Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.4% → 98.7%

Time: 9.3s

Alternatives: 15

Speedup: 55.2×

• could not determine a ground truth

  1. wj = -2.226505817388843e+301
  2. x = -4.814113774203614e-52

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: 78.4% 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: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, x, \mathsf{fma}\left(\mathsf{fma}\left(-wj, \mathsf{fma}\left(2.6666666666666665, x, 1\right), 2.5 \cdot x\right), wj, wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \mathsf{fma}\left(\frac{x}{1 + wj}, \frac{wj}{x}, \frac{\frac{x}{-1 - wj}}{e^{wj}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 2e-16)
     (fma
      (fma
       -2.0
       x
       (fma (fma (- wj) (fma 2.6666666666666665 x 1.0) (* 2.5 x)) wj wj))
      wj
      x)
     (- wj (fma (/ x (+ 1.0 wj)) (/ wj x) (/ (/ x (- -1.0 wj)) (exp wj)))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 2e-16) {
		tmp = fma(fma(-2.0, x, fma(fma(-wj, fma(2.6666666666666665, x, 1.0), (2.5 * x)), wj, wj)), wj, x);
	} else {
		tmp = wj - fma((x / (1.0 + wj)), (wj / x), ((x / (-1.0 - wj)) / exp(wj)));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 2e-16)
		tmp = fma(fma(-2.0, x, fma(fma(Float64(-wj), fma(2.6666666666666665, x, 1.0), Float64(2.5 * x)), wj, wj)), wj, x);
	else
		tmp = Float64(wj - fma(Float64(x / Float64(1.0 + wj)), Float64(wj / x), Float64(Float64(x / Float64(-1.0 - wj)) / exp(wj))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-16], N[(N[(-2.0 * x + N[(N[((-wj) * N[(2.6666666666666665 * x + 1.0), $MachinePrecision] + N[(2.5 * x), $MachinePrecision]), $MachinePrecision] * wj + wj), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(x / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision] * N[(wj / x), $MachinePrecision] + N[(N[(x / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision] / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))))) < 2e-16

   1. Initial program 67.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.3%

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

   7. Applied rewrites 87.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, \frac{1 - {\left(wj \cdot \mathsf{fma}\left(x, 2.6666666666666665, 1\right)\right)}^{3}}{\mathsf{fma}\left(wj \cdot \mathsf{fma}\left(x, 2.6666666666666665, 1\right), \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.6666666666666665, 1\right), 1\right), 1\right)}\right), wj, -2 \cdot x\right), wj, x\right) \]

   8. Taylor expanded in wj around 0

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

   10. Applied rewrites 98.3%

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

   if 2e-16 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 97.2%

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

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

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

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. mul-1-neg N/A

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

   5. Applied rewrites 99.6%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, x, \mathsf{fma}\left(\mathsf{fma}\left(-wj, \mathsf{fma}\left(2.6666666666666665, x, 1\right), 2.5 \cdot x\right), wj, wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \mathsf{fma}\left(\frac{x}{1 + wj}, \frac{wj}{x}, \frac{\frac{x}{-1 - wj}}{e^{wj}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, x, \mathsf{fma}\left(\mathsf{fma}\left(-wj, \mathsf{fma}\left(2.6666666666666665, x, 1\right), 2.5 \cdot x\right), wj, wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{wj}{\mathsf{fma}\left(wj, x, x\right)} - \frac{e^{-wj}}{1 + wj}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 2e-16)
     (fma
      (fma
       -2.0
       x
       (fma (fma (- wj) (fma 2.6666666666666665 x 1.0) (* 2.5 x)) wj wj))
      wj
      x)
     (- wj (* (- (/ wj (fma wj x x)) (/ (exp (- wj)) (+ 1.0 wj))) x)))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 2e-16) {
		tmp = fma(fma(-2.0, x, fma(fma(-wj, fma(2.6666666666666665, x, 1.0), (2.5 * x)), wj, wj)), wj, x);
	} else {
		tmp = wj - (((wj / fma(wj, x, x)) - (exp(-wj) / (1.0 + wj))) * x);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 2e-16)
		tmp = fma(fma(-2.0, x, fma(fma(Float64(-wj), fma(2.6666666666666665, x, 1.0), Float64(2.5 * x)), wj, wj)), wj, x);
	else
		tmp = Float64(wj - Float64(Float64(Float64(wj / fma(wj, x, x)) - Float64(exp(Float64(-wj)) / Float64(1.0 + wj))) * 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[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-16], N[(N[(-2.0 * x + N[(N[((-wj) * N[(2.6666666666666665 * x + 1.0), $MachinePrecision] + N[(2.5 * x), $MachinePrecision]), $MachinePrecision] * wj + wj), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(N[(wj / N[(wj * x + x), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[(-wj)], $MachinePrecision] / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $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))))) < 2e-16

   1. Initial program 67.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.3%

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

   7. Applied rewrites 87.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, \frac{1 - {\left(wj \cdot \mathsf{fma}\left(x, 2.6666666666666665, 1\right)\right)}^{3}}{\mathsf{fma}\left(wj \cdot \mathsf{fma}\left(x, 2.6666666666666665, 1\right), \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.6666666666666665, 1\right), 1\right), 1\right)}\right), wj, -2 \cdot x\right), wj, x\right) \]

   8. Taylor expanded in wj around 0

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

   10. Applied rewrites 98.3%

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

   if 2e-16 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 97.2%

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

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

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

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. mul-1-neg N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.6%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, x, \mathsf{fma}\left(\mathsf{fma}\left(-wj, \mathsf{fma}\left(2.6666666666666665, x, 1\right), 2.5 \cdot x\right), wj, wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{wj}{\mathsf{fma}\left(wj, x, x\right)} - \frac{e^{-wj}}{1 + wj}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0085:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, x, \mathsf{fma}\left(\mathsf{fma}\left(-wj, \mathsf{fma}\left(2.6666666666666665, x, 1\right), 2.5 \cdot x\right), wj, wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(1 + wj\right)}^{-1} \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0085)
   (fma
    (fma
     -2.0
     x
     (fma (fma (- wj) (fma 2.6666666666666665 x 1.0) (* 2.5 x)) wj wj))
    wj
    x)
   (- wj (* (pow (+ 1.0 wj) -1.0) wj))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0085) {
		tmp = fma(fma(-2.0, x, fma(fma(-wj, fma(2.6666666666666665, x, 1.0), (2.5 * x)), wj, wj)), wj, x);
	} else {
		tmp = wj - (pow((1.0 + wj), -1.0) * wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0085)
		tmp = fma(fma(-2.0, x, fma(fma(Float64(-wj), fma(2.6666666666666665, x, 1.0), Float64(2.5 * x)), wj, wj)), wj, x);
	else
		tmp = Float64(wj - Float64((Float64(1.0 + wj) ^ -1.0) * wj));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0085], N[(N[(-2.0 * x + N[(N[((-wj) * N[(2.6666666666666665 * x + 1.0), $MachinePrecision] + N[(2.5 * x), $MachinePrecision]), $MachinePrecision] * wj + wj), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[Power[N[(1.0 + wj), $MachinePrecision], -1.0], $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.0085000000000000006

   1. Initial program 76.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.3%

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

   7. Applied rewrites 86.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, \frac{1 - {\left(wj \cdot \mathsf{fma}\left(x, 2.6666666666666665, 1\right)\right)}^{3}}{\mathsf{fma}\left(wj \cdot \mathsf{fma}\left(x, 2.6666666666666665, 1\right), \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.6666666666666665, 1\right), 1\right), 1\right)}\right), wj, -2 \cdot x\right), wj, x\right) \]

   8. Taylor expanded in wj around 0

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

   10. Applied rewrites 98.3%

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

   if 0.0085000000000000006 < wj

   1. Initial program 59.4%

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

   3. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-lft-identity N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-/r* N/A

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

      7. *-inverses N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 99.4

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

   8. Applied rewrites 99.4%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.0085:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, x, \mathsf{fma}\left(\mathsf{fma}\left(-wj, \mathsf{fma}\left(2.6666666666666665, x, 1\right), 2.5 \cdot x\right), wj, wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(1 + wj\right)}^{-1} \cdot wj\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0085:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), wj, -2\right), wj, 1\right), x, \left(1 - wj\right) \cdot \left(wj \cdot wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(1 + wj\right)}^{-1} \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0085)
   (fma
    (fma (fma (fma -2.6666666666666665 wj 2.5) wj -2.0) wj 1.0)
    x
    (* (- 1.0 wj) (* wj wj)))
   (- wj (* (pow (+ 1.0 wj) -1.0) wj))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0085) {
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), wj, -2.0), wj, 1.0), x, ((1.0 - wj) * (wj * wj)));
	} else {
		tmp = wj - (pow((1.0 + wj), -1.0) * wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0085)
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), wj, -2.0), wj, 1.0), x, Float64(Float64(1.0 - wj) * Float64(wj * wj)));
	else
		tmp = Float64(wj - Float64((Float64(1.0 + wj) ^ -1.0) * wj));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0085], N[(N[(N[(N[(-2.6666666666666665 * wj + 2.5), $MachinePrecision] * wj + -2.0), $MachinePrecision] * wj + 1.0), $MachinePrecision] * x + N[(N[(1.0 - wj), $MachinePrecision] * N[(wj * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[Power[N[(1.0 + wj), $MachinePrecision], -1.0], $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.0085000000000000006

   1. Initial program 76.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 98.3%

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

   if 0.0085000000000000006 < wj

   1. Initial program 59.4%

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

   3. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-lft-identity N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-/r* N/A

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

      7. *-inverses N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 99.4

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

   8. Applied rewrites 99.4%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.0085:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), wj, -2\right), wj, 1\right), x, \left(1 - wj\right) \cdot \left(wj \cdot wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(1 + wj\right)}^{-1} \cdot wj\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00135:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, x, \mathsf{fma}\left(-wj, wj, wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(1 + wj\right)}^{-1} \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00135)
   (fma (fma -2.0 x (fma (- wj) wj wj)) wj x)
   (- wj (* (pow (+ 1.0 wj) -1.0) wj))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00135) {
		tmp = fma(fma(-2.0, x, fma(-wj, wj, wj)), wj, x);
	} else {
		tmp = wj - (pow((1.0 + wj), -1.0) * wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00135)
		tmp = fma(fma(-2.0, x, fma(Float64(-wj), wj, wj)), wj, x);
	else
		tmp = Float64(wj - Float64((Float64(1.0 + wj) ^ -1.0) * wj));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00135], N[(N[(-2.0 * x + N[((-wj) * wj + wj), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[Power[N[(1.0 + wj), $MachinePrecision], -1.0], $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.0013500000000000001

   1. Initial program 76.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.3%

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

   7. Applied rewrites 86.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, \frac{1 - {\left(wj \cdot \mathsf{fma}\left(x, 2.6666666666666665, 1\right)\right)}^{3}}{\mathsf{fma}\left(wj \cdot \mathsf{fma}\left(x, 2.6666666666666665, 1\right), \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.6666666666666665, 1\right), 1\right), 1\right)}\right), wj, -2 \cdot x\right), wj, x\right) \]

   8. Taylor expanded in wj around 0

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

   10. Applied rewrites 98.3%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 97.7%

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

   if 0.0013500000000000001 < wj

   1. Initial program 59.4%

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

   3. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-lft-identity N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-/r* N/A

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

      7. *-inverses N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 99.4

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

   8. Applied rewrites 99.4%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.00135:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, x, \mathsf{fma}\left(-wj, wj, wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(1 + wj\right)}^{-1} \cdot wj\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.4% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00135:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, x, \mathsf{fma}\left(-wj, wj, wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00135)
   (fma (fma -2.0 x (fma (- wj) wj wj)) wj x)
   (- wj (/ wj (+ 1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00135) {
		tmp = fma(fma(-2.0, x, fma(-wj, wj, wj)), wj, x);
	} else {
		tmp = wj - (wj / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00135)
		tmp = fma(fma(-2.0, x, fma(Float64(-wj), wj, wj)), wj, x);
	else
		tmp = Float64(wj - Float64(wj / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00135], N[(N[(-2.0 * x + N[((-wj) * wj + wj), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.0013500000000000001

   1. Initial program 76.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.3%

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

   7. Applied rewrites 86.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, \frac{1 - {\left(wj \cdot \mathsf{fma}\left(x, 2.6666666666666665, 1\right)\right)}^{3}}{\mathsf{fma}\left(wj \cdot \mathsf{fma}\left(x, 2.6666666666666665, 1\right), \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.6666666666666665, 1\right), 1\right), 1\right)}\right), wj, -2 \cdot x\right), wj, x\right) \]

   8. Taylor expanded in wj around 0

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

   10. Applied rewrites 98.3%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 97.7%

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

   if 0.0013500000000000001 < wj

   1. Initial program 59.4%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

Alternative 7: 97.1% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00135:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, wj, -2\right), x, wj\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00135)
   (fma (fma (fma 2.5 wj -2.0) x wj) wj x)
   (- wj (/ wj (+ 1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00135) {
		tmp = fma(fma(fma(2.5, wj, -2.0), x, wj), wj, x);
	} else {
		tmp = wj - (wj / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00135)
		tmp = fma(fma(fma(2.5, wj, -2.0), x, wj), wj, x);
	else
		tmp = Float64(wj - Float64(wj / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00135], N[(N[(N[(2.5 * wj + -2.0), $MachinePrecision] * x + wj), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.0013500000000000001

   1. Initial program 76.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.3%

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

   6. Taylor expanded in wj around 0

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

   8. Applied rewrites 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(wj + x \cdot \mathsf{fma}\left(2.5, wj, -2\right), wj, x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 97.6%

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

   if 0.0013500000000000001 < wj

   1. Initial program 59.4%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

Alternative 8: 96.9% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.000115:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - wj\right) \cdot wj, wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.000115) (fma (* (- 1.0 wj) wj) wj x) (- wj (/ wj (+ 1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.000115) {
		tmp = fma(((1.0 - wj) * wj), wj, x);
	} else {
		tmp = wj - (wj / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.000115)
		tmp = fma(Float64(Float64(1.0 - wj) * wj), wj, x);
	else
		tmp = Float64(wj - Float64(wj / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.000115], N[(N[(N[(1.0 - wj), $MachinePrecision] * wj), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.15e-4

   1. Initial program 76.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.3%

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

   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 96.9%

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

   if 1.15e-4 < wj

   1. Initial program 59.4%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

Alternative 9: 95.8% 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 76.3%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.5%

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

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 95.1%

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

Alternative 10: 84.8% 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 76.3%

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

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

   5. lower-*.f64 83.2

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

5. Applied rewrites 83.2%

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

Alternative 11: 84.8% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.3%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.5%

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

6. Taylor expanded in wj around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. +-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 83.2

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

8. Applied rewrites 83.2%

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

Alternative 12: 84.3% accurate, 55.2× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.5%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 96.0%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 83.6%

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

12. Taylor expanded in wj around 0

    \[\leadsto 1 \cdot x \]
13. Step-by-step derivation

14. Applied rewrites 82.5%

    \[\leadsto 1 \cdot x \]
15. Add Preprocessing

Alternative 13: 13.9% 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 76.3%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.5%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 96.0%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 14.5%

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

Alternative 14: 4.1% accurate, 82.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

3. Taylor expanded in wj around inf

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. distribute-lft-neg-out N/A

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

   5. lft-mult-inverse N/A

      \[\leadsto wj + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 4.0

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

5. Applied rewrites 4.0%

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

Alternative 15: 3.4% accurate, 331.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return -1.0

Julia

function code(wj, x)
	return -1.0
end

MATLAB

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

Wolfram

code[wj_, x_] := -1.0

Derivation

1. Initial program 76.3%

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

3. Taylor expanded in wj around inf

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. distribute-lft-neg-out N/A

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

   5. lft-mult-inverse N/A

      \[\leadsto wj + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right) \]

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 4.0

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

5. Applied rewrites 4.0%

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

6. Taylor expanded in wj around 0

   \[\leadsto -1 \]
7. Step-by-step derivation

8. Applied rewrites 3.1%

   \[\leadsto -1 \]
9. Add Preprocessing

Developer Target 1: 79.1% 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 2024317 
(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))))))