Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.8% → 99.4%

Time: 9.1s

Alternatives: 17

Speedup: 27.6×

• could not determine a ground truth

  1. wj = -1.068796459089224e+242
  2. x = -7.872897524555104e+35

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: 77.8% 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: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - wj \cdot wj\\ \mathbf{if}\;wj \leq -1.02 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{t\_0}, \left(1 - wj\right) \cdot \frac{\mathsf{fma}\left(e^{wj}, wj, -x\right)}{e^{wj}}, wj\right)\\ \mathbf{elif}\;wj \leq 5.6 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, wj, -2\right), x, wj\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj - 1, \frac{wj}{t\_0 \cdot x}, \frac{wj}{x} + \frac{1 - wj}{t\_0 \cdot e^{wj}}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (- 1.0 (* wj wj))))
   (if (<= wj -1.02e-7)
     (fma (/ -1.0 t_0) (* (- 1.0 wj) (/ (fma (exp wj) wj (- x)) (exp wj))) wj)
     (if (<= wj 5.6e-14)
       (fma (fma (fma 2.5 wj -2.0) x wj) wj x)
       (*
        (fma
         (- wj 1.0)
         (/ wj (* t_0 x))
         (+ (/ wj x) (/ (- 1.0 wj) (* t_0 (exp wj)))))
        x)))))

C

double code(double wj, double x) {
	double t_0 = 1.0 - (wj * wj);
	double tmp;
	if (wj <= -1.02e-7) {
		tmp = fma((-1.0 / t_0), ((1.0 - wj) * (fma(exp(wj), wj, -x) / exp(wj))), wj);
	} else if (wj <= 5.6e-14) {
		tmp = fma(fma(fma(2.5, wj, -2.0), x, wj), wj, x);
	} else {
		tmp = fma((wj - 1.0), (wj / (t_0 * x)), ((wj / x) + ((1.0 - wj) / (t_0 * exp(wj))))) * x;
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(1.0 - Float64(wj * wj))
	tmp = 0.0
	if (wj <= -1.02e-7)
		tmp = fma(Float64(-1.0 / t_0), Float64(Float64(1.0 - wj) * Float64(fma(exp(wj), wj, Float64(-x)) / exp(wj))), wj);
	elseif (wj <= 5.6e-14)
		tmp = fma(fma(fma(2.5, wj, -2.0), x, wj), wj, x);
	else
		tmp = Float64(fma(Float64(wj - 1.0), Float64(wj / Float64(t_0 * x)), Float64(Float64(wj / x) + Float64(Float64(1.0 - wj) / Float64(t_0 * exp(wj))))) * x);
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(1.0 - N[(wj * wj), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[wj, -1.02e-7], N[(N[(-1.0 / t$95$0), $MachinePrecision] * N[(N[(1.0 - wj), $MachinePrecision] * N[(N[(N[Exp[wj], $MachinePrecision] * wj + (-x)), $MachinePrecision] / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision], If[LessEqual[wj, 5.6e-14], N[(N[(N[(2.5 * wj + -2.0), $MachinePrecision] * x + wj), $MachinePrecision] * wj + x), $MachinePrecision], N[(N[(N[(wj - 1.0), $MachinePrecision] * N[(wj / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] + N[(N[(wj / x), $MachinePrecision] + N[(N[(1.0 - wj), $MachinePrecision] / N[(t$95$0 * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if wj < -1.02e-7

   1. Initial program 42.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 98.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 97.9

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

   6. Applied rewrites 97.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.0

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lower-fma.f64 99.0

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

   8. Applied rewrites 99.0%

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

   if -1.02e-7 < wj < 5.6000000000000001e-14

   1. Initial program 82.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 100.0%

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

   6. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in wj around 0

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

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

       5. metadata-eval N/A

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

       6. *-commutative N/A

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

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

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

       8. metadata-eval N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

   11. Applied rewrites 100.0%

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

   if 5.6000000000000001e-14 < wj

   1. Initial program 76.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 76.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 76.7

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

   6. Applied rewrites 76.7%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 96.9%

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

Alternative 2: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := wj - \frac{t\_0 - x}{e^{wj} + t\_0}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-281} \lor \neg \left(t\_1 \leq 5 \cdot 10^{-302}\right):\\ \;\;\;\;wj - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))) (t_1 (- wj (/ (- t_0 x) (+ (exp wj) t_0)))))
   (if (or (<= t_1 -2e-281) (not (<= t_1 5e-302))) (- wj (- x)) (* wj wj))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double t_1 = wj - ((t_0 - x) / (exp(wj) + t_0));
	double tmp;
	if ((t_1 <= -2e-281) || !(t_1 <= 5e-302)) {
		tmp = wj - -x;
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = wj * exp(wj)
    t_1 = wj - ((t_0 - x) / (exp(wj) + t_0))
    if ((t_1 <= (-2d-281)) .or. (.not. (t_1 <= 5d-302))) then
        tmp = wj - -x
    else
        tmp = wj * wj
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double t_1 = wj - ((t_0 - x) / (Math.exp(wj) + t_0));
	double tmp;
	if ((t_1 <= -2e-281) || !(t_1 <= 5e-302)) {
		tmp = wj - -x;
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	t_1 = wj - ((t_0 - x) / (math.exp(wj) + t_0))
	tmp = 0
	if (t_1 <= -2e-281) or not (t_1 <= 5e-302):
		tmp = wj - -x
	else:
		tmp = wj * wj
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	t_1 = Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0)))
	tmp = 0.0
	if ((t_1 <= -2e-281) || !(t_1 <= 5e-302))
		tmp = Float64(wj - Float64(-x));
	else
		tmp = Float64(wj * wj);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	t_1 = wj - ((t_0 - x) / (exp(wj) + t_0));
	tmp = 0.0;
	if ((t_1 <= -2e-281) || ~((t_1 <= 5e-302)))
		tmp = wj - -x;
	else
		tmp = wj * wj;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-281], N[Not[LessEqual[t$95$1, 5e-302]], $MachinePrecision]], N[(wj - (-x)), $MachinePrecision], N[(wj * wj), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < -2e-281 or 5.00000000000000033e-302 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 94.6%

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

   3. Taylor expanded in wj around 0

      \[\leadsto wj - \color{blue}{-1 \cdot x} \]
   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 87.2

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

   5. Applied rewrites 87.2%

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

   if -2e-281 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 5.00000000000000033e-302

   1. Initial program 7.1%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 100.0%

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

   6. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 42.5%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq -2 \cdot 10^{-281} \lor \neg \left(wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-302}\right):\\ \;\;\;\;wj - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 2 \cdot 10^{-200}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5 + \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(-0.5, x, -0.5\right)\right), wj, -0.5 \cdot x\right), wj, 1 + x\right), wj, -x\right)}{-1 + wj \cdot wj}, 1 - wj, wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 2e-200)
     (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)
     (fma
      (/
       (fma
        (fma
         (fma
          (+ 0.5 (fma 0.6666666666666666 x (fma -0.5 x -0.5)))
          wj
          (* -0.5 x))
         wj
         (+ 1.0 x))
        wj
        (- x))
       (+ -1.0 (* wj wj)))
      (- 1.0 wj)
      wj))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 2e-200) {
		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 = fma((fma(fma(fma((0.5 + fma(0.6666666666666666, x, fma(-0.5, x, -0.5))), wj, (-0.5 * x)), wj, (1.0 + x)), wj, -x) / (-1.0 + (wj * wj))), (1.0 - wj), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 2e-200)
		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 = fma(Float64(fma(fma(fma(Float64(0.5 + fma(0.6666666666666666, x, fma(-0.5, x, -0.5))), wj, Float64(-0.5 * x)), wj, Float64(1.0 + x)), wj, Float64(-x)) / Float64(-1.0 + Float64(wj * wj))), Float64(1.0 - wj), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-200], 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[(N[(N[(N[(N[(N[(0.5 + N[(0.6666666666666666 * x + N[(-0.5 * x + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * wj + N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * wj + (-x)), $MachinePrecision] / N[(-1.0 + N[(wj * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - wj), $MachinePrecision] + wj), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 2e-200

   1. Initial program 70.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 96.5%

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

   if 2e-200 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 96.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 97.3%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{1 + wj}, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x, -0.5 \cdot \left(1 + x\right)\right) + 0.5, wj, -0.5 \cdot x\right), wj, 1 + x\right), wj, -x\right)}, wj\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

   9. Applied rewrites 97.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5 + \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(-0.5, x, -0.5\right)\right), wj, -0.5 \cdot x\right), wj, 1 + x\right), wj, -x\right)}{1 - wj \cdot wj}, 1 - wj, wj\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 2 \cdot 10^{-200}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5 + \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(-0.5, x, -0.5\right)\right), wj, -0.5 \cdot x\right), wj, 1 + x\right), wj, -x\right)}{-1 + wj \cdot wj}, 1 - wj, wj\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 2 \cdot 10^{-200}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{1 + wj}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, wj, -0.5\right) \cdot x, wj, 1 + x\right), wj, -x\right), wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 2e-200)
     (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)
     (fma
      (/ -1.0 (+ 1.0 wj))
      (fma (fma (* (fma 0.16666666666666666 wj -0.5) x) wj (+ 1.0 x)) wj (- x))
      wj))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 2e-200) {
		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 = fma((-1.0 / (1.0 + wj)), fma(fma((fma(0.16666666666666666, wj, -0.5) * x), wj, (1.0 + x)), wj, -x), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 2e-200)
		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 = fma(Float64(-1.0 / Float64(1.0 + wj)), fma(fma(Float64(fma(0.16666666666666666, wj, -0.5) * x), wj, Float64(1.0 + x)), wj, Float64(-x)), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-200], 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[(N[(-1.0 / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.16666666666666666 * wj + -0.5), $MachinePrecision] * x), $MachinePrecision] * wj + N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * wj + (-x)), $MachinePrecision] + wj), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 2e-200

   1. Initial program 70.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 96.5%

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

   if 2e-200 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 96.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 97.3%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{1 + wj}, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x, -0.5 \cdot \left(1 + x\right)\right) + 0.5, wj, -0.5 \cdot x\right), wj, 1 + x\right), wj, -x\right)}, wj\right) \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.3%

       \[\leadsto \mathsf{fma}\left(\frac{-1}{1 + wj}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, wj, -0.5\right) \cdot x, wj, 1 + x\right), wj, -x\right), wj\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 2 \cdot 10^{-200}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, wj, -2\right), x, wj\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{1 + wj}, \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(wj, -0.5, 1\right), 1\right), wj, -x\right), wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 2e-200)
     (fma (fma (fma 2.5 wj -2.0) x wj) wj x)
     (fma
      (/ -1.0 (+ 1.0 wj))
      (fma (fma x (fma wj -0.5 1.0) 1.0) wj (- x))
      wj))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 2e-200) {
		tmp = fma(fma(fma(2.5, wj, -2.0), x, wj), wj, x);
	} else {
		tmp = fma((-1.0 / (1.0 + wj)), fma(fma(x, fma(wj, -0.5, 1.0), 1.0), wj, -x), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 2e-200)
		tmp = fma(fma(fma(2.5, wj, -2.0), x, wj), wj, x);
	else
		tmp = fma(Float64(-1.0 / Float64(1.0 + wj)), fma(fma(x, fma(wj, -0.5, 1.0), 1.0), wj, Float64(-x)), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-200], N[(N[(N[(2.5 * wj + -2.0), $MachinePrecision] * x + wj), $MachinePrecision] * wj + x), $MachinePrecision], N[(N[(-1.0 / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision] * N[(N[(x * N[(wj * -0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * wj + (-x)), $MachinePrecision] + wj), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 2e-200

   1. Initial program 70.8%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 96.5%

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

   6. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 96.5%

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

   9. Taylor expanded in wj around 0

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

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

       5. metadata-eval N/A

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

       6. *-commutative N/A

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

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

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

       8. metadata-eval N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

   11. Applied rewrites 96.5%

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

   if 2e-200 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 96.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate--l+ N/A

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

      5. +-commutative N/A

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

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

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

      7. distribute-rgt-out-- N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-out N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 96.8

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

   7. Applied rewrites 96.8%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{1 + wj}, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(wj, -0.5, 1\right), 1\right), wj, -x\right)}, wj\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - wj \cdot wj\\ \mathbf{if}\;wj \leq -1.02 \cdot 10^{-7} \lor \neg \left(wj \leq 5.6 \cdot 10^{-14}\right):\\ \;\;\;\;\mathsf{fma}\left(wj - 1, \frac{wj}{t\_0 \cdot x}, \frac{wj}{x} + \frac{1 - wj}{t\_0 \cdot e^{wj}}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, wj, -2\right), x, wj\right), wj, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (- 1.0 (* wj wj))))
   (if (or (<= wj -1.02e-7) (not (<= wj 5.6e-14)))
     (*
      (fma
       (- wj 1.0)
       (/ wj (* t_0 x))
       (+ (/ wj x) (/ (- 1.0 wj) (* t_0 (exp wj)))))
      x)
     (fma (fma (fma 2.5 wj -2.0) x wj) wj x))))

C

double code(double wj, double x) {
	double t_0 = 1.0 - (wj * wj);
	double tmp;
	if ((wj <= -1.02e-7) || !(wj <= 5.6e-14)) {
		tmp = fma((wj - 1.0), (wj / (t_0 * x)), ((wj / x) + ((1.0 - wj) / (t_0 * exp(wj))))) * x;
	} else {
		tmp = fma(fma(fma(2.5, wj, -2.0), x, wj), wj, x);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(1.0 - Float64(wj * wj))
	tmp = 0.0
	if ((wj <= -1.02e-7) || !(wj <= 5.6e-14))
		tmp = Float64(fma(Float64(wj - 1.0), Float64(wj / Float64(t_0 * x)), Float64(Float64(wj / x) + Float64(Float64(1.0 - wj) / Float64(t_0 * exp(wj))))) * x);
	else
		tmp = fma(fma(fma(2.5, wj, -2.0), x, wj), wj, x);
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(1.0 - N[(wj * wj), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[wj, -1.02e-7], N[Not[LessEqual[wj, 5.6e-14]], $MachinePrecision]], N[(N[(N[(wj - 1.0), $MachinePrecision] * N[(wj / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] + N[(N[(wj / x), $MachinePrecision] + N[(N[(1.0 - wj), $MachinePrecision] / N[(t$95$0 * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(2.5 * wj + -2.0), $MachinePrecision] * x + wj), $MachinePrecision] * wj + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if wj < -1.02e-7 or 5.6000000000000001e-14 < wj

   1. Initial program 60.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 86.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 86.7

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

   6. Applied rewrites 86.7%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 97.8%

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

   if -1.02e-7 < wj < 5.6000000000000001e-14

   1. Initial program 82.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 100.0%

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

   6. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in wj around 0

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

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

       5. metadata-eval N/A

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

       6. *-commutative N/A

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

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

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

       8. metadata-eval N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

   11. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -1.02 \cdot 10^{-7} \lor \neg \left(wj \leq 5.6 \cdot 10^{-14}\right):\\ \;\;\;\;\mathsf{fma}\left(wj - 1, \frac{wj}{\left(1 - wj \cdot wj\right) \cdot x}, \frac{wj}{x} + \frac{1 - wj}{\left(1 - wj \cdot wj\right) \cdot e^{wj}}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, wj, -2\right), x, wj\right), wj, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -1.02 \cdot 10^{-7} \lor \neg \left(wj \leq 1.1 \cdot 10^{-8}\right):\\ \;\;\;\;wj - \mathsf{fma}\left(\frac{x}{1 + wj}, \frac{wj}{x}, \frac{\frac{x}{-1 - wj}}{e^{wj}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, wj, -2\right), x, wj\right), wj, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= wj -1.02e-7) (not (<= wj 1.1e-8)))
   (- wj (fma (/ x (+ 1.0 wj)) (/ wj x) (/ (/ x (- -1.0 wj)) (exp wj))))
   (fma (fma (fma 2.5 wj -2.0) x wj) wj x)))

C

double code(double wj, double x) {
	double tmp;
	if ((wj <= -1.02e-7) || !(wj <= 1.1e-8)) {
		tmp = wj - fma((x / (1.0 + wj)), (wj / x), ((x / (-1.0 - wj)) / exp(wj)));
	} else {
		tmp = fma(fma(fma(2.5, wj, -2.0), x, wj), wj, x);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if ((wj <= -1.02e-7) || !(wj <= 1.1e-8))
		tmp = Float64(wj - fma(Float64(x / Float64(1.0 + wj)), Float64(wj / x), Float64(Float64(x / Float64(-1.0 - wj)) / exp(wj))));
	else
		tmp = fma(fma(fma(2.5, wj, -2.0), x, wj), wj, x);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[wj, -1.02e-7], N[Not[LessEqual[wj, 1.1e-8]], $MachinePrecision]], N[(wj - N[(N[(x / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision] * N[(wj / x), $MachinePrecision] + N[(N[(x / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision] / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.5 * wj + -2.0), $MachinePrecision] * x + wj), $MachinePrecision] * wj + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < -1.02e-7 or 1.0999999999999999e-8 < wj

   1. Initial program 55.9%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

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

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. mul-1-neg N/A

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

   5. Applied rewrites 96.8%

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

   if -1.02e-7 < wj < 1.0999999999999999e-8

   1. Initial program 82.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 100.0%

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

   6. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in wj around 0

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

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

       5. metadata-eval N/A

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

       6. *-commutative N/A

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

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

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

       8. metadata-eval N/A

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 N/A

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

   11. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq -1.02 \cdot 10^{-7} \lor \neg \left(wj \leq 1.1 \cdot 10^{-8}\right):\\ \;\;\;\;wj - \mathsf{fma}\left(\frac{x}{1 + wj}, \frac{wj}{x}, \frac{\frac{x}{-1 - wj}}{e^{wj}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, wj, -2\right), x, wj\right), wj, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -0.00032:\\ \;\;\;\;wj - \frac{x}{\left(-1 - wj\right) \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -0.00032)
   (- wj (/ x (* (- -1.0 wj) (exp wj))))
   (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)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -0.00032) {
		tmp = wj - (x / ((-1.0 - wj) * exp(wj)));
	} else {
		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);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -0.00032)
		tmp = Float64(wj - Float64(x / Float64(Float64(-1.0 - wj) * exp(wj))));
	else
		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);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -0.00032], N[(wj - N[(x / N[(N[(-1.0 - wj), $MachinePrecision] * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if wj < -3.20000000000000026e-4

   1. Initial program 37.5%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.8%

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

   if -3.20000000000000026e-4 < wj

   1. Initial program 82.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 97.5% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -1.0)
   (- wj (/ x (* (- wj) (exp wj))))
   (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)))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if wj < -1

   1. Initial program 16.7%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in wj around inf

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

   10. Applied rewrites 88.2%

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

   if -1 < wj

   1. Initial program 82.5%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 97.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 96.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right) \end{array} \]

FPCore

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

C

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

Julia

function code(wj, x)
	return 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)
end

Wolfram

code[wj_, x_] := 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]

Derivation

1. Initial program 80.9%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 95.2%

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

Alternative 11: 96.1% accurate, 17.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.9%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 95.2%

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

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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

8. Applied rewrites 94.9%

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

9. Taylor expanded in wj around 0

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

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

    5. metadata-eval N/A

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

    6. *-commutative N/A

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

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

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

    8. metadata-eval N/A

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

    9. +-commutative N/A

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

    10. *-commutative N/A

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

    11. lower-fma.f64 N/A

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

11. Applied rewrites 94.9%

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

Alternative 12: 95.9% accurate, 25.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.9%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 95.2%

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

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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

8. Applied rewrites 94.9%

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

9. Taylor expanded in wj around 0

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

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

    5. metadata-eval N/A

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

    6. *-commutative N/A

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

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

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

    8. metadata-eval N/A

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

    9. +-commutative N/A

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

    10. *-commutative N/A

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

    11. lower-fma.f64 N/A

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

11. Applied rewrites 94.9%

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

12. Taylor expanded in wj around 0

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

14. Applied rewrites 94.6%

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

Alternative 13: 84.8% 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 80.9%

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

3. Taylor expanded in wj around 0

   \[\leadsto \color{blue}{x + -2 \cdot \left(wj \cdot x\right)} \]
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 87.4

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

5. Applied rewrites 87.4%

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

Alternative 14: 84.8% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.9%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 95.2%

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

6. Taylor expanded in wj around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. +-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 87.4

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

8. Applied rewrites 87.4%

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

Alternative 15: 13.9% accurate, 55.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 95.2%

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

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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

8. Applied rewrites 94.9%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 9.8%

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

Alternative 16: 4.3% 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 80.9%

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

3. Taylor expanded in wj around inf

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

   5. lft-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 3.8

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

5. Applied rewrites 3.8%

   \[\leadsto \color{blue}{-1 + wj} \]
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 80.9%

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

3. Taylor expanded in wj around inf

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

   5. lft-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 3.8

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

5. Applied rewrites 3.8%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 3.3%

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

Developer Target 1: 78.8% 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 2024299 
(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))))))