Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.3% → 98.8%

Time: 10.7s

Alternatives: 10

Speedup: 27.6×

• could not determine a ground truth

  1. wj = 1.885141171662937e+195
  2. x = 5.050207789237553e-153

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: 78.3% 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: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), 1\right) - wj, x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\mathsf{fma}\left(x, e^{-wj}, -wj\right)}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 1e-14)
     (fma
      wj
      (fma wj (- (fma x (fma wj -2.6666666666666665 2.5) 1.0) wj) (* x -2.0))
      x)
     (+ wj (/ (fma x (exp (- wj)) (- wj)) (+ wj 1.0))))))

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-14) {
		tmp = fma(wj, fma(wj, (fma(x, fma(wj, -2.6666666666666665, 2.5), 1.0) - wj), (x * -2.0)), x);
	} else {
		tmp = wj + (fma(x, exp(-wj), -wj) / (wj + 1.0));
	}
	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-14)
		tmp = fma(wj, fma(wj, Float64(fma(x, fma(wj, -2.6666666666666665, 2.5), 1.0) - wj), Float64(x * -2.0)), x);
	else
		tmp = Float64(wj + Float64(fma(x, exp(Float64(-wj)), Float64(-wj)) / Float64(wj + 1.0)));
	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-14], N[(wj * N[(wj * N[(N[(x * N[(wj * -2.6666666666666665 + 2.5), $MachinePrecision] + 1.0), $MachinePrecision] - wj), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj + N[(N[(x * N[Exp[(-wj)], $MachinePrecision] + (-wj)), $MachinePrecision] / N[(wj + 1.0), $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))))) < 9.99999999999999999e-15

   1. Initial program 68.1%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

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

   if 9.99999999999999999e-15 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 93.0%

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

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 92.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. lower-fma.f64 N/A

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

   6. Applied rewrites 92.9%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. exp-neg N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. neg-mul-1 N/A

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

      8. neg-mul-1 N/A

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

      9. exp-neg N/A

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

      10. rgt-mult-inverse N/A

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

      11. *-rgt-identity N/A

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

      12. lower--.f64 N/A

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

   9. Applied rewrites 99.7%

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

       1. lift-fma.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. un-div-inv N/A

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

       5. lower-/.f64 99.7

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

   11. Applied rewrites 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), 1\right) - wj, x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\mathsf{fma}\left(x, e^{-wj}, -wj\right)}{wj + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := wj + \frac{x - t\_0}{e^{wj} + t\_0}\\ t_2 := wj - \left(-x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-260}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-229}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj)))
        (t_1 (+ wj (/ (- x t_0) (+ (exp wj) t_0))))
        (t_2 (- wj (- x))))
   (if (<= t_1 -2e-260) t_2 (if (<= t_1 2e-229) (* wj wj) t_2))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double t_1 = wj + ((x - t_0) / (exp(wj) + t_0));
	double t_2 = wj - -x;
	double tmp;
	if (t_1 <= -2e-260) {
		tmp = t_2;
	} else if (t_1 <= 2e-229) {
		tmp = wj * wj;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = wj * exp(wj)
    t_1 = wj + ((x - t_0) / (exp(wj) + t_0))
    t_2 = wj - -x
    if (t_1 <= (-2d-260)) then
        tmp = t_2
    else if (t_1 <= 2d-229) then
        tmp = wj * wj
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double t_1 = wj + ((x - t_0) / (Math.exp(wj) + t_0));
	double t_2 = wj - -x;
	double tmp;
	if (t_1 <= -2e-260) {
		tmp = t_2;
	} else if (t_1 <= 2e-229) {
		tmp = wj * wj;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	t_1 = wj + ((x - t_0) / (math.exp(wj) + t_0))
	t_2 = wj - -x
	tmp = 0
	if t_1 <= -2e-260:
		tmp = t_2
	elif t_1 <= 2e-229:
		tmp = wj * wj
	else:
		tmp = t_2
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	t_1 = Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0)))
	t_2 = Float64(wj - Float64(-x))
	tmp = 0.0
	if (t_1 <= -2e-260)
		tmp = t_2;
	elseif (t_1 <= 2e-229)
		tmp = Float64(wj * wj);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	t_1 = wj + ((x - t_0) / (exp(wj) + t_0));
	t_2 = wj - -x;
	tmp = 0.0;
	if (t_1 <= -2e-260)
		tmp = t_2;
	elseif (t_1 <= 2e-229)
		tmp = wj * wj;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(wj - (-x)), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-260], t$95$2, If[LessEqual[t$95$1, 2e-229], N[(wj * wj), $MachinePrecision], t$95$2]]]]]

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))))) < -1.99999999999999992e-260 or 2.00000000000000014e-229 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 95.3%

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

   3. Taylor expanded in wj around 0

      \[\leadsto wj - \color{blue}{-1 \cdot x} \]
   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 88.5

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

   5. Applied rewrites 88.5%

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

   if -1.99999999999999992e-260 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 2.00000000000000014e-229

   1. Initial program 6.7%

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

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.8%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq -2 \cdot 10^{-260}:\\ \;\;\;\;wj - \left(-x\right)\\ \mathbf{elif}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 2 \cdot 10^{-229}:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.004:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), 1\right) - wj, x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.004)
   (fma
    wj
    (fma wj (- (fma x (fma wj -2.6666666666666665 2.5) 1.0) wj) (* x -2.0))
    x)
   (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.004) {
		tmp = fma(wj, fma(wj, (fma(x, fma(wj, -2.6666666666666665, 2.5), 1.0) - wj), (x * -2.0)), x);
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.004)
		tmp = fma(wj, fma(wj, Float64(fma(x, fma(wj, -2.6666666666666665, 2.5), 1.0) - wj), Float64(x * -2.0)), x);
	else
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.004], N[(wj * N[(wj * N[(N[(x * N[(wj * -2.6666666666666665 + 2.5), $MachinePrecision] + 1.0), $MachinePrecision] - wj), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.0040000000000000001

   1. Initial program 76.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.8%

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

   if 0.0040000000000000001 < wj

   1. Initial program 16.3%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.6

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

   5. Applied rewrites 99.6%

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

Alternative 4: 97.5% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00034:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, 1 - wj, x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00034)
   (fma wj (fma wj (- 1.0 wj) (* x -2.0)) x)
   (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00034) {
		tmp = fma(wj, fma(wj, (1.0 - wj), (x * -2.0)), x);
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00034)
		tmp = fma(wj, fma(wj, Float64(1.0 - wj), Float64(x * -2.0)), x);
	else
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00034], N[(wj * N[(wj * N[(1.0 - wj), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 3.4e-4

   1. Initial program 76.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.7%

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

   if 3.4e-4 < wj

   1. Initial program 16.3%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.6

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

   5. Applied rewrites 99.6%

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

Alternative 5: 97.2% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0003:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(x, \mathsf{fma}\left(wj, 2.5, -2\right), wj\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0003)
   (fma wj (fma x (fma wj 2.5 -2.0) wj) x)
   (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0003) {
		tmp = fma(wj, fma(x, fma(wj, 2.5, -2.0), wj), x);
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0003)
		tmp = fma(wj, fma(x, fma(wj, 2.5, -2.0), wj), x);
	else
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0003], N[(wj * N[(x * N[(wj * 2.5 + -2.0), $MachinePrecision] + wj), $MachinePrecision] + x), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.99999999999999974e-4

   1. Initial program 76.7%

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

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.0%

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

   if 2.99999999999999974e-4 < wj

   1. Initial program 16.3%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.6

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

   5. Applied rewrites 99.6%

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

Alternative 6: 96.9% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0003:\\ \;\;\;\;\mathsf{fma}\left(wj, wj - wj \cdot wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0003) (fma wj (- wj (* wj wj)) x) (- wj (/ wj (+ wj 1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0003) {
		tmp = fma(wj, (wj - (wj * wj)), x);
	} else {
		tmp = wj - (wj / (wj + 1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0003)
		tmp = fma(wj, Float64(wj - Float64(wj * wj)), x);
	else
		tmp = Float64(wj - Float64(wj / Float64(wj + 1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0003], N[(wj * N[(wj - N[(wj * wj), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj - N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.99999999999999974e-4

   1. Initial program 76.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 97.9%

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

   if 2.99999999999999974e-4 < wj

   1. Initial program 16.3%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.6

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

   5. Applied rewrites 99.6%

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

Alternative 7: 95.5% accurate, 22.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.6%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 95.7%

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

Alternative 8: 15.0% accurate, 27.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-72}:\\ \;\;\;\;wj - 1\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (if (<= x -7.5e-72) (- wj 1.0) (* wj wj)))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -7.5e-72) {
		tmp = wj - 1.0;
	} 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) :: tmp
    if (x <= (-7.5d-72)) then
        tmp = wj - 1.0d0
    else
        tmp = wj * wj
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -7.5e-72) {
		tmp = wj - 1.0;
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -7.5e-72:
		tmp = wj - 1.0
	else:
		tmp = wj * wj
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -7.5e-72)
		tmp = Float64(wj - 1.0);
	else
		tmp = Float64(wj * wj);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -7.5e-72)
		tmp = wj - 1.0;
	else
		tmp = wj * wj;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -7.5e-72], N[(wj - 1.0), $MachinePrecision], N[(wj * wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.5000000000000004e-72

   1. Initial program 96.4%

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

   3. Taylor expanded in wj around inf

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

   5. Applied rewrites 7.8%

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

   if -7.5000000000000004e-72 < x

   1. Initial program 66.7%

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

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 21.2%

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

Alternative 9: 84.6% accurate, 27.6× speedup

Math

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

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(x, Float64(wj * -2.0), x)
end

Wolfram

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

Derivation

1. Initial program 75.3%

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

3. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 82.9

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

5. Applied rewrites 82.9%

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

Alternative 10: 4.2% accurate, 82.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

3. Taylor expanded in wj around inf

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

5. Applied rewrites 4.6%

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

Developer Target 1: 79.3% 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 2024238 
(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))))))