Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.6% → 98.7%

Time: 4.6s

Alternatives: 13

Speedup: 48.6×

• could not determine a ground truth

  1. wj = 2.0643499338907584e+231
  2. x = 6.32447738920865e-50

Specification

Math

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

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

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

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: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2.5, 1 - \mathsf{fma}\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(x, 5, 0.6666666666666666 \cdot x\right)\right), wj, wj\right)\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{wj - -1}, e^{-wj}, wj - \frac{wj}{wj - -1}\right)\\ \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 5e-18)
     (fma
      (fma
       (fma
        x
        2.5
        (- 1.0 (fma (fma -3.0 x (fma x 5.0 (* 0.6666666666666666 x))) wj wj)))
       wj
       (* -2.0 x))
      wj
      x)
     (fma (/ x (- wj -1.0)) (exp (- wj)) (- wj (/ wj (- 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))) <= 5e-18) {
		tmp = fma(fma(fma(x, 2.5, (1.0 - fma(fma(-3.0, x, fma(x, 5.0, (0.6666666666666666 * x))), wj, wj))), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = fma((x / (wj - -1.0)), exp(-wj), (wj - (wj / (wj - -1.0))));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 5e-18)
		tmp = fma(fma(fma(x, 2.5, Float64(1.0 - fma(fma(-3.0, x, fma(x, 5.0, Float64(0.6666666666666666 * x))), wj, wj))), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = fma(Float64(x / Float64(wj - -1.0)), exp(Float64(-wj)), Float64(wj - Float64(wj / Float64(wj - -1.0))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-18], N[(N[(N[(x * 2.5 + N[(1.0 - N[(N[(-3.0 * x + N[(x * 5.0 + N[(0.6666666666666666 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(N[(x / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[(-wj)], $MachinePrecision] + N[(wj - N[(wj / 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))))) < 5.00000000000000036e-18

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 96.3%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, \mathsf{fma}\left(-4, x, 1.5 \cdot x\right), 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Applied rewrites 96.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2.5, 1 - \mathsf{fma}\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(x, 5, 0.6666666666666666 \cdot x\right)\right), wj, wj\right)\right), wj, -2 \cdot x\right), \color{blue}{wj}, x\right) \]

   if 5.00000000000000036e-18 < (-.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. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate--r- N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. associate-/r* N/A

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

      11. mult-flip N/A

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

      12. lift-exp.f64 N/A

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

      13. exp-neg N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 89.3%

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

Alternative 2: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 5 \cdot 10^{-18}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-1, wj, x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right)\right) - 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{wj - -1}, e^{-wj}, wj - \frac{wj}{wj - -1}\right)\\ \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 5e-18)
     (+
      x
      (*
       wj
       (-
        (* wj (+ 1.0 (fma -1.0 wj (* x (+ 2.5 (* -2.6666666666666665 wj))))))
        (* 2.0 x))))
     (fma (/ x (- wj -1.0)) (exp (- wj)) (- wj (/ wj (- 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))) <= 5e-18) {
		tmp = x + (wj * ((wj * (1.0 + fma(-1.0, wj, (x * (2.5 + (-2.6666666666666665 * wj)))))) - (2.0 * x)));
	} else {
		tmp = fma((x / (wj - -1.0)), exp(-wj), (wj - (wj / (wj - -1.0))));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 5e-18)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(1.0 + fma(-1.0, wj, Float64(x * Float64(2.5 + Float64(-2.6666666666666665 * wj)))))) - Float64(2.0 * x))));
	else
		tmp = fma(Float64(x / Float64(wj - -1.0)), exp(Float64(-wj)), Float64(wj - Float64(wj / Float64(wj - -1.0))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-18], N[(x + N[(wj * N[(N[(wj * N[(1.0 + N[(-1.0 * wj + N[(x * N[(2.5 + N[(-2.6666666666666665 * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[(-wj)], $MachinePrecision] + N[(wj - N[(wj / 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))))) < 5.00000000000000036e-18

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 96.3%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, \mathsf{fma}\left(-4, x, 1.5 \cdot x\right), 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 + \left(-1 \cdot wj + x \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)\right) - 2 \cdot x\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-1, wj, x \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)\right) - 2 \cdot x\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-1, wj, x \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)\right) - 2 \cdot x\right) \]

      4. lower-+.f64 N/A

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-1, wj, x \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)\right) - 2 \cdot x\right) \]

      5. lower-*.f64 96.3

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

   7. Applied rewrites 96.3%

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

   if 5.00000000000000036e-18 < (-.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. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate--r- N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. associate-/r* N/A

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

      11. mult-flip N/A

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

      12. lift-exp.f64 N/A

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

      13. exp-neg N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 89.3%

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

Alternative 3: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 5 \cdot 10^{-18}:\\ \;\;\;\;x + wj \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-1, wj, x \cdot \left(2.5 + -2.6666666666666665 \cdot wj\right)\right)\right) - 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{wj}{wj - -1} - \frac{x}{\left(wj - -1\right) \cdot e^{wj}}\right)\\ \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 5e-18)
     (+
      x
      (*
       wj
       (-
        (* wj (+ 1.0 (fma -1.0 wj (* x (+ 2.5 (* -2.6666666666666665 wj))))))
        (* 2.0 x))))
     (- wj (- (/ wj (- wj -1.0)) (/ x (* (- wj -1.0) (exp 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))) <= 5e-18) {
		tmp = x + (wj * ((wj * (1.0 + fma(-1.0, wj, (x * (2.5 + (-2.6666666666666665 * wj)))))) - (2.0 * x)));
	} else {
		tmp = wj - ((wj / (wj - -1.0)) - (x / ((wj - -1.0) * exp(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))) <= 5e-18)
		tmp = Float64(x + Float64(wj * Float64(Float64(wj * Float64(1.0 + fma(-1.0, wj, Float64(x * Float64(2.5 + Float64(-2.6666666666666665 * wj)))))) - Float64(2.0 * x))));
	else
		tmp = Float64(wj - Float64(Float64(wj / Float64(wj - -1.0)) - Float64(x / Float64(Float64(wj - -1.0) * exp(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], 5e-18], N[(x + N[(wj * N[(N[(wj * N[(1.0 + N[(-1.0 * wj + N[(x * N[(2.5 + N[(-2.6666666666666665 * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[(wj - -1.0), $MachinePrecision] * N[Exp[wj], $MachinePrecision]), $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))))) < 5.00000000000000036e-18

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 96.3%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, \mathsf{fma}\left(-4, x, 1.5 \cdot x\right), 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 + \left(-1 \cdot wj + x \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)\right) - 2 \cdot x\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-1, wj, x \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)\right) - 2 \cdot x\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-1, wj, x \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)\right) - 2 \cdot x\right) \]

      4. lower-+.f64 N/A

         \[\leadsto x + wj \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-1, wj, x \cdot \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)\right) - 2 \cdot x\right) \]

      5. lower-*.f64 96.3

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

   7. Applied rewrites 96.3%

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

   if 5.00000000000000036e-18 < (-.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

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

   3. Applied rewrites 79.7%

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

Alternative 4: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 10^{-19}:\\ \;\;\;\;x + wj \cdot \mathsf{fma}\left(-2, x, wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{wj}{wj - -1} - \frac{x}{\left(wj - -1\right) \cdot e^{wj}}\right)\\ \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 1e-19)
     (+ x (* wj (fma -2.0 x (* wj (- 1.0 (fma -4.0 x (* 1.5 x)))))))
     (- wj (- (/ wj (- wj -1.0)) (/ x (* (- wj -1.0) (exp 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))) <= 1e-19) {
		tmp = x + (wj * fma(-2.0, x, (wj * (1.0 - fma(-4.0, x, (1.5 * x))))));
	} else {
		tmp = wj - ((wj / (wj - -1.0)) - (x / ((wj - -1.0) * exp(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))) <= 1e-19)
		tmp = Float64(x + Float64(wj * fma(-2.0, x, Float64(wj * Float64(1.0 - fma(-4.0, x, Float64(1.5 * x)))))));
	else
		tmp = Float64(wj - Float64(Float64(wj / Float64(wj - -1.0)) - Float64(x / Float64(Float64(wj - -1.0) * exp(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], 1e-19], N[(x + N[(wj * N[(-2.0 * x + N[(wj * N[(1.0 - N[(-4.0 * x + N[(1.5 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[(wj - -1.0), $MachinePrecision] * N[Exp[wj], $MachinePrecision]), $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))))) < 9.9999999999999998e-20

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 96.3%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, \mathsf{fma}\left(-4, x, 1.5 \cdot x\right), 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   5. Taylor expanded in wj around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 95.8

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

   7. Applied rewrites 95.8%

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

   if 9.9999999999999998e-20 < (-.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

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

   3. Applied rewrites 79.7%

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

Alternative 5: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 5 \cdot 10^{-18}:\\ \;\;\;\;x + wj \cdot \mathsf{fma}\left(-2, x, wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{-wj}, x, \mathsf{fma}\left(wj, wj, wj\right) - wj\right)}{wj - -1}\\ \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 5e-18)
     (+ x (* wj (fma -2.0 x (* wj (- 1.0 (fma -4.0 x (* 1.5 x)))))))
     (/ (fma (exp (- wj)) x (- (fma wj wj wj) wj)) (- 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))) <= 5e-18) {
		tmp = x + (wj * fma(-2.0, x, (wj * (1.0 - fma(-4.0, x, (1.5 * x))))));
	} else {
		tmp = fma(exp(-wj), x, (fma(wj, wj, wj) - wj)) / (wj - -1.0);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 5e-18)
		tmp = Float64(x + Float64(wj * fma(-2.0, x, Float64(wj * Float64(1.0 - fma(-4.0, x, Float64(1.5 * x)))))));
	else
		tmp = Float64(fma(exp(Float64(-wj)), x, Float64(fma(wj, wj, wj) - wj)) / Float64(wj - -1.0));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-18], N[(x + N[(wj * N[(-2.0 * x + N[(wj * N[(1.0 - N[(-4.0 * x + N[(1.5 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-wj)], $MachinePrecision] * x + N[(N[(wj * wj + wj), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision] / N[(wj - -1.0), $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))))) < 5.00000000000000036e-18

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 96.3%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, \mathsf{fma}\left(-4, x, 1.5 \cdot x\right), 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   5. Taylor expanded in wj around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 95.8

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

   7. Applied rewrites 95.8%

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

   if 5.00000000000000036e-18 < (-.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. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate--r- N/A

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

      6. +-commutative N/A

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

      7. mult-flip N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 89.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sub-to-fraction N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-flip N/A

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

      13. metadata-eval N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. lower-fma.f64 89.3

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

   5. Applied rewrites 89.3%

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

Alternative 6: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 10^{-19}:\\ \;\;\;\;x + wj \cdot \mathsf{fma}\left(-2, x, wj \cdot \left(1 - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj - -1}\\ \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 1e-19)
     (+ x (* wj (fma -2.0 x (* wj (- 1.0 (fma -4.0 x (* 1.5 x)))))))
     (- wj (/ (- wj (/ x (exp wj))) (- 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))) <= 1e-19) {
		tmp = x + (wj * fma(-2.0, x, (wj * (1.0 - fma(-4.0, x, (1.5 * x))))));
	} else {
		tmp = wj - ((wj - (x / exp(wj))) / (wj - -1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 1e-19)
		tmp = Float64(x + Float64(wj * fma(-2.0, x, Float64(wj * Float64(1.0 - fma(-4.0, x, Float64(1.5 * x)))))));
	else
		tmp = Float64(wj - Float64(Float64(wj - Float64(x / exp(wj))) / Float64(wj - -1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-19], N[(x + N[(wj * N[(-2.0 * x + N[(wj * N[(1.0 - N[(-4.0 * x + N[(1.5 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj - N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 9.9999999999999998e-20

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 96.3%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, \mathsf{fma}\left(-4, x, 1.5 \cdot x\right), 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   5. Taylor expanded in wj around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 95.8

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

   7. Applied rewrites 95.8%

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

   if 9.9999999999999998e-20 < (-.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. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate--r- N/A

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

      6. +-commutative N/A

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

      7. mult-flip N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 89.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-commutative N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. exp-neg N/A

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

      13. lift-exp.f64 N/A

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

      14. mult-flip N/A

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

      15. lift-/.f64 N/A

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

      16. sub-div N/A

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

      17. lower-/.f64 N/A

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

      18. lower--.f64 79.7

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

   5. Applied rewrites 79.7%

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

Alternative 7: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 0.16:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(wj, 1, wj \cdot \left(-wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{wj - -1} \cdot e^{-wj}\\ \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 0.16)
     (fma (fma wj 1.0 (* wj (- wj))) wj x)
     (* (/ x (- wj -1.0)) (exp (- 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))) <= 0.16) {
		tmp = fma(fma(wj, 1.0, (wj * -wj)), wj, x);
	} else {
		tmp = (x / (wj - -1.0)) * exp(-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))) <= 0.16)
		tmp = fma(fma(wj, 1.0, Float64(wj * Float64(-wj))), wj, x);
	else
		tmp = Float64(Float64(x / Float64(wj - -1.0)) * exp(Float64(-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], 0.16], N[(N[(wj * 1.0 + N[(wj * (-wj)), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(N[(x / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[(-wj)], $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))))) < 0.160000000000000003

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 96.3%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, \mathsf{fma}\left(-4, x, 1.5 \cdot x\right), 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Applied rewrites 96.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2.5, 1 - \mathsf{fma}\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(x, 5, 0.6666666666666666 \cdot x\right)\right), wj, wj\right)\right), wj, -2 \cdot x\right), \color{blue}{wj}, x\right) \]

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 95.5

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

   9. Applied rewrites 95.5%

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

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. sub-flip N/A

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

       4. lift-neg.f64 N/A

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

       5. distribute-lft-in N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 95.5

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

   11. Applied rewrites 95.5%

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

   if 0.160000000000000003 < (-.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 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 85.2

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

   4. Applied rewrites 85.2%

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

      1. lift-/.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt1-in N/A

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

      6. times-frac N/A

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

      7. lift-exp.f64 N/A

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

      8. exp-neg N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 86.2

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

   6. Applied rewrites 86.2%

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

Alternative 8: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 0.16:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(wj, 1, wj \cdot \left(-wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{e^{-wj}}{wj - -1}\\ \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 0.16)
     (fma (fma wj 1.0 (* wj (- wj))) wj x)
     (* x (/ (exp (- wj)) (- 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))) <= 0.16) {
		tmp = fma(fma(wj, 1.0, (wj * -wj)), wj, x);
	} else {
		tmp = x * (exp(-wj) / (wj - -1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 0.16)
		tmp = fma(fma(wj, 1.0, Float64(wj * Float64(-wj))), wj, x);
	else
		tmp = Float64(x * Float64(exp(Float64(-wj)) / Float64(wj - -1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.16], N[(N[(wj * 1.0 + N[(wj * (-wj)), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(x * N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 0.160000000000000003

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 96.3%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, \mathsf{fma}\left(-4, x, 1.5 \cdot x\right), 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Applied rewrites 96.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2.5, 1 - \mathsf{fma}\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(x, 5, 0.6666666666666666 \cdot x\right)\right), wj, wj\right)\right), wj, -2 \cdot x\right), \color{blue}{wj}, x\right) \]

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 95.5

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

   9. Applied rewrites 95.5%

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

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. sub-flip N/A

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

       4. lift-neg.f64 N/A

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

       5. distribute-lft-in N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 95.5

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

   11. Applied rewrites 95.5%

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

   if 0.160000000000000003 < (-.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 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 85.2

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

   4. Applied rewrites 85.2%

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

      1. lift-/.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-rgt1-in N/A

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

      6. times-frac N/A

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

      7. lift-exp.f64 N/A

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

      8. exp-neg N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. add-flip N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 86.2

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

   6. Applied rewrites 86.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 86.2

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

   8. Applied rewrites 86.2%

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

Alternative 9: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 0.16:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(wj, 1, wj \cdot \left(-wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(wj - -1\right) \cdot e^{wj}}\\ \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 0.16)
     (fma (fma wj 1.0 (* wj (- wj))) wj x)
     (/ x (* (- wj -1.0) (exp 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))) <= 0.16) {
		tmp = fma(fma(wj, 1.0, (wj * -wj)), wj, x);
	} else {
		tmp = x / ((wj - -1.0) * exp(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))) <= 0.16)
		tmp = fma(fma(wj, 1.0, Float64(wj * Float64(-wj))), wj, x);
	else
		tmp = Float64(x / Float64(Float64(wj - -1.0) * exp(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], 0.16], N[(N[(wj * 1.0 + N[(wj * (-wj)), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(x / N[(N[(wj - -1.0), $MachinePrecision] * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 0.160000000000000003

   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

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   4. Applied rewrites 96.3%

      \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, \mathsf{fma}\left(-4, x, 1.5 \cdot x\right), 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

   6. Applied rewrites 96.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2.5, 1 - \mathsf{fma}\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(x, 5, 0.6666666666666666 \cdot x\right)\right), wj, wj\right)\right), wj, -2 \cdot x\right), \color{blue}{wj}, x\right) \]

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 95.5

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

   9. Applied rewrites 95.5%

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

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. sub-flip N/A

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

       4. lift-neg.f64 N/A

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

       5. distribute-lft-in N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 95.5

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

   11. Applied rewrites 95.5%

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

   if 0.160000000000000003 < (-.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 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 85.2

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

   4. Applied rewrites 85.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. lower-*.f64 N/A

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 86.2

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

   6. Applied rewrites 86.2%

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

Alternative 10: 95.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

4. Applied rewrites 96.3%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, \mathsf{fma}\left(-4, x, 1.5 \cdot x\right), 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

6. Applied rewrites 96.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2.5, 1 - \mathsf{fma}\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(x, 5, 0.6666666666666666 \cdot x\right)\right), wj, wj\right)\right), wj, -2 \cdot x\right), \color{blue}{wj}, x\right) \]

7. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 95.5

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

9. Applied rewrites 95.5%

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

    1. lift-*.f64 N/A

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

    2. lift--.f64 N/A

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

    3. sub-flip N/A

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

    4. lift-neg.f64 N/A

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

    5. distribute-lft-in N/A

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

    6. lower-fma.f64 N/A

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

    7. lower-*.f64 95.5

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

11. Applied rewrites 95.5%

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

Alternative 11: 95.5% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

4. Applied rewrites 96.3%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, \mathsf{fma}\left(-4, x, 1.5 \cdot x\right), 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

6. Applied rewrites 96.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2.5, 1 - \mathsf{fma}\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(x, 5, 0.6666666666666666 \cdot x\right)\right), wj, wj\right)\right), wj, -2 \cdot x\right), \color{blue}{wj}, x\right) \]

7. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 95.5

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

9. Applied rewrites 95.5%

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

Alternative 12: 95.1% accurate, 8.2× speedup

Math

\[\mathsf{fma}\left(wj, wj, x\right) \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

2. Taylor expanded in wj around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

4. Applied rewrites 96.3%

   \[\leadsto \color{blue}{x + wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \mathsf{fma}\left(-3, x, \mathsf{fma}\left(-2, \mathsf{fma}\left(-4, x, 1.5 \cdot x\right), 0.6666666666666666 \cdot x\right)\right)\right)\right)\right) - \mathsf{fma}\left(-4, x, 1.5 \cdot x\right)\right) - 2 \cdot x\right)} \]

6. Applied rewrites 96.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2.5, 1 - \mathsf{fma}\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(x, 5, 0.6666666666666666 \cdot x\right)\right), wj, wj\right)\right), wj, -2 \cdot x\right), \color{blue}{wj}, x\right) \]

7. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 95.5

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

9. Applied rewrites 95.5%

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

10. Taylor expanded in wj around 0

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

12. Applied rewrites 95.1%

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

Alternative 13: 83.9% accurate, 48.6× speedup

Math

\[x \]

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 83.9%

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

Developer Target 1: 78.7% accurate, 1.2× speedup

Math

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

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 2025175 
(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))))))