Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.6% → 97.5%

Time: 8.9s

Alternatives: 10

Speedup: 27.6×

• could not determine a ground truth

  1. wj = 1.6378790483378747e+206
  2. x = 6.160503439421526e-173

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: 78.6% 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: 97.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - wj\right) \cdot wj, wj, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), wj, -2\right), x \cdot wj, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.6666666666666666, 1.5 \cdot wj, -\frac{wj}{1 + wj}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 7.2e-6)
   (fma
    (* (- 1.0 wj) wj)
    wj
    (fma (fma (fma -2.6666666666666665 wj 2.5) wj -2.0) (* x wj) x))
   (fma 0.6666666666666666 (* 1.5 wj) (- (/ wj (+ 1.0 wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 7.2e-6) {
		tmp = fma(((1.0 - wj) * wj), wj, fma(fma(fma(-2.6666666666666665, wj, 2.5), wj, -2.0), (x * wj), x));
	} else {
		tmp = fma(0.6666666666666666, (1.5 * wj), -(wj / (1.0 + wj)));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 7.2e-6)
		tmp = fma(Float64(Float64(1.0 - wj) * wj), wj, fma(fma(fma(-2.6666666666666665, wj, 2.5), wj, -2.0), Float64(x * wj), x));
	else
		tmp = fma(0.6666666666666666, Float64(1.5 * wj), Float64(-Float64(wj / Float64(1.0 + wj))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 7.2e-6], N[(N[(N[(1.0 - wj), $MachinePrecision] * wj), $MachinePrecision] * wj + N[(N[(N[(-2.6666666666666665 * wj + 2.5), $MachinePrecision] * wj + -2.0), $MachinePrecision] * N[(x * wj), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(0.6666666666666666 * N[(1.5 * wj), $MachinePrecision] + (-N[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 7.19999999999999967e-6

   1. Initial program 79.4%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.8%

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

   if 7.19999999999999967e-6 < wj

   1. Initial program 50.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. lower-+.f64 72.5

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

   5. Applied rewrites 72.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

         \[\leadsto wj \cdot \color{blue}{\left(\frac{2}{3} \cdot \frac{3}{2}\right)} + \left(\mathsf{neg}\left(\frac{wj}{1 + wj}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(wj \cdot \frac{2}{3}\right) \cdot \frac{3}{2}} + \left(\mathsf{neg}\left(\frac{wj}{1 + wj}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} \cdot wj\right)} \cdot \frac{3}{2} + \left(\mathsf{neg}\left(\frac{wj}{1 + wj}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{2}{3} \cdot \left(wj \cdot \frac{3}{2}\right)} + \left(\mathsf{neg}\left(\frac{wj}{1 + wj}\right)\right) \]

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 72.8

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

   7. Applied rewrites 72.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.6666666666666666, wj \cdot 1.5, -\frac{wj}{wj + 1}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - wj\right) \cdot wj, wj, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), wj, -2\right), x \cdot wj, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.6666666666666666, 1.5 \cdot wj, -\frac{wj}{1 + wj}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.2% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(1 - wj, wj, -2 \cdot x\right) \cdot wj + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.6666666666666666, 1.5 \cdot wj, -\frac{wj}{1 + wj}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 7.2e-6)
   (+ (* (fma (- 1.0 wj) wj (* -2.0 x)) wj) x)
   (fma 0.6666666666666666 (* 1.5 wj) (- (/ wj (+ 1.0 wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 7.2e-6) {
		tmp = (fma((1.0 - wj), wj, (-2.0 * x)) * wj) + x;
	} else {
		tmp = fma(0.6666666666666666, (1.5 * wj), -(wj / (1.0 + wj)));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 7.2e-6)
		tmp = Float64(Float64(fma(Float64(1.0 - wj), wj, Float64(-2.0 * x)) * wj) + x);
	else
		tmp = fma(0.6666666666666666, Float64(1.5 * wj), Float64(-Float64(wj / Float64(1.0 + wj))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 7.2e-6], N[(N[(N[(N[(1.0 - wj), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision] + x), $MachinePrecision], N[(0.6666666666666666 * N[(1.5 * wj), $MachinePrecision] + (-N[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 7.19999999999999967e-6

   1. Initial program 79.4%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.7%

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

   10. Applied rewrites 98.7%

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

   if 7.19999999999999967e-6 < wj

   1. Initial program 50.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. lower-+.f64 72.5

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

   5. Applied rewrites 72.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

         \[\leadsto wj \cdot \color{blue}{\left(\frac{2}{3} \cdot \frac{3}{2}\right)} + \left(\mathsf{neg}\left(\frac{wj}{1 + wj}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left(wj \cdot \frac{2}{3}\right) \cdot \frac{3}{2}} + \left(\mathsf{neg}\left(\frac{wj}{1 + wj}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} \cdot wj\right)} \cdot \frac{3}{2} + \left(\mathsf{neg}\left(\frac{wj}{1 + wj}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{2}{3} \cdot \left(wj \cdot \frac{3}{2}\right)} + \left(\mathsf{neg}\left(\frac{wj}{1 + wj}\right)\right) \]

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 72.8

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

   7. Applied rewrites 72.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.6666666666666666, wj \cdot 1.5, -\frac{wj}{wj + 1}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(1 - wj, wj, -2 \cdot x\right) \cdot wj + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.6666666666666666, 1.5 \cdot wj, -\frac{wj}{1 + wj}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.2% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(1 - wj, wj, -2 \cdot x\right) \cdot wj + x\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 7.2e-6)
   (+ (* (fma (- 1.0 wj) wj (* -2.0 x)) wj) x)
   (- wj (/ wj (+ 1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 7.2e-6) {
		tmp = (fma((1.0 - wj), wj, (-2.0 * x)) * wj) + x;
	} else {
		tmp = wj - (wj / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 7.2e-6)
		tmp = Float64(Float64(fma(Float64(1.0 - wj), wj, Float64(-2.0 * x)) * wj) + x);
	else
		tmp = Float64(wj - Float64(wj / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 7.2e-6], N[(N[(N[(N[(1.0 - wj), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision] + x), $MachinePrecision], N[(wj - N[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 7.19999999999999967e-6

   1. Initial program 79.4%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.7%

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

   10. Applied rewrites 98.7%

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

   if 7.19999999999999967e-6 < wj

   1. Initial program 50.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. lower-+.f64 72.5

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

   5. Applied rewrites 72.5%

      \[\leadsto wj - \color{blue}{\frac{wj}{1 + wj}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.2% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1 - wj, wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 7.2e-6)
   (fma (fma (- 1.0 wj) wj (* -2.0 x)) wj x)
   (- wj (/ wj (+ 1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 7.2e-6) {
		tmp = fma(fma((1.0 - wj), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - (wj / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 7.2e-6)
		tmp = fma(fma(Float64(1.0 - wj), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - Float64(wj / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 7.2e-6], N[(N[(N[(1.0 - wj), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 7.19999999999999967e-6

   1. Initial program 79.4%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.7%

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

   if 7.19999999999999967e-6 < wj

   1. Initial program 50.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. lower-+.f64 72.5

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

   5. Applied rewrites 72.5%

      \[\leadsto wj - \color{blue}{\frac{wj}{1 + wj}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.8% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - wj\right) \cdot wj, wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 7.2e-6) (fma (* (- 1.0 wj) wj) wj x) (- wj (/ wj (+ 1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 7.2e-6) {
		tmp = fma(((1.0 - wj) * wj), wj, x);
	} else {
		tmp = wj - (wj / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 7.2e-6)
		tmp = fma(Float64(Float64(1.0 - wj) * wj), wj, x);
	else
		tmp = Float64(wj - Float64(wj / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 7.2e-6], N[(N[(N[(1.0 - wj), $MachinePrecision] * wj), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 7.19999999999999967e-6

   1. Initial program 79.4%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 98.8%

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

   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 98.0%

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

   if 7.19999999999999967e-6 < wj

   1. Initial program 50.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. lower-+.f64 72.5

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

   5. Applied rewrites 72.5%

      \[\leadsto wj - \color{blue}{\frac{wj}{1 + wj}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 95.5% 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 78.4%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.0%

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

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 95.4%

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

Alternative 7: 85.0% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.0%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 85.0%

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

Alternative 8: 84.5% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-exp.f64 86.0

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

5. Applied rewrites 86.0%

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

6. Taylor expanded in wj around 0

   \[\leadsto \frac{x}{1} \]
7. Step-by-step derivation

8. Applied rewrites 84.5%

   \[\leadsto \frac{x}{1} \]
9. Add Preprocessing

Alternative 9: 73.3% 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 78.4%

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

3. Taylor expanded in wj around 0

   \[\leadsto wj - \color{blue}{-1 \cdot x} \]
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 74.7

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

5. Applied rewrites 74.7%

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

Alternative 10: 4.1% 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 78.4%

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

3. Taylor expanded in wj around inf

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

5. Applied rewrites 4.3%

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

Developer Target 1: 79.6% 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 2024264 
(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))))))