Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.6% → 96.9%

Time: 4.5s

Alternatives: 10

Speedup: 48.6×

• could not determine a ground truth

  1. wj = 2.3914886819169226e+270
  2. x = -1.230453500903031e-308

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(wj, x)
use fmin_fmax_functions
    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: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(wj + x \cdot \left(2.5 \cdot wj - 2\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{\frac{wj}{wj - -1} - \frac{x}{\left(wj - -1\right) \cdot e^{wj}}}{wj}\right) \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 2e+305)
     (fma (+ wj (* x (- (* 2.5 wj) 2.0))) wj x)
     (*
      (- 1.0 (/ (- (/ wj (- wj -1.0)) (/ x (* (- wj -1.0) (exp wj)))) 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+305) {
		tmp = fma((wj + (x * ((2.5 * wj) - 2.0))), wj, x);
	} else {
		tmp = (1.0 - (((wj / (wj - -1.0)) - (x / ((wj - -1.0) * exp(wj)))) / 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+305)
		tmp = fma(Float64(wj + Float64(x * Float64(Float64(2.5 * wj) - 2.0))), wj, x);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(wj / Float64(wj - -1.0)) - Float64(x / Float64(Float64(wj - -1.0) * exp(wj)))) / 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+305], N[(N[(wj + N[(x * N[(N[(2.5 * wj), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(wj / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[(wj - -1.0), $MachinePrecision] * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / wj), $MachinePrecision]), $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))))) < 1.9999999999999999e305

   1. Initial program 77.6%

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

   2. Taylor expanded in wj around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 95.7

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot \color{blue}{x}\right) \]

   4. Applied rewrites 95.7%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 95.7

         \[\leadsto \mathsf{fma}\left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x, \color{blue}{wj}, x\right) \]

   6. Applied rewrites 95.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 95.7

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

   9. Applied rewrites 95.7%

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

   if 1.9999999999999999e305 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 77.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. 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-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\left(1 - \frac{\frac{wj}{wj - -1} - \frac{x}{\left(wj - -1\right) \cdot e^{wj}}}{wj}\right) \cdot wj} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(wj + x \cdot \left(2.5 \cdot wj - 2\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(1 + -1 \cdot \frac{1 + \frac{x}{e^{wj}}}{wj}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 2e+305)
     (fma (+ wj (* x (- (* 2.5 wj) 2.0))) wj x)
     (- wj (+ 1.0 (* -1.0 (/ (+ 1.0 (/ x (exp 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+305) {
		tmp = fma((wj + (x * ((2.5 * wj) - 2.0))), wj, x);
	} else {
		tmp = wj - (1.0 + (-1.0 * ((1.0 + (x / exp(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+305)
		tmp = fma(Float64(wj + Float64(x * Float64(Float64(2.5 * wj) - 2.0))), wj, x);
	else
		tmp = Float64(wj - Float64(1.0 + Float64(-1.0 * Float64(Float64(1.0 + Float64(x / exp(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+305], N[(N[(wj + N[(x * N[(N[(2.5 * wj), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(1.0 + N[(-1.0 * N[(N[(1.0 + N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 1.9999999999999999e305

   1. Initial program 77.6%

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

   2. Taylor expanded in wj around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 95.7

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot \color{blue}{x}\right) \]

   4. Applied rewrites 95.7%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 95.7

         \[\leadsto \mathsf{fma}\left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x, \color{blue}{wj}, x\right) \]

   6. Applied rewrites 95.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 95.7

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

   9. Applied rewrites 95.7%

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

   if 1.9999999999999999e305 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 77.6%

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

   2. Taylor expanded in wj around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-exp.f64 5.2

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

   4. Applied rewrites 5.2%

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

Alternative 3: 96.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 2 \cdot 10^{-146}:\\ \;\;\;\;\mathsf{fma}\left(wj + x \cdot \left(2.5 \cdot wj - 2\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\left(wj - -1\right) \cdot wj - x}{\left(wj - -1\right) \cdot \left(wj - -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 2e-146)
     (fma (+ wj (* x (- (* 2.5 wj) 2.0))) wj x)
     (- wj (/ (- (* (- wj -1.0) wj) x) (* (- wj -1.0) (- wj -1.0)))))))

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-146) {
		tmp = fma((wj + (x * ((2.5 * wj) - 2.0))), wj, x);
	} else {
		tmp = wj - ((((wj - -1.0) * wj) - x) / ((wj - -1.0) * (wj - -1.0)));
	}
	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-146)
		tmp = fma(Float64(wj + Float64(x * Float64(Float64(2.5 * wj) - 2.0))), wj, x);
	else
		tmp = Float64(wj - Float64(Float64(Float64(Float64(wj - -1.0) * wj) - x) / Float64(Float64(wj - -1.0) * Float64(wj - -1.0))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-146], N[(N[(wj + N[(x * N[(N[(2.5 * wj), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(N[(N[(wj - -1.0), $MachinePrecision] * wj), $MachinePrecision] - x), $MachinePrecision] / N[(N[(wj - -1.0), $MachinePrecision] * N[(wj - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 2.00000000000000005e-146

   1. Initial program 77.6%

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

   2. Taylor expanded in wj around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 95.7

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot \color{blue}{x}\right) \]

   4. Applied rewrites 95.7%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 95.7

         \[\leadsto \mathsf{fma}\left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x, \color{blue}{wj}, x\right) \]

   6. Applied rewrites 95.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 95.7

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

   9. Applied rewrites 95.7%

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

   if 2.00000000000000005e-146 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 77.6%

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

   2. Taylor expanded in wj around 0

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

      1. lower-+.f64 77.2

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in wj around 0

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

      1. lower-+.f64 76.4

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

   7. Applied rewrites 76.4%

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

   8. Taylor expanded in wj around 0

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

      1. lower-+.f64 77.0

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

   10. Applied rewrites 77.0%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 77.0

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

       4. lift-+.f64 N/A

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

       5. +-commutative N/A

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

       6. metadata-eval N/A

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

       7. sub-flip N/A

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

       8. lift--.f64 77.0

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

       9. lift--.f64 N/A

          \[\leadsto wj - \frac{\left(wj - \color{blue}{-1}\right) \cdot wj - x}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(\left(1 + wj\right) + wj \cdot \left(1 + wj\right)\right)\right)} \]

       10. lift--.f64 N/A

           \[\leadsto wj - \frac{\left(wj - \color{blue}{-1}\right) \cdot wj - x}{\left(1 + wj\right) + \mathsf{Rewrite=>}\left(lift-*.f64, \left(wj \cdot \left(1 + wj\right)\right)\right)} \]

       11. lift--.f64 N/A

           \[\leadsto wj - \frac{\left(wj - \color{blue}{-1}\right) \cdot wj - x}{\mathsf{Rewrite=>}\left(distribute-rgt1-in, \left(\left(wj + 1\right) \cdot \left(1 + wj\right)\right)\right)} \]

       12. lift--.f64 N/A

           \[\leadsto wj - \frac{\left(wj - \color{blue}{-1}\right) \cdot wj - x}{\left(wj + \mathsf{Rewrite<=}\left(metadata-eval, \left(\mathsf{neg}\left(-1\right)\right)\right)\right) \cdot \left(1 + wj\right)} \]

       13. lift--.f64 N/A

           \[\leadsto wj - \frac{\left(wj - \color{blue}{-1}\right) \cdot wj - x}{\mathsf{Rewrite<=}\left(sub-flip, \left(wj - -1\right)\right) \cdot \left(1 + wj\right)} \]

       14. lift--.f64 N/A

           \[\leadsto wj - \frac{\left(wj - \color{blue}{-1}\right) \cdot wj - x}{\mathsf{Rewrite<=}\left(lift--.f64, \left(wj - -1\right)\right) \cdot \left(1 + wj\right)} \]

       15. lift--.f64 N/A

           \[\leadsto wj - \frac{\left(wj - \color{blue}{-1}\right) \cdot wj - x}{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\left(wj - -1\right) \cdot \left(1 + wj\right)\right)\right)} \]

       16. lift--.f64 N/A

           \[\leadsto wj - \frac{\left(wj - \color{blue}{-1}\right) \cdot wj - x}{\left(wj - -1\right) \cdot \mathsf{Rewrite=>}\left(lift-+.f64, \left(1 + wj\right)\right)} \]

       17. lift--.f64 N/A

           \[\leadsto wj - \frac{\left(wj - \color{blue}{-1}\right) \cdot wj - x}{\left(wj - -1\right) \cdot \mathsf{Rewrite=>}\left(+-commutative, \left(wj + 1\right)\right)} \]

       18. lift--.f64 N/A

           \[\leadsto wj - \frac{\left(wj - \color{blue}{-1}\right) \cdot wj - x}{\left(wj - -1\right) \cdot \left(wj + \mathsf{Rewrite<=}\left(metadata-eval, \left(\mathsf{neg}\left(-1\right)\right)\right)\right)} \]

       19. lift--.f64 N/A

           \[\leadsto wj - \frac{\left(wj - \color{blue}{-1}\right) \cdot wj - x}{\left(wj - -1\right) \cdot \mathsf{Rewrite<=}\left(sub-flip, \left(wj - -1\right)\right)} \]

       20. lift--.f64 N/A

           \[\leadsto wj - \frac{\left(wj - \color{blue}{-1}\right) \cdot wj - x}{\left(wj - -1\right) \cdot \mathsf{Rewrite<=}\left(lift--.f64, \left(wj - -1\right)\right)} \]

   12. Applied rewrites 77.0%

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

Alternative 4: 96.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 2 \cdot 10^{-146}:\\ \;\;\;\;\mathsf{fma}\left(wj + x \cdot \left(2.5 \cdot wj - 2\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj \cdot 1 - x}{1 + wj \cdot 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 2e-146)
     (fma (+ wj (* x (- (* 2.5 wj) 2.0))) wj x)
     (- wj (/ (- (* wj 1.0) x) (+ 1.0 (* wj 1.0)))))))

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-146) {
		tmp = fma((wj + (x * ((2.5 * wj) - 2.0))), wj, x);
	} else {
		tmp = wj - (((wj * 1.0) - x) / (1.0 + (wj * 1.0)));
	}
	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-146)
		tmp = fma(Float64(wj + Float64(x * Float64(Float64(2.5 * wj) - 2.0))), wj, x);
	else
		tmp = Float64(wj - Float64(Float64(Float64(wj * 1.0) - x) / Float64(1.0 + Float64(wj * 1.0))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-146], N[(N[(wj + N[(x * N[(N[(2.5 * wj), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(N[(wj * 1.0), $MachinePrecision] - x), $MachinePrecision] / N[(1.0 + N[(wj * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 2.00000000000000005e-146

   1. Initial program 77.6%

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

   2. Taylor expanded in wj around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 95.7

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot \color{blue}{x}\right) \]

   4. Applied rewrites 95.7%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 95.7

         \[\leadsto \mathsf{fma}\left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x, \color{blue}{wj}, x\right) \]

   6. Applied rewrites 95.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 95.7

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

   9. Applied rewrites 95.7%

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

   if 2.00000000000000005e-146 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 77.6%

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

   2. Taylor expanded in wj around 0

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

   4. Applied rewrites 76.7%

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

   5. Taylor expanded in wj around 0

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

   7. Applied rewrites 75.6%

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

   8. Taylor expanded in wj around 0

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

   10. Applied rewrites 76.5%

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

Alternative 5: 96.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(wj + x \cdot \left(2.5 \cdot wj - 2\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + -1 \cdot \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 4.2e-5)
   (fma (+ wj (* x (- (* 2.5 wj) 2.0))) wj x)
   (+ wj (* -1.0 (/ wj (+ 1.0 wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 4.2e-5) {
		tmp = fma((wj + (x * ((2.5 * wj) - 2.0))), wj, x);
	} else {
		tmp = wj + (-1.0 * (wj / (1.0 + wj)));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 4.2e-5)
		tmp = fma(Float64(wj + Float64(x * Float64(Float64(2.5 * wj) - 2.0))), wj, x);
	else
		tmp = Float64(wj + Float64(-1.0 * Float64(wj / Float64(1.0 + wj))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 4.2e-5], N[(N[(wj + N[(x * N[(N[(2.5 * wj), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj + N[(-1.0 * N[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 4.19999999999999977e-5

   1. Initial program 77.6%

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

   2. Taylor expanded in wj around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 95.7

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot \color{blue}{x}\right) \]

   4. Applied rewrites 95.7%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 95.7

         \[\leadsto \mathsf{fma}\left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x, \color{blue}{wj}, x\right) \]

   6. Applied rewrites 95.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 95.7

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

   9. Applied rewrites 95.7%

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

   if 4.19999999999999977e-5 < wj

   1. Initial program 77.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 78.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 77.6%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 6.3

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

   9. Applied rewrites 6.3%

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

Alternative 6: 96.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(wj + x \cdot -2, wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + -1 \cdot \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 4.2e-5)
   (fma (+ wj (* x -2.0)) wj x)
   (+ wj (* -1.0 (/ wj (+ 1.0 wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 4.2e-5) {
		tmp = fma((wj + (x * -2.0)), wj, x);
	} else {
		tmp = wj + (-1.0 * (wj / (1.0 + wj)));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 4.2e-5)
		tmp = fma(Float64(wj + Float64(x * -2.0)), wj, x);
	else
		tmp = Float64(wj + Float64(-1.0 * Float64(wj / Float64(1.0 + wj))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 4.2e-5], N[(N[(wj + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj + N[(-1.0 * N[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 4.19999999999999977e-5

   1. Initial program 77.6%

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

   2. Taylor expanded in wj around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 95.7

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot \color{blue}{x}\right) \]

   4. Applied rewrites 95.7%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 95.7

         \[\leadsto \mathsf{fma}\left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x, \color{blue}{wj}, x\right) \]

   6. Applied rewrites 95.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 95.7

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

   9. Applied rewrites 95.7%

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

   10. Taylor expanded in wj around 0

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

   12. Applied rewrites 95.5%

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

   if 4.19999999999999977e-5 < wj

   1. Initial program 77.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 78.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 77.6%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 6.3

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

   9. Applied rewrites 6.3%

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

Alternative 7: 95.5% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

2. Taylor expanded in wj around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 95.7

      \[\leadsto x + wj \cdot \left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot \color{blue}{x}\right) \]

4. Applied rewrites 95.7%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 95.7

      \[\leadsto \mathsf{fma}\left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x, \color{blue}{wj}, x\right) \]

6. Applied rewrites 95.7%

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

7. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 95.7

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

9. Applied rewrites 95.7%

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

10. Taylor expanded in wj around 0

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

12. Applied rewrites 95.5%

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

Alternative 8: 84.4% accurate, 5.2× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 (/ x (fma 2.0 wj 1.0)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-exp.f64 85.3

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

4. Applied rewrites 85.3%

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

5. Taylor expanded in wj around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 84.6

      \[\leadsto \frac{x}{1 + wj \cdot \left(2 + 1.5 \cdot wj\right)} \]

7. Applied rewrites 84.6%

   \[\leadsto \frac{x}{1 + \color{blue}{wj \cdot \left(2 + 1.5 \cdot wj\right)}} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 84.6

      \[\leadsto \frac{x}{\mathsf{fma}\left(2 + 1.5 \cdot wj, wj, 1\right)} \]

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 84.6

      \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(1.5, wj, 2\right), wj, 1\right)} \]

9. Applied rewrites 84.6%

   \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(1.5, wj, 2\right), wj, 1\right)}} \]

10. Taylor expanded in wj around 0

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

12. Applied rewrites 84.4%

    \[\leadsto \frac{x}{\mathsf{fma}\left(2, wj, 1\right)} \]
13. Add Preprocessing

Alternative 9: 84.3% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

2. Taylor expanded in wj around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 95.7

      \[\leadsto x + wj \cdot \left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot \color{blue}{x}\right) \]

4. Applied rewrites 95.7%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 95.7

      \[\leadsto \mathsf{fma}\left(wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x, \color{blue}{wj}, x\right) \]

6. Applied rewrites 95.7%

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

7. Taylor expanded in wj around 0

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

   1. lower-*.f64 84.3

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

9. Applied rewrites 84.3%

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

Alternative 10: 83.8% accurate, 48.6× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 x)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

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

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 77.6%

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

2. Taylor expanded in wj around 0

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

4. Applied rewrites 83.8%

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

Developer Target 1: 78.5% accurate, 1.2× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(wj, x)
use fmin_fmax_functions
    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 2025159 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :alt
  (! :herbie-platform c (let ((ew (exp wj))) (- wj (- (/ wj (+ wj 1)) (/ x (+ ew (* wj ew)))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))