Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.1% → 98.3%

Time: 9.0s

Alternatives: 11

Speedup: 55.2×

• could not determine a ground truth

  1. wj = 7.773185709655008e+39
  2. x = -3.194700886320209e-292

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: 78.1% 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: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot wj\\ \mathbf{if}\;wj - \frac{t\_0 - x}{t\_0 + e^{wj}} \leq 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, x, \mathsf{fma}\left(\mathsf{fma}\left(-wj, \mathsf{fma}\left(x, 2.6666666666666665, 1\right), 2.5 \cdot x\right), wj, wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{e^{-wj}}{wj - -1} + \frac{wj}{x}\right) - \frac{wj}{\mathsf{fma}\left(x, wj, x\right)}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) wj)))
   (if (<= (- wj (/ (- t_0 x) (+ t_0 (exp wj)))) 1e-17)
     (fma
      (fma
       -2.0
       x
       (fma (fma (- wj) (fma x 2.6666666666666665 1.0) (* 2.5 x)) wj wj))
      wj
      x)
     (* (- (+ (/ (exp (- wj)) (- wj -1.0)) (/ wj x)) (/ wj (fma x wj x))) x))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * wj;
	double tmp;
	if ((wj - ((t_0 - x) / (t_0 + exp(wj)))) <= 1e-17) {
		tmp = fma(fma(-2.0, x, fma(fma(-wj, fma(x, 2.6666666666666665, 1.0), (2.5 * x)), wj, wj)), wj, x);
	} else {
		tmp = (((exp(-wj) / (wj - -1.0)) + (wj / x)) - (wj / fma(x, wj, x))) * x;
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(exp(wj) * wj)
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(t_0 + exp(wj)))) <= 1e-17)
		tmp = fma(fma(-2.0, x, fma(fma(Float64(-wj), fma(x, 2.6666666666666665, 1.0), Float64(2.5 * x)), wj, wj)), wj, x);
	else
		tmp = Float64(Float64(Float64(Float64(exp(Float64(-wj)) / Float64(wj - -1.0)) + Float64(wj / x)) - Float64(wj / fma(x, wj, x))) * x);
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[Exp[wj], $MachinePrecision] * wj), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(t$95$0 + N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-17], N[(N[(-2.0 * x + N[(N[((-wj) * N[(x * 2.6666666666666665 + 1.0), $MachinePrecision] + N[(2.5 * x), $MachinePrecision]), $MachinePrecision] * wj + wj), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(N[(N[(N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision] + N[(wj / x), $MachinePrecision]), $MachinePrecision] - N[(wj / N[(x * wj + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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))))) < 1.00000000000000007e-17

   1. Initial program 72.1%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.5%

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

   6. Taylor expanded in wj around 0

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in wj around 0

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

   11. Applied rewrites 99.6%

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

   if 1.00000000000000007e-17 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 94.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in wj around 0

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

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{1 + wj}, \color{blue}{wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 \cdot x + wj \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot x\right) - \left(-1 \cdot x + \left(\frac{-1}{6} \cdot x + \frac{1}{2} \cdot \left(1 - -1 \cdot x\right)\right)\right)\right)\right) - \frac{-1}{2} \cdot x\right)\right) - -1 \cdot x\right) + -1 \cdot x}, wj\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{1 + wj}, \color{blue}{\left(\left(1 + wj \cdot \left(\left(-1 \cdot x + wj \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot x\right) - \left(-1 \cdot x + \left(\frac{-1}{6} \cdot x + \frac{1}{2} \cdot \left(1 - -1 \cdot x\right)\right)\right)\right)\right) - \frac{-1}{2} \cdot x\right)\right) - -1 \cdot x\right) \cdot wj} + -1 \cdot x, wj\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{1 + wj}, \color{blue}{\mathsf{fma}\left(\left(1 + wj \cdot \left(\left(-1 \cdot x + wj \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot x\right) - \left(-1 \cdot x + \left(\frac{-1}{6} \cdot x + \frac{1}{2} \cdot \left(1 - -1 \cdot x\right)\right)\right)\right)\right) - \frac{-1}{2} \cdot x\right)\right) - -1 \cdot x, wj, -1 \cdot x\right)}, wj\right) \]

   7. Applied rewrites 95.6%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{1 + wj}, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.5, x, 0.5 - \mathsf{fma}\left(x, -1.1666666666666667, \left(1 + x\right) \cdot 0.5\right)\right), wj, -0.5 \cdot x\right), wj, 1 + x\right), wj, -x\right)}, wj\right) \]

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}} \leq 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, x, \mathsf{fma}\left(\mathsf{fma}\left(-wj, \mathsf{fma}\left(x, 2.6666666666666665, 1\right), 2.5 \cdot x\right), wj, wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{e^{-wj}}{wj - -1} + \frac{wj}{x}\right) - \frac{wj}{\mathsf{fma}\left(x, wj, x\right)}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{wj} \cdot wj\\ \mathbf{if}\;wj - \frac{t\_0 - x}{t\_0 + e^{wj}} \leq 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, x, \mathsf{fma}\left(\mathsf{fma}\left(-wj, \mathsf{fma}\left(x, 2.6666666666666665, 1\right), 2.5 \cdot x\right), wj, wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{\frac{wj}{wj - -1}}{x} - \frac{e^{-wj}}{wj - -1}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* (exp wj) wj)))
   (if (<= (- wj (/ (- t_0 x) (+ t_0 (exp wj)))) 1e-17)
     (fma
      (fma
       -2.0
       x
       (fma (fma (- wj) (fma x 2.6666666666666665 1.0) (* 2.5 x)) wj wj))
      wj
      x)
     (- wj (* (- (/ (/ wj (- wj -1.0)) x) (/ (exp (- wj)) (- wj -1.0))) x)))))

C

double code(double wj, double x) {
	double t_0 = exp(wj) * wj;
	double tmp;
	if ((wj - ((t_0 - x) / (t_0 + exp(wj)))) <= 1e-17) {
		tmp = fma(fma(-2.0, x, fma(fma(-wj, fma(x, 2.6666666666666665, 1.0), (2.5 * x)), wj, wj)), wj, x);
	} else {
		tmp = wj - ((((wj / (wj - -1.0)) / x) - (exp(-wj) / (wj - -1.0))) * x);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(exp(wj) * wj)
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(t_0 + exp(wj)))) <= 1e-17)
		tmp = fma(fma(-2.0, x, fma(fma(Float64(-wj), fma(x, 2.6666666666666665, 1.0), Float64(2.5 * x)), wj, wj)), wj, x);
	else
		tmp = Float64(wj - Float64(Float64(Float64(Float64(wj / Float64(wj - -1.0)) / x) - Float64(exp(Float64(-wj)) / Float64(wj - -1.0))) * x));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[Exp[wj], $MachinePrecision] * wj), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(t$95$0 + N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-17], N[(N[(-2.0 * x + N[(N[((-wj) * N[(x * 2.6666666666666665 + 1.0), $MachinePrecision] + N[(2.5 * x), $MachinePrecision]), $MachinePrecision] * wj + wj), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(N[(N[(wj / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $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))))) < 1.00000000000000007e-17

   1. Initial program 72.1%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.5%

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

   6. Taylor expanded in wj around 0

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in wj around 0

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

   11. Applied rewrites 99.6%

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

   if 1.00000000000000007e-17 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 94.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto wj - \color{blue}{x \cdot \left(\frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)} - \frac{1}{e^{wj} + wj \cdot e^{wj}}\right)} \]
   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 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}} \leq 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, x, \mathsf{fma}\left(\mathsf{fma}\left(-wj, \mathsf{fma}\left(x, 2.6666666666666665, 1\right), 2.5 \cdot x\right), wj, wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{\frac{wj}{wj - -1}}{x} - \frac{e^{-wj}}{wj - -1}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 97.4%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 96.9%

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

9. Taylor expanded in wj around 0

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

11. Applied rewrites 97.4%

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

Alternative 4: 95.7% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. cancel-sign-sub-inv N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-out N/A

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

   11. *-commutative N/A

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

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

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{5}{2}}, x, 1\right), wj, -2 \cdot x\right), wj, x\right) \]

   16. lower-*.f64 96.9

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

5. Applied rewrites 96.9%

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

Alternative 5: 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 79.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 97.4%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 96.9%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 96.6%

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

Alternative 6: 84.7% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 86.3

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

5. Applied rewrites 86.3%

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

Alternative 7: 84.7% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 97.4%

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

6. Taylor expanded in wj around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. lower-*.f64 N/A

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

   4. lower-fma.f64 86.3

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

8. Applied rewrites 86.3%

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

Alternative 8: 84.2% accurate, 55.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 97.4%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 86.5%

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

9. Taylor expanded in wj around 0

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

11. Applied rewrites 85.8%

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

Alternative 9: 13.6% 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 79.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 97.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 13.1%

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

9. Taylor expanded in wj around 0

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

11. Applied rewrites 12.8%

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

Alternative 10: 4.2% accurate, 82.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Taylor expanded in wj around inf

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

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

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

   5. rgt-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. *-rgt-identity N/A

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

   8. lower-+.f64 3.8

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

5. Applied rewrites 3.8%

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

Alternative 11: 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 79.1%

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

3. Taylor expanded in wj around inf

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

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

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

   5. rgt-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. *-rgt-identity N/A

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

   8. lower-+.f64 3.8

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

5. Applied rewrites 3.8%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 3.3%

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

Developer Target 1: 79.0% 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 2024304 
(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))))))