Jmat.Real.lambertw, newton loop step

Percentage Accurate: 41.1% → 98.3%

Time: 7.6s

Alternatives: 8

Speedup: 51.6×

• Could not converge on a ground truth

  1. wj = 1.7533746046869366e+195
  2. x = -1.7030234586554808e+258

Specification

Math

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

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)
use fmin_fmax_functions
    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: 41.1% accurate, 1.0× speedup

Math

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

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)
use fmin_fmax_functions
    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: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := 0 - 0 \cdot \left(1 - -2 \cdot x\right)\\ \mathbf{if}\;wj \leq -0.102:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;wj \leq 155000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2.5, 1 - \mathsf{fma}\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(x, 5, 0.6666666666666666 \cdot x\right)\right), wj, wj\right)\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (wj x)
  :precision binary64
  (let* ((t_0 (- 0.0 (* 0.0 (- 1.0 (* -2.0 x))))))
  (if (<= wj -0.102)
    t_0
    (if (<= wj 155000.0)
      (fma
       (fma
        (fma
         x
         2.5
         (-
          1.0
          (fma
           (fma -3.0 x (fma x 5.0 (* 0.6666666666666666 x)))
           wj
           wj)))
        wj
        (* -2.0 x))
       wj
       x)
      t_0))))

C

double code(double wj, double x) {
	double t_0 = 0.0 - (0.0 * (1.0 - (-2.0 * x)));
	double tmp;
	if (wj <= -0.102) {
		tmp = t_0;
	} else if (wj <= 155000.0) {
		tmp = fma(fma(fma(x, 2.5, (1.0 - fma(fma(-3.0, x, fma(x, 5.0, (0.6666666666666666 * x))), wj, wj))), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(0.0 - Float64(0.0 * Float64(1.0 - Float64(-2.0 * x))))
	tmp = 0.0
	if (wj <= -0.102)
		tmp = t_0;
	elseif (wj <= 155000.0)
		tmp = fma(fma(fma(x, 2.5, Float64(1.0 - fma(fma(-3.0, x, fma(x, 5.0, Float64(0.6666666666666666 * x))), wj, wj))), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(0.0 - N[(0.0 * N[(1.0 - N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -0.102], t$95$0, If[LessEqual[wj, 155000.0], N[(N[(N[(x * 2.5 + N[(1.0 - N[(N[(-3.0 * x + N[(x * 5.0 + N[(0.6666666666666666 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if wj < -0.10199999999999999 or 155000 < wj

   1. Initial program 41.1%

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

   2. Taylor expanded in wj around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 62.6%

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

   4. Applied rewrites 62.6%

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

   5. Taylor expanded in wj around 0

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

      1. lower-*.f64 39.0%

         \[\leadsto wj - -1 \cdot x \]

   7. Applied rewrites 39.0%

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

   8. Taylor expanded in wj around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 25.3%

         \[\leadsto wj - wj \cdot \left(1 - -2 \cdot x\right) \]

   10. Applied rewrites 25.3%

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

   11. Taylor expanded in undef-var around zero

       \[\leadsto \color{blue}{0} - wj \cdot \left(1 - -2 \cdot x\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 3.1%

       \[\leadsto \color{blue}{0} - wj \cdot \left(1 - -2 \cdot x\right) \]

   14. Taylor expanded in undef-var around zero

       \[\leadsto 0 - 0 \cdot \left(1 - \color{blue}{-2} \cdot x\right) \]
   15. Step-by-step derivation

   16. Applied rewrites 49.8%

       \[\leadsto 0 - 0 \cdot \left(1 - \color{blue}{-2} \cdot x\right) \]

   if -0.10199999999999999 < wj < 155000

   1. Initial program 41.1%

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

   2. Taylor expanded in wj around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 51.1%

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

   6. Applied rewrites 51.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2.5, 1 - \mathsf{fma}\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(x, 5, 0.6666666666666666 \cdot x\right)\right), wj, wj\right)\right), wj, -2 \cdot x\right), \color{blue}{wj}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := 0 - 0 \cdot \left(1 - -2 \cdot x\right)\\ \mathbf{if}\;wj \leq -0.102:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;wj \leq 155000:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 + -1 \cdot wj\right) - 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (wj x)
  :precision binary64
  (let* ((t_0 (- 0.0 (* 0.0 (- 1.0 (* -2.0 x))))))
  (if (<= wj -0.102)
    t_0
    (if (<= wj 155000.0)
      (+ x (* wj (- (* wj (+ 1.0 (* -1.0 wj))) (* 2.0 x))))
      t_0))))

C

double code(double wj, double x) {
	double t_0 = 0.0 - (0.0 * (1.0 - (-2.0 * x)));
	double tmp;
	if (wj <= -0.102) {
		tmp = t_0;
	} else if (wj <= 155000.0) {
		tmp = x + (wj * ((wj * (1.0 + (-1.0 * wj))) - (2.0 * x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
use fmin_fmax_functions
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.0d0 - (0.0d0 * (1.0d0 - ((-2.0d0) * x)))
    if (wj <= (-0.102d0)) then
        tmp = t_0
    else if (wj <= 155000.0d0) then
        tmp = x + (wj * ((wj * (1.0d0 + ((-1.0d0) * wj))) - (2.0d0 * x)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = 0.0 - (0.0 * (1.0 - (-2.0 * x)));
	double tmp;
	if (wj <= -0.102) {
		tmp = t_0;
	} else if (wj <= 155000.0) {
		tmp = x + (wj * ((wj * (1.0 + (-1.0 * wj))) - (2.0 * x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = 0.0 - (0.0 * (1.0 - (-2.0 * x)))
	tmp = 0
	if wj <= -0.102:
		tmp = t_0
	elif wj <= 155000.0:
		tmp = x + (wj * ((wj * (1.0 + (-1.0 * wj))) - (2.0 * x)))
	else:
		tmp = t_0
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(0.0 - Float64(0.0 * Float64(1.0 - Float64(-2.0 * x))))
	tmp = 0.0
	if (wj <= -0.102)
		tmp = t_0;
	elseif (wj <= 155000.0)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(1.0 + Float64(-1.0 * wj))) - Float64(2.0 * x))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = 0.0 - (0.0 * (1.0 - (-2.0 * x)));
	tmp = 0.0;
	if (wj <= -0.102)
		tmp = t_0;
	elseif (wj <= 155000.0)
		tmp = x + (wj * ((wj * (1.0 + (-1.0 * wj))) - (2.0 * x)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(0.0 - N[(0.0 * N[(1.0 - N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -0.102], t$95$0, If[LessEqual[wj, 155000.0], N[(x + N[(wj * N[(N[(wj * N[(1.0 + N[(-1.0 * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if wj < -0.10199999999999999 or 155000 < wj

   1. Initial program 41.1%

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

   2. Taylor expanded in wj around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 62.6%

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

   4. Applied rewrites 62.6%

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

   5. Taylor expanded in wj around 0

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

      1. lower-*.f64 39.0%

         \[\leadsto wj - -1 \cdot x \]

   7. Applied rewrites 39.0%

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

   8. Taylor expanded in wj around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 25.3%

         \[\leadsto wj - wj \cdot \left(1 - -2 \cdot x\right) \]

   10. Applied rewrites 25.3%

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

   11. Taylor expanded in undef-var around zero

       \[\leadsto \color{blue}{0} - wj \cdot \left(1 - -2 \cdot x\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 3.1%

       \[\leadsto \color{blue}{0} - wj \cdot \left(1 - -2 \cdot x\right) \]

   14. Taylor expanded in undef-var around zero

       \[\leadsto 0 - 0 \cdot \left(1 - \color{blue}{-2} \cdot x\right) \]
   15. Step-by-step derivation

   16. Applied rewrites 49.8%

       \[\leadsto 0 - 0 \cdot \left(1 - \color{blue}{-2} \cdot x\right) \]

   if -0.10199999999999999 < wj < 155000

   1. Initial program 41.1%

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

   2. Taylor expanded in wj around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 + -1 \cdot wj\right) - 2 \cdot x\right) \]

      2. lower-+.f64 N/A

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 + -1 \cdot wj\right) - 2 \cdot x\right) \]

      3. lower-*.f64 51.0%

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 + -1 \cdot wj\right) - 2 \cdot x\right) \]

   7. Applied rewrites 51.0%

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

Alternative 3: 97.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} t_0 := 0 - 0 \cdot \left(1 - -2 \cdot x\right)\\ \mathbf{if}\;wj \leq -0.102:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;wj \leq 0.00021:\\ \;\;\;\;\mathsf{fma}\left(wj \cdot \left(1 - wj\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (wj x)
  :precision binary64
  (let* ((t_0 (- 0.0 (* 0.0 (- 1.0 (* -2.0 x))))))
  (if (<= wj -0.102)
    t_0
    (if (<= wj 0.00021) (fma (* wj (- 1.0 wj)) wj x) t_0))))

C

double code(double wj, double x) {
	double t_0 = 0.0 - (0.0 * (1.0 - (-2.0 * x)));
	double tmp;
	if (wj <= -0.102) {
		tmp = t_0;
	} else if (wj <= 0.00021) {
		tmp = fma((wj * (1.0 - wj)), wj, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(0.0 - Float64(0.0 * Float64(1.0 - Float64(-2.0 * x))))
	tmp = 0.0
	if (wj <= -0.102)
		tmp = t_0;
	elseif (wj <= 0.00021)
		tmp = fma(Float64(wj * Float64(1.0 - wj)), wj, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(0.0 - N[(0.0 * N[(1.0 - N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -0.102], t$95$0, If[LessEqual[wj, 0.00021], N[(N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if wj < -0.10199999999999999 or 2.1000000000000001e-4 < wj

   1. Initial program 41.1%

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

   2. Taylor expanded in wj around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 62.6%

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

   4. Applied rewrites 62.6%

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

   5. Taylor expanded in wj around 0

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

      1. lower-*.f64 39.0%

         \[\leadsto wj - -1 \cdot x \]

   7. Applied rewrites 39.0%

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

   8. Taylor expanded in wj around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 25.3%

         \[\leadsto wj - wj \cdot \left(1 - -2 \cdot x\right) \]

   10. Applied rewrites 25.3%

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

   11. Taylor expanded in undef-var around zero

       \[\leadsto \color{blue}{0} - wj \cdot \left(1 - -2 \cdot x\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 3.1%

       \[\leadsto \color{blue}{0} - wj \cdot \left(1 - -2 \cdot x\right) \]

   14. Taylor expanded in undef-var around zero

       \[\leadsto 0 - 0 \cdot \left(1 - \color{blue}{-2} \cdot x\right) \]
   15. Step-by-step derivation

   16. Applied rewrites 49.8%

       \[\leadsto 0 - 0 \cdot \left(1 - \color{blue}{-2} \cdot x\right) \]

   if -0.10199999999999999 < wj < 2.1000000000000001e-4

   1. Initial program 41.1%

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

   2. Taylor expanded in wj around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 51.1%

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

   6. Applied rewrites 51.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2.5, 1 - \mathsf{fma}\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(x, 5, 0.6666666666666666 \cdot x\right)\right), wj, wj\right)\right), wj, -2 \cdot x\right), \color{blue}{wj}, x\right) \]

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 50.7%

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

   9. Applied rewrites 50.7%

      \[\leadsto \mathsf{fma}\left(wj \cdot \left(1 - wj\right), wj, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;wj \leq -0.102:\\ \;\;\;\;0 - 0 \cdot \left(1 - -2 \cdot x\right)\\ \mathbf{elif}\;wj \leq 400:\\ \;\;\;\;\mathsf{fma}\left(wj \cdot \left(1 - wj\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{wj \cdot e^{wj}}\\ \end{array} \]

FPCore

(FPCore (wj x)
  :precision binary64
  (if (<= wj -0.102)
  (- 0.0 (* 0.0 (- 1.0 (* -2.0 x))))
  (if (<= wj 400.0)
    (fma (* wj (- 1.0 wj)) wj x)
    (/ x (* wj (exp wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.102) {
		tmp = 0.0 - (0.0 * (1.0 - (-2.0 * x)));
	} else if (wj <= 400.0) {
		tmp = fma((wj * (1.0 - wj)), wj, x);
	} else {
		tmp = x / (wj * exp(wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.102)
		tmp = Float64(0.0 - Float64(0.0 * Float64(1.0 - Float64(-2.0 * x))));
	elseif (wj <= 400.0)
		tmp = fma(Float64(wj * Float64(1.0 - wj)), wj, x);
	else
		tmp = Float64(x / Float64(wj * exp(wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.102], N[(0.0 - N[(0.0 * N[(1.0 - N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 400.0], N[(N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(x / N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -0.10199999999999999

   1. Initial program 41.1%

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

   2. Taylor expanded in wj around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 62.6%

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

   4. Applied rewrites 62.6%

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

   5. Taylor expanded in wj around 0

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

      1. lower-*.f64 39.0%

         \[\leadsto wj - -1 \cdot x \]

   7. Applied rewrites 39.0%

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

   8. Taylor expanded in wj around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 25.3%

         \[\leadsto wj - wj \cdot \left(1 - -2 \cdot x\right) \]

   10. Applied rewrites 25.3%

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

   11. Taylor expanded in undef-var around zero

       \[\leadsto \color{blue}{0} - wj \cdot \left(1 - -2 \cdot x\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 3.1%

       \[\leadsto \color{blue}{0} - wj \cdot \left(1 - -2 \cdot x\right) \]

   14. Taylor expanded in undef-var around zero

       \[\leadsto 0 - 0 \cdot \left(1 - \color{blue}{-2} \cdot x\right) \]
   15. Step-by-step derivation

   16. Applied rewrites 49.8%

       \[\leadsto 0 - 0 \cdot \left(1 - \color{blue}{-2} \cdot x\right) \]

   if -0.10199999999999999 < wj < 400

   1. Initial program 41.1%

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

   2. Taylor expanded in wj around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 51.1%

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

   6. Applied rewrites 51.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2.5, 1 - \mathsf{fma}\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(x, 5, 0.6666666666666666 \cdot x\right)\right), wj, wj\right)\right), wj, -2 \cdot x\right), \color{blue}{wj}, x\right) \]

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 50.7%

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

   9. Applied rewrites 50.7%

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

   if 400 < wj

   1. Initial program 41.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 69.4%

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in wj around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{x}{e^{wj} + wj \cdot \left(1 + \color{blue}{wj \cdot \left(1 + wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{x}{e^{wj} + wj \cdot \left(1 + wj \cdot \color{blue}{\left(1 + wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{x}{e^{wj} + wj \cdot \left(1 + wj \cdot \left(1 + \color{blue}{wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)}\right)\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{x}{e^{wj} + wj \cdot \left(1 + wj \cdot \left(1 + wj \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot wj\right)}\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{x}{e^{wj} + wj \cdot \left(1 + wj \cdot \left(1 + wj \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{6} \cdot wj}\right)\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{x}{e^{wj} + wj \cdot \left(1 + wj \cdot \left(1 + wj \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \color{blue}{wj}\right)\right)\right)} \]

      7. lower-*.f64 88.4%

         \[\leadsto \frac{x}{e^{wj} + wj \cdot \left(1 + wj \cdot \left(1 + wj \cdot \left(0.5 + 0.16666666666666666 \cdot wj\right)\right)\right)} \]

   7. Applied rewrites 88.4%

      \[\leadsto \frac{x}{e^{wj} + wj \cdot \color{blue}{\left(1 + wj \cdot \left(1 + wj \cdot \left(0.5 + 0.16666666666666666 \cdot wj\right)\right)\right)}} \]

   8. Taylor expanded in wj around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 27.0%

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

   10. Applied rewrites 27.0%

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

Alternative 5: 73.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_0 := wj - \left(1 + \left(x + x\right)\right) \cdot wj\\ \mathbf{if}\;wj \leq -0.49:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;wj \leq 95000:\\ \;\;\;\;\mathsf{fma}\left(wj \cdot \left(1 - wj\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (wj x)
  :precision binary64
  (let* ((t_0 (- wj (* (+ 1.0 (+ x x)) wj))))
  (if (<= wj -0.49)
    t_0
    (if (<= wj 95000.0) (fma (* wj (- 1.0 wj)) wj x) t_0))))

C

double code(double wj, double x) {
	double t_0 = wj - ((1.0 + (x + x)) * wj);
	double tmp;
	if (wj <= -0.49) {
		tmp = t_0;
	} else if (wj <= 95000.0) {
		tmp = fma((wj * (1.0 - wj)), wj, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj - Float64(Float64(1.0 + Float64(x + x)) * wj))
	tmp = 0.0
	if (wj <= -0.49)
		tmp = t_0;
	elseif (wj <= 95000.0)
		tmp = fma(Float64(wj * Float64(1.0 - wj)), wj, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj - N[(N[(1.0 + N[(x + x), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -0.49], t$95$0, If[LessEqual[wj, 95000.0], N[(N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if wj < -0.48999999999999999 or 95000 < wj

   1. Initial program 41.1%

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

   2. Taylor expanded in wj around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 62.6%

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

   4. Applied rewrites 62.6%

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

   5. Taylor expanded in wj around 0

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

      1. lower-*.f64 39.0%

         \[\leadsto wj - -1 \cdot x \]

   7. Applied rewrites 39.0%

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

   8. Taylor expanded in wj around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 25.3%

         \[\leadsto wj - wj \cdot \left(1 - -2 \cdot x\right) \]

   10. Applied rewrites 25.3%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

          \[\leadsto wj - \left(1 - -2 \cdot x\right) \cdot wj \]

       3. lower-*.f64 25.3%

          \[\leadsto wj - \left(1 - -2 \cdot x\right) \cdot wj \]

       4. lift--.f64 N/A

          \[\leadsto wj - \left(1 - -2 \cdot x\right) \cdot wj \]

       5. lift-*.f64 N/A

          \[\leadsto wj - \left(1 - -2 \cdot x\right) \cdot wj \]

       6. fp-cancel-sub-sign-inv N/A

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

       7. metadata-eval N/A

          \[\leadsto wj - \left(1 + 2 \cdot x\right) \cdot wj \]

       8. lower-+.f64 N/A

          \[\leadsto wj - \left(1 + 2 \cdot x\right) \cdot wj \]

       9. count-2-rev N/A

          \[\leadsto wj - \left(1 + \left(x + x\right)\right) \cdot wj \]

       10. lower-+.f64 25.3%

           \[\leadsto wj - \left(1 + \left(x + x\right)\right) \cdot wj \]

   12. Applied rewrites 25.3%

       \[\leadsto wj - \left(1 + \left(x + x\right)\right) \cdot wj \]

   if -0.48999999999999999 < wj < 95000

   1. Initial program 41.1%

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

   2. Taylor expanded in wj around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 51.1%

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

   6. Applied rewrites 51.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2.5, 1 - \mathsf{fma}\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(x, 5, 0.6666666666666666 \cdot x\right)\right), wj, wj\right)\right), wj, -2 \cdot x\right), \color{blue}{wj}, x\right) \]

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 50.7%

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

   9. Applied rewrites 50.7%

      \[\leadsto \mathsf{fma}\left(wj \cdot \left(1 - wj\right), wj, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 63.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := wj - \frac{t\_0 - x}{e^{wj} + t\_0}\\ t_2 := \frac{x}{1 + 2 \cdot wj}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+76}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(wj \cdot \left(1 - wj\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (wj x)
  :precision binary64
  (let* ((t_0 (* wj (exp wj)))
       (t_1 (- wj (/ (- t_0 x) (+ (exp wj) t_0))))
       (t_2 (/ x (+ 1.0 (* 2.0 wj)))))
  (if (<= t_1 -2e+76)
    t_2
    (if (<= t_1 5e-8) (fma (* wj (- 1.0 wj)) wj x) t_2))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double t_1 = wj - ((t_0 - x) / (exp(wj) + t_0));
	double t_2 = x / (1.0 + (2.0 * wj));
	double tmp;
	if (t_1 <= -2e+76) {
		tmp = t_2;
	} else if (t_1 <= 5e-8) {
		tmp = fma((wj * (1.0 - wj)), wj, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	t_1 = Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0)))
	t_2 = Float64(x / Float64(1.0 + Float64(2.0 * wj)))
	tmp = 0.0
	if (t_1 <= -2e+76)
		tmp = t_2;
	elseif (t_1 <= 5e-8)
		tmp = fma(Float64(wj * Float64(1.0 - wj)), wj, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(1.0 + N[(2.0 * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+76], t$95$2, If[LessEqual[t$95$1, 5e-8], N[(N[(wj * N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] * wj + x), $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))))) < -2.0000000000000001e76 or 4.9999999999999998e-8 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 41.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 69.4%

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in wj around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 58.1%

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

   7. Applied rewrites 58.1%

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

   if -2.0000000000000001e76 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 4.9999999999999998e-8

   1. Initial program 41.1%

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

   2. Taylor expanded in wj around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 51.1%

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

   6. Applied rewrites 51.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2.5, 1 - \mathsf{fma}\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(x, 5, 0.6666666666666666 \cdot x\right)\right), wj, wj\right)\right), wj, -2 \cdot x\right), \color{blue}{wj}, x\right) \]

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 50.7%

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

   9. Applied rewrites 50.7%

      \[\leadsto \mathsf{fma}\left(wj \cdot \left(1 - wj\right), wj, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 58.1% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (wj x)
  :precision binary64
  (/ x (+ 1.0 (* 2.0 wj))))

C

double code(double wj, double x) {
	return x / (1.0 + (2.0 * wj));
}

Fortran

real(8) function code(wj, x)
use fmin_fmax_functions
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x / (1.0d0 + (2.0d0 * wj))
end function

Java

public static double code(double wj, double x) {
	return x / (1.0 + (2.0 * wj));
}

Python

def code(wj, x):
	return x / (1.0 + (2.0 * wj))

Julia

function code(wj, x)
	return Float64(x / Float64(1.0 + Float64(2.0 * wj)))
end

MATLAB

function tmp = code(wj, x)
	tmp = x / (1.0 + (2.0 * wj));
end

Wolfram

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

Derivation

1. Initial program 41.1%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-exp.f64 69.4%

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

4. Applied rewrites 69.4%

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

5. Taylor expanded in wj around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 58.1%

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

7. Applied rewrites 58.1%

   \[\leadsto \frac{x}{1 + \color{blue}{2 \cdot wj}} \]
8. Add Preprocessing

Alternative 8: 45.9% accurate, 51.6× speedup

Math

\[x \]

FPCore

(FPCore (wj x)
  :precision binary64
  x)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

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

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 41.1%

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

2. Taylor expanded in wj around 0

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 45.9%

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

Developer Target 1: 42.1% accurate, 1.2× speedup

Math

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

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)
use fmin_fmax_functions
    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 2025313 -o setup:search
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

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

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