Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.9% → 97.9%

Time: 10.9s

Alternatives: 9

Speedup: 47.3×

• could not determine a ground truth

  1. wj = 7.651187167493154e+121
  2. x = 170488.4559264945

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: 97.9% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -5e-6)
   (fma (/ -1.0 (exp wj)) (/ (fma wj (exp wj) (- x)) (+ 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 <= -5e-6) {
		tmp = fma((-1.0 / exp(wj)), (fma(wj, exp(wj), -x) / (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 <= -5e-6)
		tmp = fma(Float64(-1.0 / exp(wj)), Float64(fma(wj, exp(wj), Float64(-x)) / Float64(wj + 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, -5e-6], N[(N[(-1.0 / N[Exp[wj], $MachinePrecision]), $MachinePrecision] * N[(N[(wj * N[Exp[wj], $MachinePrecision] + (-x)), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + wj), $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 < -5.00000000000000041e-6

   1. Initial program 61.2%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. sub-neg N/A

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

      10. +-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. Applied rewrites 94.1%

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

   if -5.00000000000000041e-6 < wj

   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 99.0%

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

Alternative 2: 80.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := wj + \frac{x - t\_0}{e^{wj} + t\_0}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-244}:\\ \;\;\;\;wj + x\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))) (t_1 (+ wj (/ (- x t_0) (+ (exp wj) t_0)))))
   (if (<= t_1 -1e-244) (+ wj x) (if (<= t_1 0.0) (* wj wj) (+ wj x)))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double t_1 = wj + ((x - t_0) / (exp(wj) + t_0));
	double tmp;
	if (t_1 <= -1e-244) {
		tmp = wj + x;
	} else if (t_1 <= 0.0) {
		tmp = wj * wj;
	} else {
		tmp = wj + x;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = wj * exp(wj)
    t_1 = wj + ((x - t_0) / (exp(wj) + t_0))
    if (t_1 <= (-1d-244)) then
        tmp = wj + x
    else if (t_1 <= 0.0d0) then
        tmp = wj * wj
    else
        tmp = wj + x
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double t_1 = wj + ((x - t_0) / (Math.exp(wj) + t_0));
	double tmp;
	if (t_1 <= -1e-244) {
		tmp = wj + x;
	} else if (t_1 <= 0.0) {
		tmp = wj * wj;
	} else {
		tmp = wj + x;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	t_1 = wj + ((x - t_0) / (math.exp(wj) + t_0))
	tmp = 0
	if t_1 <= -1e-244:
		tmp = wj + x
	elif t_1 <= 0.0:
		tmp = wj * wj
	else:
		tmp = wj + x
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	t_1 = Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0)))
	tmp = 0.0
	if (t_1 <= -1e-244)
		tmp = Float64(wj + x);
	elseif (t_1 <= 0.0)
		tmp = Float64(wj * wj);
	else
		tmp = Float64(wj + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	t_1 = wj + ((x - t_0) / (exp(wj) + t_0));
	tmp = 0.0;
	if (t_1 <= -1e-244)
		tmp = wj + x;
	elseif (t_1 <= 0.0)
		tmp = wj * wj;
	else
		tmp = wj + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-244], N[(wj + x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(wj * wj), $MachinePrecision], N[(wj + x), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < -9.9999999999999993e-245 or 0.0 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 95.9%

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

   3. Taylor expanded in wj around 0

      \[\leadsto wj - \color{blue}{-1 \cdot x} \]
   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 90.8

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

   5. Applied rewrites 90.8%

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

   if -9.9999999999999993e-245 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 0.0

   1. Initial program 7.5%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 6.7

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

   5. Applied rewrites 6.7%

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

   6. Taylor expanded in wj around 0

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

      1. unpow2 N/A

         \[\leadsto \color{blue}{wj \cdot wj} \]

      2. lower-*.f64 48.2

         \[\leadsto \color{blue}{wj \cdot wj} \]

   8. Applied rewrites 48.2%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq -1 \cdot 10^{-244}:\\ \;\;\;\;wj + x\\ \mathbf{elif}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% 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 76.2%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 97.2%

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

Alternative 4: 95.7% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.2%

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

3. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

5. Applied rewrites 96.8%

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

6. Final simplification 96.8%

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

Alternative 5: 95.5% accurate, 22.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.2%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 97.2%

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

6. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-neg.f64 96.7

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

8. Applied rewrites 96.7%

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

Alternative 6: 95.0% accurate, 47.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(wj, x)
	return fma(wj, wj, x)
end

Wolfram

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

Derivation

1. Initial program 76.2%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 97.2%

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

6. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-neg.f64 96.7

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

8. Applied rewrites 96.7%

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

9. Taylor expanded in wj around 0

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

    1. +-commutative N/A

       \[\leadsto \color{blue}{{wj}^{2} + x} \]

    2. unpow2 N/A

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

    3. lower-fma.f64 96.4

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

11. Applied rewrites 96.4%

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

Alternative 7: 13.7% 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 76.2%

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

3. Taylor expanded in x around 0

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

   1. distribute-rgt1-in N/A

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

   2. +-commutative N/A

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

   3. times-frac N/A

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

   4. *-inverses N/A

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

   5. associate-*l/ N/A

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

   6. *-rgt-identity N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 6.0

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

5. Applied rewrites 6.0%

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

6. Taylor expanded in wj around 0

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

   1. unpow2 N/A

      \[\leadsto \color{blue}{wj \cdot wj} \]

   2. lower-*.f64 15.2

      \[\leadsto \color{blue}{wj \cdot wj} \]

8. Applied rewrites 15.2%

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

Alternative 8: 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 76.2%

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

3. Taylor expanded in wj around inf

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

5. Applied rewrites 3.5%

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

6. Final simplification 3.5%

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

Alternative 9: 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 76.2%

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

3. Taylor expanded in wj around inf

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

5. Applied rewrites 3.5%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 3.1%

   \[\leadsto \color{blue}{-1} \]
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 2024219 
(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))))))