Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.0% → 98.2%

Time: 8.6s

Alternatives: 12

Speedup: 27.6×

• could not determine a ground truth

  1. wj = -2.475225432863384e+225
  2. x = 3.878497227118807e-248

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: 78.0% 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: 98.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{1 + wj}, \frac{e^{wj} \cdot wj - x}{e^{wj}}, wj\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), x, 1 - wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -5.5e-6)
   (fma (/ -1.0 (+ 1.0 wj)) (/ (- (* (exp wj) wj) x) (exp wj)) wj)
   (fma
    (fma (fma (fma -2.6666666666666665 wj 2.5) x (- 1.0 wj)) wj (* -2.0 x))
    wj
    x)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -5.5e-6) {
		tmp = fma((-1.0 / (1.0 + wj)), (((exp(wj) * wj) - x) / exp(wj)), wj);
	} else {
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), x, (1.0 - wj)), wj, (-2.0 * x)), wj, x);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -5.5e-6)
		tmp = fma(Float64(-1.0 / Float64(1.0 + wj)), Float64(Float64(Float64(exp(wj) * wj) - x) / exp(wj)), wj);
	else
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), x, Float64(1.0 - wj)), wj, Float64(-2.0 * x)), wj, x);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -5.5e-6], N[(N[(-1.0 / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Exp[wj], $MachinePrecision] * wj), $MachinePrecision] - x), $MachinePrecision] / N[Exp[wj], $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision], N[(N[(N[(N[(-2.6666666666666665 * wj + 2.5), $MachinePrecision] * x + N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -5.4999999999999999e-6

   1. Initial program 74.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.6%

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

   if -5.4999999999999999e-6 < wj

   1. Initial program 77.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), x, 1 - wj\right), wj, -2 \cdot x\right), wj, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 82.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := wj - \frac{t\_0 - x}{e^{wj} + t\_0}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-273} \lor \neg \left(t\_1 \leq 10^{-306}\right):\\ \;\;\;\;wj - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))) (t_1 (- wj (/ (- t_0 x) (+ (exp wj) t_0)))))
   (if (or (<= t_1 -1e-273) (not (<= t_1 1e-306))) (- wj (- x)) (* wj wj))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double t_1 = wj - ((t_0 - x) / (exp(wj) + t_0));
	double tmp;
	if ((t_1 <= -1e-273) || !(t_1 <= 1e-306)) {
		tmp = wj - -x;
	} else {
		tmp = wj * wj;
	}
	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 - ((t_0 - x) / (exp(wj) + t_0))
    if ((t_1 <= (-1d-273)) .or. (.not. (t_1 <= 1d-306))) then
        tmp = wj - -x
    else
        tmp = wj * wj
    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 - ((t_0 - x) / (Math.exp(wj) + t_0));
	double tmp;
	if ((t_1 <= -1e-273) || !(t_1 <= 1e-306)) {
		tmp = wj - -x;
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	t_1 = wj - ((t_0 - x) / (math.exp(wj) + t_0))
	tmp = 0
	if (t_1 <= -1e-273) or not (t_1 <= 1e-306):
		tmp = wj - -x
	else:
		tmp = wj * wj
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	t_1 = Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0)))
	tmp = 0.0
	if ((t_1 <= -1e-273) || !(t_1 <= 1e-306))
		tmp = Float64(wj - Float64(-x));
	else
		tmp = Float64(wj * wj);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	t_1 = wj - ((t_0 - x) / (exp(wj) + t_0));
	tmp = 0.0;
	if ((t_1 <= -1e-273) || ~((t_1 <= 1e-306)))
		tmp = wj - -x;
	else
		tmp = wj * wj;
	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[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-273], N[Not[LessEqual[t$95$1, 1e-306]], $MachinePrecision]], N[(wj - (-x)), $MachinePrecision], N[(wj * wj), $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))))) < -1e-273 or 1.00000000000000003e-306 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 96.0%

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

   3. Taylor expanded in wj around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 88.1

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

   5. Applied rewrites 88.1%

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

   if -1e-273 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 1.00000000000000003e-306

   1. Initial program 7.0%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in wj around 0

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)} \]
   7. 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

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 46.8%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq -1 \cdot 10^{-273} \lor \neg \left(wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 10^{-306}\right):\\ \;\;\;\;wj - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -5.9 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(e^{wj}, wj, -x\right), e^{-wj} \cdot \frac{-1}{1 + wj}, wj\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), x, 1 - wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -5.9e-6)
   (fma (fma (exp wj) wj (- x)) (* (exp (- wj)) (/ -1.0 (+ 1.0 wj))) wj)
   (fma
    (fma (fma (fma -2.6666666666666665 wj 2.5) x (- 1.0 wj)) wj (* -2.0 x))
    wj
    x)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -5.9e-6) {
		tmp = fma(fma(exp(wj), wj, -x), (exp(-wj) * (-1.0 / (1.0 + wj))), wj);
	} else {
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), x, (1.0 - wj)), wj, (-2.0 * x)), wj, x);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -5.9e-6)
		tmp = fma(fma(exp(wj), wj, Float64(-x)), Float64(exp(Float64(-wj)) * Float64(-1.0 / Float64(1.0 + wj))), wj);
	else
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), x, Float64(1.0 - wj)), wj, Float64(-2.0 * x)), wj, x);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -5.9e-6], N[(N[(N[Exp[wj], $MachinePrecision] * wj + (-x)), $MachinePrecision] * N[(N[Exp[(-wj)], $MachinePrecision] * N[(-1.0 / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision], N[(N[(N[(N[(-2.6666666666666665 * wj + 2.5), $MachinePrecision] * x + N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -5.90000000000000026e-6

   1. Initial program 74.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.6%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. lift-*.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. rec-exp N/A

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

      15. lower-exp.f64 N/A

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

      16. lower-neg.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if -5.90000000000000026e-6 < wj

   1. Initial program 77.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), x, 1 - wj\right), wj, -2 \cdot x\right), wj, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.0068:\\ \;\;\;\;wj - \frac{\frac{x}{-1 - wj}}{e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), x, 1 - wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.0068)
   (- wj (/ (/ x (- -1.0 wj)) (exp wj)))
   (fma
    (fma (fma (fma -2.6666666666666665 wj 2.5) x (- 1.0 wj)) wj (* -2.0 x))
    wj
    x)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.0068) {
		tmp = wj - ((x / (-1.0 - wj)) / exp(wj));
	} else {
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), x, (1.0 - wj)), wj, (-2.0 * x)), wj, x);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.0068)
		tmp = Float64(wj - Float64(Float64(x / Float64(-1.0 - wj)) / exp(wj)));
	else
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), x, Float64(1.0 - wj)), wj, Float64(-2.0 * x)), wj, x);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.0068], N[(wj - N[(N[(x / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision] / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.6666666666666665 * wj + 2.5), $MachinePrecision] * x + N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -0.00679999999999999962

   1. Initial program 74.4%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. associate-/r* N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-exp.f64 89.5

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

   5. Applied rewrites 89.5%

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

   if -0.00679999999999999962 < wj

   1. Initial program 77.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), x, 1 - wj\right), wj, -2 \cdot x\right), wj, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.0068:\\ \;\;\;\;wj - \frac{\frac{x}{-1 - wj}}{e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), x, 1 - wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.0056:\\ \;\;\;\;\frac{x}{\left(wj + 1\right) \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), x, 1 - wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.0056)
   (/ x (* (+ wj 1.0) (exp wj)))
   (fma
    (fma (fma (fma -2.6666666666666665 wj 2.5) x (- 1.0 wj)) wj (* -2.0 x))
    wj
    x)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.0056) {
		tmp = x / ((wj + 1.0) * exp(wj));
	} else {
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), x, (1.0 - wj)), wj, (-2.0 * x)), wj, x);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.0056)
		tmp = Float64(x / Float64(Float64(wj + 1.0) * exp(wj)));
	else
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), x, Float64(1.0 - wj)), wj, Float64(-2.0 * x)), wj, x);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.0056], N[(x / N[(N[(wj + 1.0), $MachinePrecision] * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.6666666666666665 * wj + 2.5), $MachinePrecision] * x + N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -0.00559999999999999994

   1. Initial program 74.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 62.6

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

   5. Applied rewrites 62.6%

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

   7. Applied rewrites 87.6%

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

   if -0.00559999999999999994 < wj

   1. Initial program 77.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), x, 1 - wj\right), wj, -2 \cdot x\right), wj, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.6% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.4%

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

Alternative 7: 96.2% accurate, 17.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.4%

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

6. Taylor expanded in wj around 0

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)} \]
7. 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

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

8. Applied rewrites 95.8%

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

Alternative 8: 96.0% accurate, 25.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.4%

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

6. Taylor expanded in wj around 0

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)} \]
7. 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

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

8. Applied rewrites 95.8%

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

9. Taylor expanded in wj around 0

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

11. Applied rewrites 95.5%

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

Alternative 9: 84.9% 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 77.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.4%

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

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 84.5

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

8. Applied rewrites 84.5%

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

Alternative 10: 13.9% 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 77.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.4%

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

6. Taylor expanded in wj around 0

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)} \]
7. 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

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

8. Applied rewrites 95.8%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 13.4%

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

Alternative 11: 4.1% 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 77.6%

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

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

   3. *-lft-identity N/A

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

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

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

   5. lft-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 3.4

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

5. Applied rewrites 3.4%

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

Alternative 12: 3.4% 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 77.6%

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

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

   3. *-lft-identity N/A

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

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

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

   5. lft-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 3.4

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

5. Applied rewrites 3.4%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 3.1%

   \[\leadsto -1 \]
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 2024313 
(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))))))