Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.4% → 98.5%

Time: 4.9s

Alternatives: 12

Speedup: 331.0×

• could not determine a ground truth

  1. wj = 3.425462789334075e+33
  2. x = 6.495528946252539e+191

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{e^{wj}}\\ t_1 := wj \cdot e^{wj}\\ t_2 := t\_1 - x\\ t_3 := wj - \frac{t\_2}{e^{wj} + t\_1}\\ t_4 := t\_0 + 1\\ \mathbf{if}\;t\_3 \leq -20:\\ \;\;\;\;wj - \frac{t\_2}{\left(1 + wj\right) \cdot e^{wj}}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1, wj, 1\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\left(-\frac{t\_4}{wj}\right) - \left(-t\_4\right)}{wj}, -1, t\_0\right) + 1}{wj}, -1, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ x (exp wj)))
        (t_1 (* wj (exp wj)))
        (t_2 (- t_1 x))
        (t_3 (- wj (/ t_2 (+ (exp wj) t_1))))
        (t_4 (+ t_0 1.0)))
   (if (<= t_3 -20.0)
     (- wj (/ t_2 (* (+ 1.0 wj) (exp wj))))
     (if (<= t_3 5e+307)
       (fma (fma (fma -1.0 wj 1.0) wj (* -2.0 x)) wj x)
       (-
        wj
        (fma
         (/ (+ (fma (/ (- (- (/ t_4 wj)) (- t_4)) wj) -1.0 t_0) 1.0) wj)
         -1.0
         1.0))))))

C

double code(double wj, double x) {
	double t_0 = x / exp(wj);
	double t_1 = wj * exp(wj);
	double t_2 = t_1 - x;
	double t_3 = wj - (t_2 / (exp(wj) + t_1));
	double t_4 = t_0 + 1.0;
	double tmp;
	if (t_3 <= -20.0) {
		tmp = wj - (t_2 / ((1.0 + wj) * exp(wj)));
	} else if (t_3 <= 5e+307) {
		tmp = fma(fma(fma(-1.0, wj, 1.0), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - fma(((fma(((-(t_4 / wj) - -t_4) / wj), -1.0, t_0) + 1.0) / wj), -1.0, 1.0);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(x / exp(wj))
	t_1 = Float64(wj * exp(wj))
	t_2 = Float64(t_1 - x)
	t_3 = Float64(wj - Float64(t_2 / Float64(exp(wj) + t_1)))
	t_4 = Float64(t_0 + 1.0)
	tmp = 0.0
	if (t_3 <= -20.0)
		tmp = Float64(wj - Float64(t_2 / Float64(Float64(1.0 + wj) * exp(wj))));
	elseif (t_3 <= 5e+307)
		tmp = fma(fma(fma(-1.0, wj, 1.0), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - fma(Float64(Float64(fma(Float64(Float64(Float64(-Float64(t_4 / wj)) - Float64(-t_4)) / wj), -1.0, t_0) + 1.0) / wj), -1.0, 1.0));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - x), $MachinePrecision]}, Block[{t$95$3 = N[(wj - N[(t$95$2 / N[(N[Exp[wj], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 + 1.0), $MachinePrecision]}, If[LessEqual[t$95$3, -20.0], N[(wj - N[(t$95$2 / N[(N[(1.0 + wj), $MachinePrecision] * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+307], N[(N[(N[(-1.0 * wj + 1.0), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(N[(N[(N[(N[((-N[(t$95$4 / wj), $MachinePrecision]) - (-t$95$4)), $MachinePrecision] / wj), $MachinePrecision] * -1.0 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / wj), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 98.0%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-exp.f64 100.0

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

   3. Applied rewrites 100.0%

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

   if -20 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 5e307

   1. Initial program 73.0%

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

   2. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 98.0

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

   10. Applied rewrites 98.0%

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

   if 5e307 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 2.5%

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

   2. Taylor expanded in wj around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 98.3%

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

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{e^{wj}}\\ t_1 := wj \cdot e^{wj}\\ t_2 := t\_1 - x\\ t_3 := wj - \frac{t\_2}{e^{wj} + t\_1}\\ \mathbf{if}\;t\_3 \leq -20:\\ \;\;\;\;wj - \frac{t\_2}{\left(1 + wj\right) \cdot e^{wj}}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1, wj, 1\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{t\_0 + 1}{wj}, -1, t\_0\right) + 1}{wj}, -1, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ x (exp wj)))
        (t_1 (* wj (exp wj)))
        (t_2 (- t_1 x))
        (t_3 (- wj (/ t_2 (+ (exp wj) t_1)))))
   (if (<= t_3 -20.0)
     (- wj (/ t_2 (* (+ 1.0 wj) (exp wj))))
     (if (<= t_3 5e+307)
       (fma (fma (fma -1.0 wj 1.0) wj (* -2.0 x)) wj x)
       (-
        wj
        (fma (/ (+ (fma (/ (+ t_0 1.0) wj) -1.0 t_0) 1.0) wj) -1.0 1.0))))))

C

double code(double wj, double x) {
	double t_0 = x / exp(wj);
	double t_1 = wj * exp(wj);
	double t_2 = t_1 - x;
	double t_3 = wj - (t_2 / (exp(wj) + t_1));
	double tmp;
	if (t_3 <= -20.0) {
		tmp = wj - (t_2 / ((1.0 + wj) * exp(wj)));
	} else if (t_3 <= 5e+307) {
		tmp = fma(fma(fma(-1.0, wj, 1.0), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - fma(((fma(((t_0 + 1.0) / wj), -1.0, t_0) + 1.0) / wj), -1.0, 1.0);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(x / exp(wj))
	t_1 = Float64(wj * exp(wj))
	t_2 = Float64(t_1 - x)
	t_3 = Float64(wj - Float64(t_2 / Float64(exp(wj) + t_1)))
	tmp = 0.0
	if (t_3 <= -20.0)
		tmp = Float64(wj - Float64(t_2 / Float64(Float64(1.0 + wj) * exp(wj))));
	elseif (t_3 <= 5e+307)
		tmp = fma(fma(fma(-1.0, wj, 1.0), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - fma(Float64(Float64(fma(Float64(Float64(t_0 + 1.0) / wj), -1.0, t_0) + 1.0) / wj), -1.0, 1.0));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - x), $MachinePrecision]}, Block[{t$95$3 = N[(wj - N[(t$95$2 / N[(N[Exp[wj], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -20.0], N[(wj - N[(t$95$2 / N[(N[(1.0 + wj), $MachinePrecision] * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+307], N[(N[(N[(-1.0 * wj + 1.0), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(N[(N[(N[(N[(t$95$0 + 1.0), $MachinePrecision] / wj), $MachinePrecision] * -1.0 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / wj), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 98.0%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-exp.f64 100.0

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

   3. Applied rewrites 100.0%

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

   if -20 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 5e307

   1. Initial program 73.0%

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

   2. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 98.0

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

   10. Applied rewrites 98.0%

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

   if 5e307 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 2.5%

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

   2. Taylor expanded in wj around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 97.4%

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

Alternative 3: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := wj - \frac{t\_0 - x}{e^{wj} + t\_0}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-257}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))) (t_1 (- wj (/ (- t_0 x) (+ (exp wj) t_0)))))
   (if (<= t_1 -1e-257) x (if (<= t_1 0.0) (* wj wj) x))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double t_1 = wj - ((t_0 - x) / (exp(wj) + t_0));
	double tmp;
	if (t_1 <= -1e-257) {
		tmp = x;
	} else if (t_1 <= 0.0) {
		tmp = wj * wj;
	} else {
		tmp = x;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = wj * exp(wj)
    t_1 = wj - ((t_0 - x) / (exp(wj) + t_0))
    if (t_1 <= (-1d-257)) then
        tmp = x
    else if (t_1 <= 0.0d0) then
        tmp = wj * wj
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double t_1 = wj - ((t_0 - x) / (Math.exp(wj) + t_0));
	double tmp;
	if (t_1 <= -1e-257) {
		tmp = x;
	} else if (t_1 <= 0.0) {
		tmp = wj * wj;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	t_1 = wj - ((t_0 - x) / (math.exp(wj) + t_0))
	tmp = 0
	if t_1 <= -1e-257:
		tmp = x
	elif t_1 <= 0.0:
		tmp = wj * wj
	else:
		tmp = x
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	t_1 = Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0)))
	tmp = 0.0
	if (t_1 <= -1e-257)
		tmp = x;
	elseif (t_1 <= 0.0)
		tmp = Float64(wj * wj);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	t_1 = wj - ((t_0 - x) / (exp(wj) + t_0));
	tmp = 0.0;
	if (t_1 <= -1e-257)
		tmp = x;
	elseif (t_1 <= 0.0)
		tmp = wj * wj;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-257], x, If[LessEqual[t$95$1, 0.0], N[(wj * wj), $MachinePrecision], x]]]]

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-258 or 0.0 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 95.4%

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

   2. Taylor expanded in wj around 0

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

   4. Applied rewrites 92.0%

      \[\leadsto \color{blue}{x} \]

   if -9.9999999999999998e-258 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 0.0

   1. Initial program 8.3%

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

   2. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

         \[\leadsto wj \cdot wj \]

      2. lower-*.f64 49.4

         \[\leadsto wj \cdot wj \]

   7. Applied rewrites 49.4%

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

Alternative 4: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1, wj, 1\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\left(-\frac{\left(-\frac{\frac{-x}{e^{wj}} - 1}{wj}\right) - 1}{wj}\right) - 1\right) \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 5e+307)
     (fma (fma (fma -1.0 wj 1.0) wj (* -2.0 x)) wj x)
     (-
      (*
       (- (- (/ (- (- (/ (- (/ (- x) (exp wj)) 1.0) wj)) 1.0) wj)) 1.0)
       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+307) {
		tmp = fma(fma(fma(-1.0, wj, 1.0), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = -((-((-(((-x / exp(wj)) - 1.0) / wj) - 1.0) / wj) - 1.0) * 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+307)
		tmp = fma(fma(fma(-1.0, wj, 1.0), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(Float64(-x) / exp(wj)) - 1.0) / wj)) - 1.0) / wj)) - 1.0) * 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+307], N[(N[(N[(-1.0 * wj + 1.0), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], (-N[(N[((-N[(N[((-N[(N[(N[((-x) / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / wj), $MachinePrecision]) - 1.0), $MachinePrecision] / wj), $MachinePrecision]) - 1.0), $MachinePrecision] * wj), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 5e307

   1. Initial program 79.5%

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

   2. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 97.2

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 97.4

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

   10. Applied rewrites 97.4%

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

   if 5e307 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 2.5%

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

   2. Taylor expanded in wj around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 94.4%

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

Alternative 5: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1, wj, 1\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \mathsf{fma}\left(\frac{\frac{x}{e^{wj}} + 1}{wj}, -1, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 5e+307)
     (fma (fma (fma -1.0 wj 1.0) wj (* -2.0 x)) wj x)
     (- wj (fma (/ (+ (/ x (exp wj)) 1.0) wj) -1.0 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+307) {
		tmp = fma(fma(fma(-1.0, wj, 1.0), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - fma((((x / exp(wj)) + 1.0) / wj), -1.0, 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+307)
		tmp = fma(fma(fma(-1.0, wj, 1.0), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - fma(Float64(Float64(Float64(x / exp(wj)) + 1.0) / wj), -1.0, 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+307], N[(N[(N[(-1.0 * wj + 1.0), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / wj), $MachinePrecision] * -1.0 + 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))))) < 5e307

   1. Initial program 79.5%

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

   2. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 97.2

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 97.4

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

   10. Applied rewrites 97.4%

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

   if 5e307 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 2.5%

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

   2. Taylor expanded in wj around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-exp.f64 94.5

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

   4. Applied rewrites 94.5%

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

Alternative 6: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq -1.8 \cdot 10^{-7}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{\left(1 + wj\right) \cdot e^{wj}}\\ \mathbf{elif}\;wj \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1, wj, 1\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\mathsf{fma}\left(wj - 1, wj, 1\right)}{wj \cdot wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj -1.8e-7)
   (- wj (/ (- (* wj (exp wj)) x) (* (+ 1.0 wj) (exp wj))))
   (if (<= wj 1.0)
     (fma (fma (fma -1.0 wj 1.0) wj (* -2.0 x)) wj x)
     (- wj (/ (fma (- wj 1.0) wj 1.0) (* wj wj))))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= -1.8e-7) {
		tmp = wj - (((wj * exp(wj)) - x) / ((1.0 + wj) * exp(wj)));
	} else if (wj <= 1.0) {
		tmp = fma(fma(fma(-1.0, wj, 1.0), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - (fma((wj - 1.0), wj, 1.0) / (wj * wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= -1.8e-7)
		tmp = Float64(wj - Float64(Float64(Float64(wj * exp(wj)) - x) / Float64(Float64(1.0 + wj) * exp(wj))));
	elseif (wj <= 1.0)
		tmp = fma(fma(fma(-1.0, wj, 1.0), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - Float64(fma(Float64(wj - 1.0), wj, 1.0) / Float64(wj * wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, -1.8e-7], N[(wj - N[(N[(N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(N[(1.0 + wj), $MachinePrecision] * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[wj, 1.0], N[(N[(N[(-1.0 * wj + 1.0), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(N[(wj - 1.0), $MachinePrecision] * wj + 1.0), $MachinePrecision] / N[(wj * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if wj < -1.79999999999999997e-7

   1. Initial program 59.5%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-exp.f64 96.1

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

   3. Applied rewrites 96.1%

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

   if -1.79999999999999997e-7 < wj < 1

   1. Initial program 79.6%

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

   2. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 99.4

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

   10. Applied rewrites 99.4%

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

   if 1 < wj

   1. Initial program 33.5%

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

   2. Taylor expanded in wj around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 80.0%

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

   5. Taylor expanded in wj around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 47.9%

      \[\leadsto wj - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot x - x, -1, 1\right), wj, -\mathsf{fma}\left(2, x, 1\right)\right), wj, x\right) + 1}{\color{blue}{wj \cdot wj}} \]

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 76.7

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

   10. Applied rewrites 76.7%

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

Alternative 7: 97.1% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if wj < 1

   1. Initial program 79.0%

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

   2. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 97.1

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 97.4

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

   10. Applied rewrites 97.4%

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

   if 1 < wj

   1. Initial program 33.5%

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

   2. Taylor expanded in wj around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 80.0%

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

   5. Taylor expanded in wj around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 47.9%

      \[\leadsto wj - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot x - x, -1, 1\right), wj, -\mathsf{fma}\left(2, x, 1\right)\right), wj, x\right) + 1}{\color{blue}{wj \cdot wj}} \]

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 76.7

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

   10. Applied rewrites 76.7%

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

Alternative 8: 96.0% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

2. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. fp-cancel-sub-sign-inv N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. lower--.f64 N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 95.8

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

4. Applied rewrites 95.8%

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

5. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 96.3%

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

8. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. lower-fma.f64 96.0

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

10. Applied rewrites 96.0%

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

Alternative 9: 95.6% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

2. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. fp-cancel-sub-sign-inv N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. lower--.f64 N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 95.8

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

4. Applied rewrites 95.8%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 95.6%

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

Alternative 10: 95.0% accurate, 47.3× speedup

Math

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

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 78.4%

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

2. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. fp-cancel-sub-sign-inv N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. lower--.f64 N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 95.8

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

4. Applied rewrites 95.8%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 95.0%

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

Alternative 11: 84.1% accurate, 331.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 x)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

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

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 78.4%

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

2. Taylor expanded in wj around 0

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

4. Applied rewrites 84.1%

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

Alternative 12: 4.4% accurate, 331.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

double code(double wj, double x) {
	return 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
end function

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 78.4%

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

2. Taylor expanded in wj around inf

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

4. Applied rewrites 4.4%

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

Developer Target 1: 79.3% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (wj x)
 :precision binary64
 (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj)))))))

C

double code(double wj, double x) {
	return wj - ((wj / (wj + 1.0)) - (x / (exp(wj) + (wj * exp(wj)))));
}

Fortran

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 2025093 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

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

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