))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * wj;
	double tmp;
	if ((wj - ((t_0 - x) / (t_0 + exp(wj)))) <= 1e-13) {
		tmp = fma(fma(fma(2.5, x, (1.0 - (fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = fma((((exp(-wj) * x) - wj) / (1.0 - (wj * wj))), (1.0 - wj), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(exp(wj) * wj)
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(t_0 + exp(wj)))) <= 1e-13)
		tmp = fma(fma(fma(2.5, x, Float64(1.0 - Float64(fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = fma(Float64(Float64(Float64(exp(Float64(-wj)) * x) - wj) / Float64(1.0 - Float64(wj * wj))), Float64(1.0 - wj), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[Exp[wj], $MachinePrecision] * wj), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(t$95$0 + N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-13], N[(N[(N[(2.5 * x + N[(1.0 - N[(N[(0.6666666666666666 * x + N[(2.0 * x + 1.0), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(N[(N[(N[(N[Exp[(-wj)], $MachinePrecision] * x), $MachinePrecision] - wj), $MachinePrecision] / N[(1.0 - N[(wj * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - wj), $MachinePrecision] + wj), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 1e-13

   1. Initial program 72.5%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.8%

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

   if 1e-13 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 99.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. sub-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-/r/ N/A

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

      11. lower-fma.f64 N/A

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

   6. Applied rewrites 99.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}} \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-wj} \cdot x - wj}{1 - wj \cdot wj}, 1 - wj, wj\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot wj\\ \mathbf{if}\;wj - \frac{t\_0 - x}{t\_0 + e^{wj}} \leq 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{1 + wj}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(-0.5, wj, 1\right), 1\right), wj, -x\right), wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) wj)))
   (if (<= (- wj (/ (- t_0 x) (+ t_0 (exp wj)))) 1e-27)
     (fma (fma (fma 2.5 x 1.0) wj (* -2.0 x)) wj x)
     (fma
      (/ -1.0 (+ 1.0 wj))
      (fma (fma x (fma -0.5 wj 1.0) 1.0) wj (- x))
      wj))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * wj;
	double tmp;
	if ((wj - ((t_0 - x) / (t_0 + exp(wj)))) <= 1e-27) {
		tmp = fma(fma(fma(2.5, x, 1.0), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = fma((-1.0 / (1.0 + wj)), fma(fma(x, fma(-0.5, wj, 1.0), 1.0), wj, -x), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(exp(wj) * wj)
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(t_0 + exp(wj)))) <= 1e-27)
		tmp = fma(fma(fma(2.5, x, 1.0), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = fma(Float64(-1.0 / Float64(1.0 + wj)), fma(fma(x, fma(-0.5, wj, 1.0), 1.0), wj, Float64(-x)), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[Exp[wj], $MachinePrecision] * wj), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(t$95$0 + N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-27], N[(N[(N[(2.5 * x + 1.0), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(N[(-1.0 / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision] * N[(N[(x * N[(-0.5 * wj + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * wj + (-x)), $MachinePrecision] + wj), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 1e-27

   1. Initial program 71.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 1e-27 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 98.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in wj around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.4

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

   7. Applied rewrites 90.4%

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

   8. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate--l+ N/A

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

      5. distribute-rgt-out-- N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out-- N/A

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

      12. lower-fma.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 97.8

          \[\leadsto \mathsf{fma}\left(\frac{-1}{1 + wj}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(-0.5, wj, 1\right), 1\right), wj, \color{blue}{-x}\right), wj\right) \]

   10. Applied rewrites 97.8%

       \[\leadsto \mathsf{fma}\left(\frac{-1}{1 + wj}, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(-0.5, wj, 1\right), 1\right), wj, -x\right)}, wj\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}} \leq 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{1 + wj}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(-0.5, wj, 1\right), 1\right), wj, -x\right), wj\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot wj\\ \mathbf{if}\;wj - \frac{t\_0 - x}{t\_0 + e^{wj}} \leq 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{1 + wj}, \mathsf{fma}\left(1 + x, wj, -x\right), wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) wj)))
   (if (<= (- wj (/ (- t_0 x) (+ t_0 (exp wj)))) 1e-27)
     (fma (fma (fma 2.5 x 1.0) wj (* -2.0 x)) wj x)
     (fma (/ -1.0 (+ 1.0 wj)) (fma (+ 1.0 x) wj (- x)) wj))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * wj;
	double tmp;
	if ((wj - ((t_0 - x) / (t_0 + exp(wj)))) <= 1e-27) {
		tmp = fma(fma(fma(2.5, x, 1.0), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = fma((-1.0 / (1.0 + wj)), fma((1.0 + x), wj, -x), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(exp(wj) * wj)
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(t_0 + exp(wj)))) <= 1e-27)
		tmp = fma(fma(fma(2.5, x, 1.0), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = fma(Float64(-1.0 / Float64(1.0 + wj)), fma(Float64(1.0 + x), wj, Float64(-x)), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[Exp[wj], $MachinePrecision] * wj), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(t$95$0 + N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-27], N[(N[(N[(2.5 * x + 1.0), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(N[(-1.0 / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] * wj + (-x)), $MachinePrecision] + wj), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 1e-27

   1. Initial program 71.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 1e-27 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 98.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 97.8

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

   7. Applied rewrites 97.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}} \leq 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{1 + wj}, \mathsf{fma}\left(1 + x, wj, -x\right), wj\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (fma
  (fma
   (fma 2.5 x (- 1.0 (* (fma 0.6666666666666666 x (fma 2.0 x 1.0)) wj)))
   wj
   (* -2.0 x))
  wj
  x))

C

double code(double wj, double x) {
	return fma(fma(fma(2.5, x, (1.0 - (fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, (-2.0 * x)), wj, x);
}

Julia

function code(wj, x)
	return fma(fma(fma(2.5, x, Float64(1.0 - Float64(fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, Float64(-2.0 * x)), wj, x)
end

Wolfram

code[wj_, x_] := N[(N[(N[(2.5 * x + N[(1.0 - N[(N[(0.6666666666666666 * x + N[(2.0 * x + 1.0), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision]

Derivation

1. Initial program 80.9%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 97.1%

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

Alternative 7: 95.7% accurate, 22.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.9%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 97.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.8%

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

Alternative 8: 84.5% accurate, 27.6× speedup

Math

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

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(Float64(x * wj), -2.0, x)
end

Wolfram

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

Derivation

1. Initial program 80.9%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 85.5

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

5. Applied rewrites 85.5%

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

6. Final simplification 85.5%

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

Alternative 9: 14.1% accurate, 55.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

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

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-out N/A

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

   11. *-commutative N/A

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

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

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f64 96.7

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

5. Applied rewrites 96.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 14.0%

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

Alternative 10: 4.1% 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 80.9%

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

3. Taylor expanded in wj around inf

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

5. Applied rewrites 3.3%

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

Developer Target 1: 79.0% 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 2024240 
(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))))))