Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.6% → 98.7%

Time: 12.5s

Alternatives: 9

Speedup: 331.0×

• could not determine a ground truth

  1. wj = 7.311529345014676e+36
  2. x = -6.2970572017377586e-142

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.6% 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: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(x, 0.6666666666666666, 1\right)\right), 0 - wj, \mathsf{fma}\left(x, 2.5, 1\right)\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - x \cdot e^{0 - wj}}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 1e-16)
     (fma
      wj
      (fma
       wj
       (fma
        (fma x 2.0 (fma x 0.6666666666666666 1.0))
        (- 0.0 wj)
        (fma x 2.5 1.0))
       (* x -2.0))
      x)
     (+ wj (/ (- wj (* x (exp (- 0.0 wj)))) (- -1.0 wj))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1e-16) {
		tmp = fma(wj, fma(wj, fma(fma(x, 2.0, fma(x, 0.6666666666666666, 1.0)), (0.0 - wj), fma(x, 2.5, 1.0)), (x * -2.0)), x);
	} else {
		tmp = wj + ((wj - (x * exp((0.0 - wj)))) / (-1.0 - wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 1e-16)
		tmp = fma(wj, fma(wj, fma(fma(x, 2.0, fma(x, 0.6666666666666666, 1.0)), Float64(0.0 - wj), fma(x, 2.5, 1.0)), Float64(x * -2.0)), x);
	else
		tmp = Float64(wj + Float64(Float64(wj - Float64(x * exp(Float64(0.0 - wj)))) / Float64(-1.0 - wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-16], N[(wj * N[(wj * N[(N[(x * 2.0 + N[(x * 0.6666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(0.0 - wj), $MachinePrecision] + N[(x * 2.5 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj + N[(N[(wj - N[(x * N[Exp[N[(0.0 - wj), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $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))))) < 9.9999999999999998e-17

   1. Initial program 74.3%

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

   3. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(x, 0.6666666666666666, 1\right)\right), 0 - wj, \mathsf{fma}\left(x, 2.5, 1\right)\right), x \cdot -2\right), x\right)} \]

   if 9.9999999999999998e-17 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 94.5%

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

      1. div-sub N/A

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

      2. --lowering--.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. times-frac N/A

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

      5. *-inverses N/A

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

      6. associate-*l/ N/A

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

      7. *-rgt-identity N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. distribute-rgt1-in N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. exp-lowering-exp.f64 N/A

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

      15. +-lowering-+.f64 99.7

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

   4. Applied egg-rr 99.7%

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

      1. associate-/r* N/A

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

      2. sub-div N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. div-inv N/A

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

      6. *-lowering-*.f64 N/A

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

      7. rec-exp N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. neg-sub0 N/A

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

      10. --lowering--.f64 N/A

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

      11. +-lowering-+.f64 99.7

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

   6. Applied egg-rr 99.7%

      \[\leadsto wj - \color{blue}{\frac{wj - x \cdot e^{0 - wj}}{wj + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(x, 0.6666666666666666, 1\right)\right), 0 - wj, \mathsf{fma}\left(x, 2.5, 1\right)\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj - x \cdot e^{0 - wj}}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.024:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(x, 0.6666666666666666, 1\right)\right), 0 - wj, \mathsf{fma}\left(x, 2.5, 1\right)\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.024)
   (fma
    wj
    (fma
     wj
     (fma
      (fma x 2.0 (fma x 0.6666666666666666 1.0))
      (- 0.0 wj)
      (fma x 2.5 1.0))
     (* x -2.0))
    x)
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.024) {
		tmp = fma(wj, fma(wj, fma(fma(x, 2.0, fma(x, 0.6666666666666666, 1.0)), (0.0 - wj), fma(x, 2.5, 1.0)), (x * -2.0)), x);
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.024)
		tmp = fma(wj, fma(wj, fma(fma(x, 2.0, fma(x, 0.6666666666666666, 1.0)), Float64(0.0 - wj), fma(x, 2.5, 1.0)), Float64(x * -2.0)), x);
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.024], N[(wj * N[(wj * N[(N[(x * 2.0 + N[(x * 0.6666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(0.0 - wj), $MachinePrecision] + N[(x * 2.5 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.024

   1. Initial program 81.7%

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

   3. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(x, 0.6666666666666666, 1\right)\right), 0 - wj, \mathsf{fma}\left(x, 2.5, 1\right)\right), x \cdot -2\right), x\right)} \]

   if 0.024 < wj

   1. Initial program 19.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 99.7

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

   5. Simplified 99.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.024:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(x, 0.6666666666666666, 1\right)\right), 0 - wj, \mathsf{fma}\left(x, 2.5, 1\right)\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.2% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0034:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0034)
   (fma wj (fma wj (fma x 2.5 1.0) (* x -2.0)) x)
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0034) {
		tmp = fma(wj, fma(wj, fma(x, 2.5, 1.0), (x * -2.0)), x);
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0034)
		tmp = fma(wj, fma(wj, fma(x, 2.5, 1.0), Float64(x * -2.0)), x);
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0034], N[(wj * N[(wj * N[(x * 2.5 + 1.0), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.00339999999999999981

   1. Initial program 81.7%

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

   3. Taylor expanded in wj around 0

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)} \]
   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. accelerator-lowering-fma.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-out N/A

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

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

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 97.8

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

   5. Simplified 97.8%

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

   if 0.00339999999999999981 < wj

   1. Initial program 19.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 99.7

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

   5. Simplified 99.7%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.0034:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00088:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, -2, wj\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00088) (fma (fma x -2.0 wj) wj x) (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00088) {
		tmp = fma(fma(x, -2.0, wj), wj, x);
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00088)
		tmp = fma(fma(x, -2.0, wj), wj, x);
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00088], N[(N[(x * -2.0 + 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 < 8.80000000000000031e-4

   1. Initial program 81.7%

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

   3. Taylor expanded in wj around 0

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)} \]
   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. accelerator-lowering-fma.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-out N/A

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

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

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 97.8

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

   5. Simplified 97.8%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 97.7%

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

      1. *-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 97.7

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

   10. Applied egg-rr 97.7%

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

   if 8.80000000000000031e-4 < wj

   1. Initial program 19.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 99.7

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

   5. Simplified 99.7%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.00088:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, -2, wj\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.8% accurate, 25.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.5%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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. accelerator-lowering-fma.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-rgt-out N/A

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

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

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. *-lowering-*.f64 96.0

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

5. Simplified 96.0%

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

6. Taylor expanded in x around 0

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

8. Simplified 95.9%

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

   1. *-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. *-rgt-identity N/A

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

   4. +-commutative N/A

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

   5. accelerator-lowering-fma.f64 95.9

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

10. Applied egg-rr 95.9%

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

Alternative 6: 95.3% accurate, 47.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.5%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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. accelerator-lowering-fma.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-rgt-out N/A

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

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

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. *-lowering-*.f64 96.0

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

5. Simplified 96.0%

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

6. Taylor expanded in x around 0

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

8. Simplified 95.5%

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

Alternative 7: 83.5% accurate, 331.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

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

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 80.5%

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

3. Taylor expanded in wj around 0

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

5. Simplified 85.6%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Alternative 8: 4.4% accurate, 331.0× speedup

Math

\[\begin{array}{l} \\ wj \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 80.5%

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

3. Taylor expanded in wj around inf

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

5. Simplified 4.6%

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

Alternative 9: 3.3% accurate, 331.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return -1.0

Julia

function code(wj, x)
	return -1.0
end

MATLAB

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

Wolfram

code[wj_, x_] := -1.0

Derivation

1. Initial program 80.5%

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

3. Taylor expanded in wj around inf

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

5. Simplified 4.4%

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

6. Taylor expanded in wj around 0

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

8. Simplified 3.3%

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

Developer Target 1: 78.5% 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 2024198 
(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))))))