Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.6% → 98.7%

Time: 9.3s

Alternatives: 15

Speedup: 331.0×

• could not determine a ground truth

  1. wj = 6.576053492960066e+100
  2. x = -2.4305625153478413e+51

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), x, 1 - wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{wj}{\mathsf{fma}\left(wj, x, x\right)} - \frac{e^{-wj}}{wj - -1}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 2e-16)
     (fma
      (fma (fma (fma -2.6666666666666665 wj 2.5) x (- 1.0 wj)) wj (* -2.0 x))
      wj
      x)
     (- wj (* (- (/ wj (fma wj x x)) (/ (exp (- wj)) (- wj -1.0))) x)))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 2e-16) {
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), x, (1.0 - wj)), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - (((wj / fma(wj, x, x)) - (exp(-wj) / (wj - -1.0))) * x);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 2e-16)
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), x, Float64(1.0 - wj)), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - Float64(Float64(Float64(wj / fma(wj, x, x)) - Float64(exp(Float64(-wj)) / Float64(wj - -1.0))) * x));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-16], N[(N[(N[(N[(-2.6666666666666665 * wj + 2.5), $MachinePrecision] * x + N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(N[(wj / N[(wj * x + x), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $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))))) < 2e-16

   1. Initial program 68.4%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.4%

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

   if 2e-16 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 96.3%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. sinh-+-cosh-rev N/A

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

      11. lift-exp.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lift-exp.f64 N/A

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

      14. sinh-+-cosh-rev N/A

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

      15. flip-+ N/A

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

      16. sinh-cosh N/A

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

      17. sinh---cosh-rev N/A

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

      18. associate-*r/ N/A

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

      19. *-rgt-identity N/A

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

      20. lower-/.f64 N/A

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

      21. lower-exp.f64 N/A

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

      22. lower-neg.f64 96.4

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

   4. Applied rewrites 96.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 96.3

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 96.3

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

   6. Applied rewrites 96.3%

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

   7. Taylor expanded in wj around inf

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

   9. Applied rewrites 3.4%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 99.6%

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

Alternative 2: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), x, 1 - wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\mathsf{fma}\left(\frac{x}{e^{wj}}, -1, wj\right)}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 2e-16)
     (fma
      (fma (fma (fma -2.6666666666666665 wj 2.5) x (- 1.0 wj)) wj (* -2.0 x))
      wj
      x)
     (- wj (/ (fma (/ x (exp 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))) <= 2e-16) {
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), x, (1.0 - wj)), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - (fma((x / exp(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))) <= 2e-16)
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), x, Float64(1.0 - wj)), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - Float64(fma(Float64(x / exp(wj)), -1.0, wj) / Float64(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], 2e-16], N[(N[(N[(N[(-2.6666666666666665 * wj + 2.5), $MachinePrecision] * x + N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] * -1.0 + wj), $MachinePrecision] / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 2e-16

   1. Initial program 68.4%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.4%

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

   if 2e-16 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 96.3%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. sinh-+-cosh-rev N/A

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

      11. lift-exp.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lift-exp.f64 N/A

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

      14. sinh-+-cosh-rev N/A

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

      15. flip-+ N/A

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

      16. sinh-cosh N/A

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

      17. sinh---cosh-rev N/A

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

      18. associate-*r/ N/A

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

      19. *-rgt-identity N/A

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

      20. lower-/.f64 N/A

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

      21. lower-exp.f64 N/A

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

      22. lower-neg.f64 96.4

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

   4. Applied rewrites 96.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.5%

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

Alternative 3: 97.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), x, 1 - wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, x \cdot -0.3333333333333333\right), wj, -0.5 \cdot x\right), wj, x\right), wj, wj\right) - x}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 2e-16)
     (fma
      (fma (fma (fma -2.6666666666666665 wj 2.5) x (- 1.0 wj)) wj (* -2.0 x))
      wj
      x)
     (-
      wj
      (/
       (-
        (fma
         (fma (fma (fma 0.5 x (* x -0.3333333333333333)) wj (* -0.5 x)) wj x)
         wj
         wj)
        x)
       (+ 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))) <= 2e-16) {
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), x, (1.0 - wj)), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - ((fma(fma(fma(fma(0.5, x, (x * -0.3333333333333333)), wj, (-0.5 * x)), wj, x), wj, wj) - x) / (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))) <= 2e-16)
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), x, Float64(1.0 - wj)), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - Float64(Float64(fma(fma(fma(fma(0.5, x, Float64(x * -0.3333333333333333)), wj, Float64(-0.5 * x)), wj, x), wj, wj) - x) / Float64(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], 2e-16], N[(N[(N[(N[(-2.6666666666666665 * wj + 2.5), $MachinePrecision] * x + N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(N[(N[(N[(N[(0.5 * x + N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * wj + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision] * wj + wj), $MachinePrecision] - x), $MachinePrecision] / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 2e-16

   1. Initial program 68.4%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.4%

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

   if 2e-16 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 96.3%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. sinh-+-cosh-rev N/A

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

      11. lift-exp.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lift-exp.f64 N/A

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

      14. sinh-+-cosh-rev N/A

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

      15. flip-+ N/A

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

      16. sinh-cosh N/A

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

      17. sinh---cosh-rev N/A

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

      18. associate-*r/ N/A

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

      19. *-rgt-identity N/A

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

      20. lower-/.f64 N/A

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

      21. lower-exp.f64 N/A

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

      22. lower-neg.f64 96.4

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

   4. Applied rewrites 96.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in wj around 0

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

   10. Applied rewrites 98.4%

       \[\leadsto wj - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, x \cdot -0.3333333333333333\right), wj, -0.5 \cdot x\right), wj, x\right), wj, wj\right) - x}{\color{blue}{1} + wj} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.0% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00023:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), x, 1 - wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\mathsf{fma}\left(wj, x, wj\right) - x}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00023)
   (fma
    (fma (fma (fma -2.6666666666666665 wj 2.5) x (- 1.0 wj)) wj (* -2.0 x))
    wj
    x)
   (- wj (/ (- (fma wj x wj) x) (+ 1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00023) {
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), x, (1.0 - wj)), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - ((fma(wj, x, wj) - x) / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00023)
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), x, Float64(1.0 - wj)), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - Float64(Float64(fma(wj, x, wj) - x) / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00023], N[(N[(N[(N[(-2.6666666666666665 * wj + 2.5), $MachinePrecision] * x + N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(N[(wj * x + wj), $MachinePrecision] - x), $MachinePrecision] / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.3000000000000001e-4

   1. Initial program 79.2%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.1%

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

   if 2.3000000000000001e-4 < wj

   1. Initial program 57.4%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. sinh-+-cosh-rev N/A

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

      11. lift-exp.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lift-exp.f64 N/A

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

      14. sinh-+-cosh-rev N/A

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

      15. flip-+ N/A

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

      16. sinh-cosh N/A

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

      17. sinh---cosh-rev N/A

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

      18. associate-*r/ N/A

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

      19. *-rgt-identity N/A

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

      20. lower-/.f64 N/A

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

      21. lower-exp.f64 N/A

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

      22. lower-neg.f64 58.5

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

   4. Applied rewrites 58.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in wj around 0

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

   10. Applied rewrites 94.9%

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

Alternative 5: 96.9% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.000135:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2.5, 1 - wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\mathsf{fma}\left(wj, x, wj\right) - x}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.000135)
   (fma (fma (fma x 2.5 (- 1.0 wj)) wj (* -2.0 x)) wj x)
   (- wj (/ (- (fma wj x wj) x) (+ 1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.000135) {
		tmp = fma(fma(fma(x, 2.5, (1.0 - wj)), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - ((fma(wj, x, wj) - x) / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.000135)
		tmp = fma(fma(fma(x, 2.5, Float64(1.0 - wj)), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - Float64(Float64(fma(wj, x, wj) - x) / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.000135], N[(N[(N[(x * 2.5 + N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(N[(wj * x + wj), $MachinePrecision] - x), $MachinePrecision] / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.35000000000000002e-4

   1. Initial program 79.2%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.9%

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

   10. Applied rewrites 98.9%

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

   if 1.35000000000000002e-4 < wj

   1. Initial program 57.4%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. sinh-+-cosh-rev N/A

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

      11. lift-exp.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lift-exp.f64 N/A

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

      14. sinh-+-cosh-rev N/A

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

      15. flip-+ N/A

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

      16. sinh-cosh N/A

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

      17. sinh---cosh-rev N/A

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

      18. associate-*r/ N/A

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

      19. *-rgt-identity N/A

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

      20. lower-/.f64 N/A

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

      21. lower-exp.f64 N/A

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

      22. lower-neg.f64 58.5

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

   4. Applied rewrites 58.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in wj around 0

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

   10. Applied rewrites 94.9%

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

Alternative 6: 96.9% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00023:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2.5, 1 - wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\mathsf{fma}\left(x, -1, wj\right)}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00023)
   (fma (fma (fma x 2.5 (- 1.0 wj)) wj (* -2.0 x)) wj x)
   (- wj (/ (fma x -1.0 wj) (+ 1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00023) {
		tmp = fma(fma(fma(x, 2.5, (1.0 - wj)), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - (fma(x, -1.0, wj) / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00023)
		tmp = fma(fma(fma(x, 2.5, Float64(1.0 - wj)), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - Float64(fma(x, -1.0, wj) / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00023], N[(N[(N[(x * 2.5 + N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(x * -1.0 + wj), $MachinePrecision] / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.3000000000000001e-4

   1. Initial program 79.2%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.9%

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

   10. Applied rewrites 98.9%

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

   if 2.3000000000000001e-4 < wj

   1. Initial program 57.4%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. sinh-+-cosh-rev N/A

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

      11. lift-exp.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lift-exp.f64 N/A

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

      14. sinh-+-cosh-rev N/A

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

      15. flip-+ N/A

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

      16. sinh-cosh N/A

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

      17. sinh---cosh-rev N/A

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

      18. associate-*r/ N/A

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

      19. *-rgt-identity N/A

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

      20. lower-/.f64 N/A

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

      21. lower-exp.f64 N/A

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

      22. lower-neg.f64 58.5

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

   4. Applied rewrites 58.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in wj around 0

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

   10. Applied rewrites 94.6%

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

Alternative 7: 96.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.000135:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1 - wj, wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\mathsf{fma}\left(x, -1, wj\right)}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.000135)
   (fma (fma (- 1.0 wj) wj (* -2.0 x)) wj x)
   (- wj (/ (fma x -1.0 wj) (+ 1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.000135) {
		tmp = fma(fma((1.0 - wj), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - (fma(x, -1.0, wj) / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.000135)
		tmp = fma(fma(Float64(1.0 - wj), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - Float64(fma(x, -1.0, wj) / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.000135], N[(N[(N[(1.0 - wj), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(x * -1.0 + wj), $MachinePrecision] / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.35000000000000002e-4

   1. Initial program 79.2%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.7%

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

   if 1.35000000000000002e-4 < wj

   1. Initial program 57.4%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. sinh-+-cosh-rev N/A

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

      11. lift-exp.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lift-exp.f64 N/A

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

      14. sinh-+-cosh-rev N/A

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

      15. flip-+ N/A

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

      16. sinh-cosh N/A

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

      17. sinh---cosh-rev N/A

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

      18. associate-*r/ N/A

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

      19. *-rgt-identity N/A

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

      20. lower-/.f64 N/A

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

      21. lower-exp.f64 N/A

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

      22. lower-neg.f64 58.5

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

   4. Applied rewrites 58.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in wj around 0

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

   10. Applied rewrites 94.6%

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

Alternative 8: 97.3% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00026:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1 - wj, wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00026)
   (fma (fma (- 1.0 wj) wj (* -2.0 x)) wj x)
   (- wj (/ wj (+ 1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00026) {
		tmp = fma(fma((1.0 - wj), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - (wj / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00026)
		tmp = fma(fma(Float64(1.0 - wj), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - Float64(wj / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00026], N[(N[(N[(1.0 - wj), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.59999999999999977e-4

   1. Initial program 79.2%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.7%

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

   if 2.59999999999999977e-4 < wj

   1. Initial program 57.4%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. sinh-+-cosh-rev N/A

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

      11. lift-exp.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lift-exp.f64 N/A

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

      14. sinh-+-cosh-rev N/A

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

      15. flip-+ N/A

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

      16. sinh-cosh N/A

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

      17. sinh---cosh-rev N/A

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

      18. associate-*r/ N/A

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

      19. *-rgt-identity N/A

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

      20. lower-/.f64 N/A

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

      21. lower-exp.f64 N/A

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

      22. lower-neg.f64 58.5

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

   4. Applied rewrites 58.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 92.3%

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

Alternative 9: 96.9% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00023:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, wj, -2\right), x, wj\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00023)
   (fma (fma (fma 2.5 wj -2.0) x wj) wj x)
   (- wj (/ wj (+ 1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00023) {
		tmp = fma(fma(fma(2.5, wj, -2.0), x, wj), wj, x);
	} else {
		tmp = wj - (wj / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00023)
		tmp = fma(fma(fma(2.5, wj, -2.0), x, wj), wj, x);
	else
		tmp = Float64(wj - Float64(wj / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00023], N[(N[(N[(2.5 * wj + -2.0), $MachinePrecision] * x + wj), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 2.3000000000000001e-4

   1. Initial program 79.2%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 98.4%

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

   if 2.3000000000000001e-4 < wj

   1. Initial program 57.4%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. sinh-+-cosh-rev N/A

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

      11. lift-exp.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lift-exp.f64 N/A

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

      14. sinh-+-cosh-rev N/A

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

      15. flip-+ N/A

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

      16. sinh-cosh N/A

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

      17. sinh---cosh-rev N/A

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

      18. associate-*r/ N/A

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

      19. *-rgt-identity N/A

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

      20. lower-/.f64 N/A

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

      21. lower-exp.f64 N/A

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

      22. lower-neg.f64 58.5

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

   4. Applied rewrites 58.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 92.3%

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

Alternative 10: 96.8% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00017:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, x, wj\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj}{1 + wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00017) (fma (fma -2.0 x wj) wj x) (- wj (/ wj (+ 1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00017) {
		tmp = fma(fma(-2.0, x, wj), wj, x);
	} else {
		tmp = wj - (wj / (1.0 + wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00017)
		tmp = fma(fma(-2.0, x, wj), wj, x);
	else
		tmp = Float64(wj - Float64(wj / Float64(1.0 + wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00017], N[(N[(-2.0 * x + wj), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(wj / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.7e-4

   1. Initial program 79.2%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in wj around 0

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

   11. Applied rewrites 98.2%

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

   if 1.7e-4 < wj

   1. Initial program 57.4%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. distribute-rgt-in N/A

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

      9. +-commutative N/A

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

      10. sinh-+-cosh-rev N/A

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

      11. lift-exp.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lift-exp.f64 N/A

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

      14. sinh-+-cosh-rev N/A

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

      15. flip-+ N/A

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

      16. sinh-cosh N/A

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

      17. sinh---cosh-rev N/A

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

      18. associate-*r/ N/A

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

      19. *-rgt-identity N/A

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

      20. lower-/.f64 N/A

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

      21. lower-exp.f64 N/A

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

      22. lower-neg.f64 58.5

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

   4. Applied rewrites 58.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 92.3%

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

Alternative 11: 81.8% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-266} \lor \neg \left(x \leq 1.22 \cdot 10^{-188}\right):\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (or (<= x -2.8e-266) (not (<= x 1.22e-188))) x (* wj wj)))

C

double code(double wj, double x) {
	double tmp;
	if ((x <= -2.8e-266) || !(x <= 1.22e-188)) {
		tmp = x;
	} else {
		tmp = wj * wj;
	}
	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) :: tmp
    if ((x <= (-2.8d-266)) .or. (.not. (x <= 1.22d-188))) then
        tmp = x
    else
        tmp = wj * wj
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if ((x <= -2.8e-266) || !(x <= 1.22e-188)) {
		tmp = x;
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if (x <= -2.8e-266) or not (x <= 1.22e-188):
		tmp = x
	else:
		tmp = wj * wj
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if ((x <= -2.8e-266) || !(x <= 1.22e-188))
		tmp = x;
	else
		tmp = Float64(wj * wj);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if ((x <= -2.8e-266) || ~((x <= 1.22e-188)))
		tmp = x;
	else
		tmp = wj * wj;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[Or[LessEqual[x, -2.8e-266], N[Not[LessEqual[x, 1.22e-188]], $MachinePrecision]], x, N[(wj * wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8e-266 or 1.22e-188 < x

   1. Initial program 85.5%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 90.3%

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

   if -2.8e-266 < x < 1.22e-188

   1. Initial program 22.0%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.2%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-266} \lor \neg \left(x \leq 1.22 \cdot 10^{-188}\right):\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 95.3% accurate, 25.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.0%

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

9. Taylor expanded in wj around 0

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

11. Applied rewrites 95.8%

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

Alternative 13: 94.9% 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 78.6%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 95.2%

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

Alternative 14: 83.6% 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 78.6%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 83.5%

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

Alternative 15: 4.5% 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 78.6%

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

3. Taylor expanded in wj around inf

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

5. Applied rewrites 4.5%

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

Developer Target 1: 78.7% 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 2025025 
(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))))))