Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.7% → 99.3%

Time: 9.1s

Alternatives: 17

Speedup: 55.2×

• could not determine a ground truth

  1. wj = -3.767941706879041e+299
  2. x = -1.8494680519056998e-152

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: 77.7% 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: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -2.55 \cdot 10^{-6}:\\ \;\;\;\;wj - \frac{\frac{e^{wj} \cdot wj - x}{e^{wj}}}{1 + wj}\\ \mathbf{elif}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(\frac{1 + wj}{wj}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -2.55e-6)
   (- wj (/ (/ (- (* (exp wj) wj) x) (exp wj)) (+ 1.0 wj)))
   (if (<= wj 0.011)
     (fma
      (fma
       (fma 2.5 x (- 1.0 (* (fma 0.6666666666666666 x (fma 2.0 x 1.0)) wj)))
       wj
       (* -2.0 x))
      wj
      x)
     (- wj (pow (/ (+ 1.0 wj) wj) -1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -2.55e-6) {
		tmp = wj - ((((exp(wj) * wj) - x) / exp(wj)) / (1.0 + wj));
	} else if (wj <= 0.011) {
		tmp = fma(fma(fma(2.5, x, (1.0 - (fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - pow(((1.0 + wj) / wj), -1.0);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -2.55e-6)
		tmp = Float64(wj - Float64(Float64(Float64(Float64(exp(wj) * wj) - x) / exp(wj)) / Float64(1.0 + wj)));
	elseif (wj <= 0.011)
		tmp = fma(fma(fma(2.5, x, Float64(1.0 - Float64(fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - (Float64(Float64(1.0 + wj) / wj) ^ -1.0));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -2.55e-6], N[(wj - N[(N[(N[(N[(N[Exp[wj], $MachinePrecision] * wj), $MachinePrecision] - x), $MachinePrecision] / N[Exp[wj], $MachinePrecision]), $MachinePrecision] / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.011], N[(N[(N[(2.5 * x + N[(1.0 - N[(N[(0.6666666666666666 * x + N[(2.0 * x + 1.0), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[Power[N[(N[(1.0 + wj), $MachinePrecision] / wj), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -2.5500000000000001e-6

   1. Initial program 49.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if -2.5500000000000001e-6 < wj < 0.010999999999999999

   1. Initial program 78.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.8%

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

   if 0.010999999999999999 < wj

   1. Initial program 20.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -2.55 \cdot 10^{-6}:\\ \;\;\;\;wj - \frac{\frac{e^{wj} \cdot wj - x}{e^{wj}}}{1 + wj}\\ \mathbf{elif}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(\frac{1 + wj}{wj}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{\left(1 + wj\right) \cdot e^{wj}}\\ \mathbf{elif}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(\frac{1 + wj}{wj}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -4.5e-6)
   (- wj (/ (- (* wj (exp wj)) x) (* (+ 1.0 wj) (exp wj))))
   (if (<= wj 0.011)
     (fma
      (fma
       (fma 2.5 x (- 1.0 (* (fma 0.6666666666666666 x (fma 2.0 x 1.0)) wj)))
       wj
       (* -2.0 x))
      wj
      x)
     (- wj (pow (/ (+ 1.0 wj) wj) -1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -4.5e-6) {
		tmp = wj - (((wj * exp(wj)) - x) / ((1.0 + wj) * exp(wj)));
	} else if (wj <= 0.011) {
		tmp = fma(fma(fma(2.5, x, (1.0 - (fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - pow(((1.0 + wj) / wj), -1.0);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -4.5e-6)
		tmp = Float64(wj - Float64(Float64(Float64(wj * exp(wj)) - x) / Float64(Float64(1.0 + wj) * exp(wj))));
	elseif (wj <= 0.011)
		tmp = fma(fma(fma(2.5, x, Float64(1.0 - Float64(fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - (Float64(Float64(1.0 + wj) / wj) ^ -1.0));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -4.5e-6], N[(wj - N[(N[(N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(N[(1.0 + wj), $MachinePrecision] * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.011], N[(N[(N[(2.5 * x + N[(1.0 - N[(N[(0.6666666666666666 * x + N[(2.0 * x + 1.0), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[Power[N[(N[(1.0 + wj), $MachinePrecision] / wj), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -4.50000000000000011e-6

   1. Initial program 49.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -4.50000000000000011e-6 < wj < 0.010999999999999999

   1. Initial program 78.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.8%

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

   if 0.010999999999999999 < wj

   1. Initial program 20.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{\left(1 + wj\right) \cdot e^{wj}}\\ \mathbf{elif}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(\frac{1 + wj}{wj}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.035:\\ \;\;\;\;wj - \frac{\frac{-x}{e^{wj}}}{1 + wj}\\ \mathbf{elif}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(\frac{1 + wj}{wj}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.035)
   (- wj (/ (/ (- x) (exp wj)) (+ 1.0 wj)))
   (if (<= wj 0.011)
     (fma
      (fma
       (fma 2.5 x (- 1.0 (* (fma 0.6666666666666666 x (fma 2.0 x 1.0)) wj)))
       wj
       (* -2.0 x))
      wj
      x)
     (- wj (pow (/ (+ 1.0 wj) wj) -1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.035) {
		tmp = wj - ((-x / exp(wj)) / (1.0 + wj));
	} else if (wj <= 0.011) {
		tmp = fma(fma(fma(2.5, x, (1.0 - (fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - pow(((1.0 + wj) / wj), -1.0);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.035)
		tmp = Float64(wj - Float64(Float64(Float64(-x) / exp(wj)) / Float64(1.0 + wj)));
	elseif (wj <= 0.011)
		tmp = fma(fma(fma(2.5, x, Float64(1.0 - Float64(fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - (Float64(Float64(1.0 + wj) / wj) ^ -1.0));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.035], N[(wj - N[(N[((-x) / N[Exp[wj], $MachinePrecision]), $MachinePrecision] / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.011], N[(N[(N[(2.5 * x + N[(1.0 - N[(N[(0.6666666666666666 * x + N[(2.0 * x + 1.0), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[Power[N[(N[(1.0 + wj), $MachinePrecision] / wj), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -0.035000000000000003

   1. Initial program 49.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-exp.f64 86.3

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

   7. Applied rewrites 86.3%

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

   if -0.035000000000000003 < wj < 0.010999999999999999

   1. Initial program 78.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.8%

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

   if 0.010999999999999999 < wj

   1. Initial program 20.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.035:\\ \;\;\;\;wj - \frac{\frac{-x}{e^{wj}}}{1 + wj}\\ \mathbf{elif}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(\frac{1 + wj}{wj}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.035:\\ \;\;\;\;wj - \frac{\frac{x}{-1 - wj}}{e^{wj}}\\ \mathbf{elif}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(\frac{1 + wj}{wj}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.035)
   (- wj (/ (/ x (- -1.0 wj)) (exp wj)))
   (if (<= wj 0.011)
     (fma
      (fma
       (fma 2.5 x (- 1.0 (* (fma 0.6666666666666666 x (fma 2.0 x 1.0)) wj)))
       wj
       (* -2.0 x))
      wj
      x)
     (- wj (pow (/ (+ 1.0 wj) wj) -1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.035) {
		tmp = wj - ((x / (-1.0 - wj)) / exp(wj));
	} else if (wj <= 0.011) {
		tmp = fma(fma(fma(2.5, x, (1.0 - (fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - pow(((1.0 + wj) / wj), -1.0);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.035)
		tmp = Float64(wj - Float64(Float64(x / Float64(-1.0 - wj)) / exp(wj)));
	elseif (wj <= 0.011)
		tmp = fma(fma(fma(2.5, x, Float64(1.0 - Float64(fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - (Float64(Float64(1.0 + wj) / wj) ^ -1.0));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.035], N[(wj - N[(N[(x / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision] / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 0.011], N[(N[(N[(2.5 * x + N[(1.0 - N[(N[(0.6666666666666666 * x + N[(2.0 * x + 1.0), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[Power[N[(N[(1.0 + wj), $MachinePrecision] / wj), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -0.035000000000000003

   1. Initial program 49.5%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 85.8

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

   5. Applied rewrites 85.8%

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

   if -0.035000000000000003 < wj < 0.010999999999999999

   1. Initial program 78.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.8%

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

   if 0.010999999999999999 < wj

   1. Initial program 20.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -0.035:\\ \;\;\;\;wj - \frac{\frac{x}{-1 - wj}}{e^{wj}}\\ \mathbf{elif}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(\frac{1 + wj}{wj}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(\frac{1 + wj}{wj}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.011)
   (fma
    (fma
     (fma 2.5 x (- 1.0 (* (fma 0.6666666666666666 x (fma 2.0 x 1.0)) wj)))
     wj
     (* -2.0 x))
    wj
    x)
   (- wj (pow (/ (+ 1.0 wj) wj) -1.0))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.011) {
		tmp = fma(fma(fma(2.5, x, (1.0 - (fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - pow(((1.0 + wj) / wj), -1.0);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.011)
		tmp = fma(fma(fma(2.5, x, Float64(1.0 - Float64(fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - (Float64(Float64(1.0 + wj) / wj) ^ -1.0));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.011], N[(N[(N[(2.5 * x + N[(1.0 - N[(N[(0.6666666666666666 * x + N[(2.0 * x + 1.0), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[Power[N[(N[(1.0 + wj), $MachinePrecision] / wj), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.010999999999999999

   1. Initial program 77.9%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 97.4%

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

   if 0.010999999999999999 < wj

   1. Initial program 20.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(\frac{1 + wj}{wj}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(\frac{wj}{x}, 1 - wj, \mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), wj, -2\right)\right), 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(\frac{1 + wj}{wj}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.011)
   (*
    (fma
     wj
     (fma (/ wj x) (- 1.0 wj) (fma (fma -2.6666666666666665 wj 2.5) wj -2.0))
     1.0)
    x)
   (- wj (pow (/ (+ 1.0 wj) wj) -1.0))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.011) {
		tmp = fma(wj, fma((wj / x), (1.0 - wj), fma(fma(-2.6666666666666665, wj, 2.5), wj, -2.0)), 1.0) * x;
	} else {
		tmp = wj - pow(((1.0 + wj) / wj), -1.0);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.011)
		tmp = Float64(fma(wj, fma(Float64(wj / x), Float64(1.0 - wj), fma(fma(-2.6666666666666665, wj, 2.5), wj, -2.0)), 1.0) * x);
	else
		tmp = Float64(wj - (Float64(Float64(1.0 + wj) / wj) ^ -1.0));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.011], N[(N[(wj * N[(N[(wj / x), $MachinePrecision] * N[(1.0 - wj), $MachinePrecision] + N[(N[(-2.6666666666666665 * wj + 2.5), $MachinePrecision] * wj + -2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(wj - N[Power[N[(N[(1.0 + wj), $MachinePrecision] / wj), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.010999999999999999

   1. Initial program 77.9%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.3%

      \[\leadsto \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \frac{1 - wj}{x}, \mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), wj, -2\right)\right), 1\right) \cdot \color{blue}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 97.3%

       \[\leadsto \mathsf{fma}\left(wj, \mathsf{fma}\left(1 - wj, {x}^{-1} \cdot wj, \mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), wj, -2\right)\right), 1\right) \cdot x \]
   11. Step-by-step derivation

   12. Applied rewrites 97.3%

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

   if 0.010999999999999999 < wj

   1. Initial program 20.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(\frac{wj}{x}, 1 - wj, \mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), wj, -2\right)\right), 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(\frac{1 + wj}{wj}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \frac{1 - wj}{x}, \mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), wj, -2\right)\right), 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(\frac{1 + wj}{wj}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.011)
   (*
    (fma
     wj
     (fma wj (/ (- 1.0 wj) x) (fma (fma -2.6666666666666665 wj 2.5) wj -2.0))
     1.0)
    x)
   (- wj (pow (/ (+ 1.0 wj) wj) -1.0))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.011) {
		tmp = fma(wj, fma(wj, ((1.0 - wj) / x), fma(fma(-2.6666666666666665, wj, 2.5), wj, -2.0)), 1.0) * x;
	} else {
		tmp = wj - pow(((1.0 + wj) / wj), -1.0);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.011)
		tmp = Float64(fma(wj, fma(wj, Float64(Float64(1.0 - wj) / x), fma(fma(-2.6666666666666665, wj, 2.5), wj, -2.0)), 1.0) * x);
	else
		tmp = Float64(wj - (Float64(Float64(1.0 + wj) / wj) ^ -1.0));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.011], N[(N[(wj * N[(wj * N[(N[(1.0 - wj), $MachinePrecision] / x), $MachinePrecision] + N[(N[(-2.6666666666666665 * wj + 2.5), $MachinePrecision] * wj + -2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(wj - N[Power[N[(N[(1.0 + wj), $MachinePrecision] / wj), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.010999999999999999

   1. Initial program 77.9%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.3%

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

   if 0.010999999999999999 < wj

   1. Initial program 20.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \frac{1 - wj}{x}, \mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), wj, -2\right)\right), 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(\frac{1 + wj}{wj}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(wj + x \cdot \mathsf{fma}\left(2.5, wj, -2\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(\frac{1 + wj}{wj}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.011)
   (fma (+ wj (* x (fma 2.5 wj -2.0))) wj x)
   (- wj (pow (/ (+ 1.0 wj) wj) -1.0))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.011) {
		tmp = fma((wj + (x * fma(2.5, wj, -2.0))), wj, x);
	} else {
		tmp = wj - pow(((1.0 + wj) / wj), -1.0);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.011)
		tmp = fma(Float64(wj + Float64(x * fma(2.5, wj, -2.0))), wj, x);
	else
		tmp = Float64(wj - (Float64(Float64(1.0 + wj) / wj) ^ -1.0));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.011], N[(N[(wj + N[(x * N[(2.5 * wj + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[Power[N[(N[(1.0 + wj), $MachinePrecision] / wj), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.010999999999999999

   1. Initial program 77.9%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in wj around 0

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)} \]
   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. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. distribute-rgt-out N/A

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

      6. *-commutative N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

   8. Applied rewrites 97.0%

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

   if 0.010999999999999999 < wj

   1. Initial program 20.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(wj + x \cdot \mathsf{fma}\left(2.5, wj, -2\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - {\left(\frac{1 + wj}{wj}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.4% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(wj + x \cdot \mathsf{fma}\left(2.5, wj, -2\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.011)
   (fma (+ wj (* x (fma 2.5 wj -2.0))) wj x)
   (- wj (/ wj (+ 1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.011) {
		tmp = fma((wj + (x * fma(2.5, wj, -2.0))), wj, x);
	} else {
		tmp = wj - (wj / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.011)
		tmp = fma(Float64(wj + Float64(x * fma(2.5, wj, -2.0))), wj, x);
	else
		tmp = Float64(wj - Float64(wj / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.011], N[(N[(wj + N[(x * N[(2.5 * wj + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.010999999999999999

   1. Initial program 77.9%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in wj around 0

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)} \]
   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. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. distribute-rgt-out N/A

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

      6. *-commutative N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

   8. Applied rewrites 97.0%

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

   if 0.010999999999999999 < wj

   1. Initial program 20.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 10: 96.6% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1\right) \cdot wj, wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.011) (fma (* (fma 2.5 x 1.0) wj) wj x) (- wj (/ wj (+ 1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.011) {
		tmp = fma((fma(2.5, x, 1.0) * wj), wj, x);
	} else {
		tmp = wj - (wj / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.011)
		tmp = fma(Float64(fma(2.5, x, 1.0) * wj), wj, x);
	else
		tmp = Float64(wj - Float64(wj / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.011], N[(N[(N[(2.5 * x + 1.0), $MachinePrecision] * wj), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.010999999999999999

   1. Initial program 77.9%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in wj around 0

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)} \]
   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. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. distribute-rgt-out N/A

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

      6. *-commutative N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

   8. Applied rewrites 97.0%

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

   9. Taylor expanded in wj around inf

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

   11. Applied rewrites 96.8%

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

   if 0.010999999999999999 < wj

   1. Initial program 20.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 11: 95.6% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.8%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 95.6%

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

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. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. distribute-rgt-in N/A

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

   5. distribute-rgt-out N/A

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

   6. *-commutative N/A

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

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

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. distribute-rgt-in N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

8. Applied rewrites 95.3%

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

9. Taylor expanded in wj around inf

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

11. Applied rewrites 95.1%

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

Alternative 12: 85.1% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.8%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 95.6%

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

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. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. distribute-rgt-in N/A

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

   5. distribute-rgt-out N/A

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

   6. *-commutative N/A

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

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

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. distribute-rgt-in N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

8. Applied rewrites 95.3%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 84.8%

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

Alternative 13: 15.1% accurate, 27.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-76}:\\ \;\;\;\;-1 + wj\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (if (<= x -3.8e-76) (+ -1.0 wj) (* wj wj)))

C

double code(double wj, double x) {
	double tmp;
	if (x <= -3.8e-76) {
		tmp = -1.0 + wj;
	} 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 <= (-3.8d-76)) then
        tmp = (-1.0d0) + wj
    else
        tmp = wj * wj
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (x <= -3.8e-76) {
		tmp = -1.0 + wj;
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if x <= -3.8e-76:
		tmp = -1.0 + wj
	else:
		tmp = wj * wj
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= -3.8e-76)
		tmp = Float64(-1.0 + wj);
	else
		tmp = Float64(wj * wj);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (x <= -3.8e-76)
		tmp = -1.0 + wj;
	else
		tmp = wj * wj;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[x, -3.8e-76], N[(-1.0 + wj), $MachinePrecision], N[(wj * wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8000000000000002e-76

   1. Initial program 94.5%

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

   3. Taylor expanded in wj around inf

      \[\leadsto \color{blue}{wj \cdot \left(1 - \frac{1}{wj}\right)} \]
   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 8.3

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

   5. Applied rewrites 8.3%

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

   if -3.8000000000000002e-76 < x

   1. Initial program 67.1%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 96.2%

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

   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. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-in N/A

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

      5. distribute-rgt-out N/A

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

      6. *-commutative N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

   8. Applied rewrites 95.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 18.8%

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

Alternative 14: 84.9% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.8%

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

3. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 84.8

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

5. Applied rewrites 84.8%

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

Alternative 15: 84.4% accurate, 55.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.8%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 95.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 95.6%

   \[\leadsto \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \frac{1 - wj}{x}, \mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), wj, -2\right)\right), 1\right) \cdot \color{blue}{x} \]
9. Step-by-step derivation

10. Applied rewrites 95.6%

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

11. Taylor expanded in wj around 0

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

13. Applied rewrites 84.7%

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

Alternative 16: 4.0% 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 76.8%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 5.0

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

5. Applied rewrites 5.0%

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

Alternative 17: 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 76.8%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 5.0

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

5. Applied rewrites 5.0%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 3.6%

   \[\leadsto -1 \]
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 2024321 
(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))))))