Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.4% → 96.8%

Time: 9.2s

Alternatives: 10

Speedup: 27.6×

• could not determine a ground truth

  1. wj = -3.5884505496798894e+211
  2. x = -1.2617793997712698e-12

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: 77.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), wj, -2\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(2 \cdot wj, t\_0, 1\right) - t\_0 \cdot wj, x, \left(wj \cdot wj\right) \cdot \left(1 - wj\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (fma (fma -2.6666666666666665 wj 2.5) wj -2.0)))
   (fma (- (fma (* 2.0 wj) t_0 1.0) (* t_0 wj)) x (* (* wj wj) (- 1.0 wj)))))

C

double code(double wj, double x) {
	double t_0 = fma(fma(-2.6666666666666665, wj, 2.5), wj, -2.0);
	return fma((fma((2.0 * wj), t_0, 1.0) - (t_0 * wj)), x, ((wj * wj) * (1.0 - wj)));
}

Julia

function code(wj, x)
	t_0 = fma(fma(-2.6666666666666665, wj, 2.5), wj, -2.0)
	return fma(Float64(fma(Float64(2.0 * wj), t_0, 1.0) - Float64(t_0 * wj)), x, Float64(Float64(wj * wj) * Float64(1.0 - wj)))
end

Wolfram

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

Derivation

1. Initial program 78.2%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.3%

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

7. Applied rewrites 58.5%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 96.7%

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

11. Final simplification 96.7%

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

Alternative 2: 96.8% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(wj, x)
	return fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), wj, -2.0), wj, Float64(Float64(Float64(1.0 - wj) / x) * Float64(wj * wj))), x, 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]), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 78.2%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.4%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 96.6%

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

12. Final simplification 96.6%

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

Alternative 3: 96.5% accurate, 14.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.3%

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

7. Applied rewrites 96.3%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 96.6%

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

Alternative 4: 96.5% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.4%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 96.6%

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

Alternative 5: 81.8% accurate, 16.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 2.5 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot wj, -2, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - wj\right) \cdot wj\right) \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 2.5e-68) (fma (* x wj) -2.0 x) (* (* (- 1.0 wj) wj) wj)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 2.5e-68) {
		tmp = fma((x * wj), -2.0, x);
	} else {
		tmp = ((1.0 - wj) * wj) * wj;
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 2.5e-68)
		tmp = fma(Float64(x * wj), -2.0, x);
	else
		tmp = Float64(Float64(Float64(1.0 - wj) * wj) * wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 2.5e-68], N[(N[(x * wj), $MachinePrecision] * -2.0 + x), $MachinePrecision], N[(N[(N[(1.0 - wj), $MachinePrecision] * wj), $MachinePrecision] * wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.49999999999999986e-68

   1. Initial program 82.4%

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

   3. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 91.6

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

   5. Applied rewrites 91.6%

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

   if 2.49999999999999986e-68 < wj

   1. Initial program 40.9%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 80.7%

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

   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 52.6%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 2.5 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot wj, -2, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - wj\right) \cdot wj\right) \cdot wj\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.1% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.2%

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

10. Applied rewrites 96.2%

    \[\leadsto \left(\left(1 - wj\right) \cdot wj\right) \cdot wj + \color{blue}{x} \]
11. Add Preprocessing

Alternative 7: 96.1% 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 78.2%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.2%

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

Alternative 8: 84.6% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

3. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 85.3

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

5. Applied rewrites 85.3%

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

6. Final simplification 85.3%

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

Alternative 9: 72.7% accurate, 55.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

3. Taylor expanded in wj around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 73.1

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

5. Applied rewrites 73.1%

   \[\leadsto wj - \color{blue}{\left(-x\right)} \]
6. Add Preprocessing

Alternative 10: 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 78.2%

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

3. Taylor expanded in wj around inf

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

5. Applied rewrites 4.2%

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

Developer Target 1: 78.4% 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 2024268 
(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))))))