Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.3% → 98.0%

Time: 10.5s

Alternatives: 10

Speedup: 27.6×

• could not determine a ground truth

  1. wj = 2.2638456792060495e+179
  2. x = 1.362086943778057e-257

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.3% 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: 98.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4.6 \cdot 10^{-6}:\\ \;\;\;\;wj - x \cdot \left(\frac{wj}{\mathsf{fma}\left(x, wj, x\right)} + \frac{e^{-wj}}{-1 - wj}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -4.6e-6)
   (- wj (* x (+ (/ wj (fma x wj x)) (/ (exp (- wj)) (- -1.0 wj)))))
   (fma
    wj
    (fma
     wj
     (- (fma x 2.5 1.0) (fma wj (fma x 0.6666666666666666 (* x 2.0)) wj))
     (* x -2.0))
    x)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -4.6e-6) {
		tmp = wj - (x * ((wj / fma(x, wj, x)) + (exp(-wj) / (-1.0 - wj))));
	} else {
		tmp = fma(wj, fma(wj, (fma(x, 2.5, 1.0) - fma(wj, fma(x, 0.6666666666666666, (x * 2.0)), wj)), (x * -2.0)), x);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -4.6e-6)
		tmp = Float64(wj - Float64(x * Float64(Float64(wj / fma(x, wj, x)) + Float64(exp(Float64(-wj)) / Float64(-1.0 - wj)))));
	else
		tmp = fma(wj, fma(wj, Float64(fma(x, 2.5, 1.0) - fma(wj, fma(x, 0.6666666666666666, Float64(x * 2.0)), wj)), Float64(x * -2.0)), x);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -4.6e-6], N[(wj - N[(x * N[(N[(wj / N[(x * wj + x), $MachinePrecision]), $MachinePrecision] + N[(N[Exp[(-wj)], $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj * N[(wj * N[(N[(x * 2.5 + 1.0), $MachinePrecision] - N[(wj * N[(x * 0.6666666666666666 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -4.6e-6

   1. Initial program 46.9%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

         \[\leadsto wj - x \cdot \left(\color{blue}{\left(0 - \frac{1}{e^{wj} + wj \cdot e^{wj}}\right)} + \frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)}\right) \]

      4. associate-+l- N/A

         \[\leadsto wj - x \cdot \color{blue}{\left(0 - \left(\frac{1}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)}\right)\right)} \]

      5. unsub-neg N/A

         \[\leadsto wj - x \cdot \left(0 - \color{blue}{\left(\frac{1}{e^{wj} + wj \cdot e^{wj}} + \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)}\right)\right)\right)}\right) \]

      6. mul-1-neg N/A

         \[\leadsto wj - x \cdot \left(0 - \left(\frac{1}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)}}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto wj - x \cdot \left(0 - \color{blue}{\left(-1 \cdot \frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)} + \frac{1}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]

   5. Applied rewrites 98.8%

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

   if -4.6e-6 < wj

   1. Initial program 82.8%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 98.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.6 \cdot 10^{-6}:\\ \;\;\;\;wj - x \cdot \left(\frac{wj}{\mathsf{fma}\left(x, wj, x\right)} + \frac{e^{-wj}}{-1 - wj}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.0%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.6%

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

Alternative 3: 96.5% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.0%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.4%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 96.5%

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

Alternative 4: 96.4% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.0%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.4%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 96.5%

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

12. Taylor expanded in x around 0

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

14. Applied rewrites 96.5%

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

Alternative 5: 96.2% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.0%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.4%

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

Alternative 6: 95.7% accurate, 22.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.0%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.4%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 96.2%

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

Alternative 7: 84.9% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.0%

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

3. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 87.0

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

5. Applied rewrites 87.0%

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

7. Applied rewrites 87.0%

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

8. Final simplification 87.0%

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

Alternative 8: 74.8% accurate, 33.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.0%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. div-inv N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 83.2%

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

5. Taylor expanded in wj around 0

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

   1. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(wj, e^{wj}, \mathsf{neg}\left(x\right)\right), \color{blue}{wj \cdot \left(2 + wj \cdot \left(\frac{8}{3} \cdot wj - \frac{5}{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}, wj\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(wj, e^{wj}, \mathsf{neg}\left(x\right)\right), wj \cdot \left(2 + wj \cdot \left(\frac{8}{3} \cdot wj - \frac{5}{2}\right)\right) + \color{blue}{-1}, wj\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(wj, e^{wj}, \mathsf{neg}\left(x\right)\right), wj \cdot \color{blue}{\left(wj \cdot \left(\frac{8}{3} \cdot wj - \frac{5}{2}\right) + 2\right)} + -1, wj\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(wj, e^{wj}, \mathsf{neg}\left(x\right)\right), wj \cdot \left(wj \cdot \color{blue}{\left(\frac{8}{3} \cdot wj + \left(\mathsf{neg}\left(\frac{5}{2}\right)\right)\right)} + 2\right) + -1, wj\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(wj, e^{wj}, \mathsf{neg}\left(x\right)\right), wj \cdot \left(wj \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-8}{3}\right)\right)} \cdot wj + \left(\mathsf{neg}\left(\frac{5}{2}\right)\right)\right) + 2\right) + -1, wj\right) \]

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

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(wj, e^{wj}, \mathsf{neg}\left(x\right)\right), wj \cdot \left(wj \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-8}{3} \cdot wj\right)\right)} + \left(\mathsf{neg}\left(\frac{5}{2}\right)\right)\right) + 2\right) + -1, wj\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(wj, e^{wj}, \mathsf{neg}\left(x\right)\right), wj \cdot \left(wj \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{-8}{3} \cdot wj + \frac{5}{2}\right)\right)\right)} + 2\right) + -1, wj\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(wj, e^{wj}, \mathsf{neg}\left(x\right)\right), wj \cdot \left(wj \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)}\right)\right) + 2\right) + -1, wj\right) \]

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

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(wj, e^{wj}, \mathsf{neg}\left(x\right)\right), wj \cdot \left(\color{blue}{\left(\mathsf{neg}\left(wj \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)\right)} + 2\right) + -1, wj\right) \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(wj, e^{wj}, \mathsf{neg}\left(x\right)\right), wj \cdot \left(\color{blue}{-1 \cdot \left(wj \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)} + 2\right) + -1, wj\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(wj, e^{wj}, \mathsf{neg}\left(x\right)\right), wj \cdot \color{blue}{\left(2 + -1 \cdot \left(wj \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)\right)} + -1, wj\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(wj, e^{wj}, \mathsf{neg}\left(x\right)\right), \color{blue}{\mathsf{fma}\left(wj, 2 + -1 \cdot \left(wj \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right), -1\right)}, wj\right) \]

7. Applied rewrites 80.2%

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

8. Taylor expanded in wj around 0

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

   1. lower--.f64 79.4

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

10. Applied rewrites 79.4%

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

11. Taylor expanded in wj around 0

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

13. Applied rewrites 78.7%

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

Alternative 9: 72.9% 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 82.0%

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

3. Taylor expanded in wj around 0

   \[\leadsto wj - \color{blue}{-1 \cdot x} \]
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 77.6

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

5. Applied rewrites 77.6%

   \[\leadsto wj - \color{blue}{\left(-x\right)} \]
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 82.0%

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

3. Taylor expanded in wj around inf

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

5. Applied rewrites 3.7%

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

Developer Target 1: 79.2% 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 2024234 
(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))))))