Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.9% → 98.1%

Time: 8.9s

Alternatives: 15

Speedup: 55.2×

• could not determine a ground truth

  1. wj = -5.876814699094704e+90
  2. x = -6.513113221597432e+90

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.9% 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: 98.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.1 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{1 + wj}, e^{-wj}, wj - \frac{wj}{1 + wj}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, wj, -2\right), wj, 1\right) + \frac{\left(wj \cdot wj\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - wj, wj, -1\right), wj, 1\right)}{x}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -2.1e-5)
   (fma (/ x (+ 1.0 wj)) (exp (- wj)) (- wj (/ wj (+ 1.0 wj))))
   (*
    (+
     (fma (fma 2.5 wj -2.0) wj 1.0)
     (/ (* (* wj wj) (fma (fma (- 1.0 wj) wj -1.0) wj 1.0)) x))
    x)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -2.1e-5) {
		tmp = fma((x / (1.0 + wj)), exp(-wj), (wj - (wj / (1.0 + wj))));
	} else {
		tmp = (fma(fma(2.5, wj, -2.0), wj, 1.0) + (((wj * wj) * fma(fma((1.0 - wj), wj, -1.0), wj, 1.0)) / x)) * x;
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -2.1e-5)
		tmp = fma(Float64(x / Float64(1.0 + wj)), exp(Float64(-wj)), Float64(wj - Float64(wj / Float64(1.0 + wj))));
	else
		tmp = Float64(Float64(fma(fma(2.5, wj, -2.0), wj, 1.0) + Float64(Float64(Float64(wj * wj) * fma(fma(Float64(1.0 - wj), wj, -1.0), wj, 1.0)) / x)) * x);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -2.1e-5], N[(N[(x / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision] * N[Exp[(-wj)], $MachinePrecision] + N[(wj - N[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.5 * wj + -2.0), $MachinePrecision] * wj + 1.0), $MachinePrecision] + N[(N[(N[(wj * wj), $MachinePrecision] * N[(N[(N[(1.0 - wj), $MachinePrecision] * wj + -1.0), $MachinePrecision] * wj + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -2.09999999999999988e-5

   1. Initial program 67.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 97.9%

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

   if -2.09999999999999988e-5 < wj

   1. Initial program 77.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in wj around 0

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in wj around 0

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

   11. Applied rewrites 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.1 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{1 + wj}, e^{-wj}, wj - \frac{wj}{1 + wj}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, wj, -2\right), wj, 1\right) + \frac{\left(wj \cdot wj\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - wj, wj, -1\right), wj, 1\right)}{x}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 89.0%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 98.9%

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

10. Applied rewrites 98.9%

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

11. Final simplification 98.9%

    \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(1 - wj, wj, -1\right), wj, 1\right) \cdot wj\right) \cdot wj, 1 + wj, \frac{x}{e^{wj}}\right)}{\left(1 + wj\right) \cdot x} \cdot x \]
12. Add Preprocessing

Alternative 3: 98.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 89.0%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 98.9%

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

9. Final simplification 98.9%

   \[\leadsto \left(\frac{e^{-wj}}{1 + wj} + \frac{\left(wj \cdot wj\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - wj, wj, -1\right), wj, 1\right)}{x}\right) \cdot x \]
10. Add Preprocessing

Alternative 4: 98.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 89.0%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 98.8%

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

Alternative 5: 97.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double wj, double x) {
	return (((wj / x) * wj) + (Math.exp(-wj) / (1.0 + wj))) * x;
}

Python

def code(wj, x):
	return (((wj / x) * wj) + (math.exp(-wj) / (1.0 + wj))) * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 89.0%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 98.2%

   \[\leadsto \left(\frac{wj}{x} \cdot wj + \frac{e^{-wj}}{1 + wj}\right) \cdot x \]
9. Add Preprocessing

Alternative 6: 97.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.115:\\ \;\;\;\;wj - \frac{\frac{x}{-1 - wj}}{e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, wj, -2\right), wj, 1\right) + \frac{\left(wj \cdot wj\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - wj, wj, -1\right), wj, 1\right)}{x}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.115)
   (- wj (/ (/ x (- -1.0 wj)) (exp wj)))
   (*
    (+
     (fma (fma 2.5 wj -2.0) wj 1.0)
     (/ (* (* wj wj) (fma (fma (- 1.0 wj) wj -1.0) wj 1.0)) x))
    x)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.115) {
		tmp = wj - ((x / (-1.0 - wj)) / exp(wj));
	} else {
		tmp = (fma(fma(2.5, wj, -2.0), wj, 1.0) + (((wj * wj) * fma(fma((1.0 - wj), wj, -1.0), wj, 1.0)) / x)) * x;
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.115)
		tmp = Float64(wj - Float64(Float64(x / Float64(-1.0 - wj)) / exp(wj)));
	else
		tmp = Float64(Float64(fma(fma(2.5, wj, -2.0), wj, 1.0) + Float64(Float64(Float64(wj * wj) * fma(fma(Float64(1.0 - wj), wj, -1.0), wj, 1.0)) / x)) * x);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.115], N[(wj - N[(N[(x / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision] / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.5 * wj + -2.0), $MachinePrecision] * wj + 1.0), $MachinePrecision] + N[(N[(N[(wj * wj), $MachinePrecision] * N[(N[(N[(1.0 - wj), $MachinePrecision] * wj + -1.0), $MachinePrecision] * wj + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -0.115000000000000005

   1. Initial program 40.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. associate-/r* N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-exp.f64 85.0

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

   5. Applied rewrites 85.0%

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

   if -0.115000000000000005 < wj

   1. Initial program 77.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in wj around 0

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in wj around 0

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

   11. Applied rewrites 98.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.115:\\ \;\;\;\;wj - \frac{\frac{x}{-1 - wj}}{e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, wj, -2\right), wj, 1\right) + \frac{\left(wj \cdot wj\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - wj, wj, -1\right), wj, 1\right)}{x}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.4% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 89.0%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 98.9%

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

9. Taylor expanded in wj around 0

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

11. Applied rewrites 97.0%

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

12. Final simplification 97.0%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, wj, -2\right), wj, 1\right) + \frac{\left(wj \cdot wj\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(1 - wj, wj, -1\right), wj, 1\right)}{x}\right) \cdot x \]
13. Add Preprocessing

Alternative 8: 96.1% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 96.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 96.9%

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

Alternative 9: 96.1% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 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 96.6%

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

Alternative 10: 95.4% 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 76.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.6%

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

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 96.4%

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

Alternative 11: 84.3% 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 76.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.6%

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

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. lower-*.f64 N/A

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

   4. lower-fma.f64 85.0

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

8. Applied rewrites 85.0%

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

Alternative 12: 83.7% 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 76.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 96.9%

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

9. Taylor expanded in wj around 0

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

11. Applied rewrites 84.8%

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

Alternative 13: 14.1% accurate, 55.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 96.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 14.4%

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

9. Taylor expanded in wj around 0

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

11. Applied rewrites 14.1%

    \[\leadsto wj \cdot wj \]
12. Add Preprocessing

Alternative 14: 4.2% accurate, 82.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.6%

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

3. Taylor expanded in wj around inf

   \[\leadsto \color{blue}{wj \cdot \left(1 - \frac{1}{wj}\right)} \]
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. +-commutative N/A

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

   3. distribute-lft-in N/A

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

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

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

   5. rgt-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. *-rgt-identity N/A

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

   8. lower-+.f64 3.8

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

5. Applied rewrites 3.8%

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

6. Final simplification 3.8%

   \[\leadsto wj - 1 \]
7. Add Preprocessing

Alternative 15: 3.3% 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 76.6%

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

3. Taylor expanded in wj around inf

   \[\leadsto \color{blue}{wj \cdot \left(1 - \frac{1}{wj}\right)} \]
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. +-commutative N/A

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

   3. distribute-lft-in N/A

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

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

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

   5. rgt-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. *-rgt-identity N/A

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

   8. lower-+.f64 3.8

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

5. Applied rewrites 3.8%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 3.5%

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

Developer Target 1: 78.9% 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 2024296 
(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))))))