Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.1% → 99.7%

Time: 8.2s

Alternatives: 15

Speedup: 331.0×

• could not determine a ground truth

  1. wj = -9.941747970962427e+127
  2. x = 1.3755125312099258e-50

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.1% 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: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{wj - 1}{\mathsf{fma}\left(wj, wj, -1\right)}\\ t_1 := wj \cdot e^{wj}\\ t_2 := e^{-wj} \cdot x\\ \mathbf{if}\;wj - \frac{t\_1 - x}{e^{wj} + t\_1} \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, t\_0, \left(wj \cdot wj\right) \cdot \mathsf{fma}\left(\left(1 - wj\right) \cdot wj - 1, wj, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, t\_0, wj - \frac{wj \cdot \left(wj - 1\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (/ (- wj 1.0) (fma wj wj -1.0)))
        (t_1 (* wj (exp wj)))
        (t_2 (* (exp (- wj)) x)))
   (if (<= (- wj (/ (- t_1 x) (+ (exp wj) t_1))) 1e-8)
     (fma t_2 t_0 (* (* wj wj) (fma (- (* (- 1.0 wj) wj) 1.0) wj 1.0)))
     (fma t_2 t_0 (- wj (/ (* wj (- wj 1.0)) (fma wj wj -1.0)))))))

C

double code(double wj, double x) {
	double t_0 = (wj - 1.0) / fma(wj, wj, -1.0);
	double t_1 = wj * exp(wj);
	double t_2 = exp(-wj) * x;
	double tmp;
	if ((wj - ((t_1 - x) / (exp(wj) + t_1))) <= 1e-8) {
		tmp = fma(t_2, t_0, ((wj * wj) * fma((((1.0 - wj) * wj) - 1.0), wj, 1.0)));
	} else {
		tmp = fma(t_2, t_0, (wj - ((wj * (wj - 1.0)) / fma(wj, wj, -1.0))));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(Float64(wj - 1.0) / fma(wj, wj, -1.0))
	t_1 = Float64(wj * exp(wj))
	t_2 = Float64(exp(Float64(-wj)) * x)
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_1 - x) / Float64(exp(wj) + t_1))) <= 1e-8)
		tmp = fma(t_2, t_0, Float64(Float64(wj * wj) * fma(Float64(Float64(Float64(1.0 - wj) * wj) - 1.0), wj, 1.0)));
	else
		tmp = fma(t_2, t_0, Float64(wj - Float64(Float64(wj * Float64(wj - 1.0)) / fma(wj, wj, -1.0))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(N[(wj - 1.0), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[(-wj)], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$1 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-8], N[(t$95$2 * t$95$0 + N[(N[(wj * wj), $MachinePrecision] * N[(N[(N[(N[(1.0 - wj), $MachinePrecision] * wj), $MachinePrecision] - 1.0), $MachinePrecision] * wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * t$95$0 + N[(wj - N[(N[(wj * N[(wj - 1.0), $MachinePrecision]), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 1e-8

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. distribute-rgt1-in N/A

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

      6. remove-double-neg N/A

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

      7. flip3-+ N/A

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

      8. lift-exp.f64 N/A

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

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

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\left(\cosh wj + \sinh wj\right)}} \]

      10. flip-+ N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\frac{\cosh wj \cdot \cosh wj - \sinh wj \cdot \sinh wj}{\cosh wj - \sinh wj}}} \]

      11. sinh-cosh N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{\color{blue}{1}}{\cosh wj - \sinh wj}} \]

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

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{1}{\color{blue}{e^{\mathsf{neg}\left(wj\right)}}}} \]

      13. flip3-+ N/A

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

      14. flip-+ N/A

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

      15. frac-times N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 87.0%

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

   8. Taylor expanded in wj around 0

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

   10. Applied rewrites 100.0%

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

   if 1e-8 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 90.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. distribute-rgt1-in N/A

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

      6. remove-double-neg N/A

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

      7. flip3-+ N/A

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

      8. lift-exp.f64 N/A

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

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

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\left(\cosh wj + \sinh wj\right)}} \]

      10. flip-+ N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\frac{\cosh wj \cdot \cosh wj - \sinh wj \cdot \sinh wj}{\cosh wj - \sinh wj}}} \]

      11. sinh-cosh N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{\color{blue}{1}}{\cosh wj - \sinh wj}} \]

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

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{1}{\color{blue}{e^{\mathsf{neg}\left(wj\right)}}}} \]

      13. flip3-+ N/A

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

      14. flip-+ N/A

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

      15. frac-times N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 92.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.6%

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

4. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(e^{-wj} \cdot x, \frac{wj - 1}{\mathsf{fma}\left(wj, wj, -1\right)}, \left(wj \cdot wj\right) \cdot \mathsf{fma}\left(\left(1 - wj\right) \cdot wj - 1, wj, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\mathsf{fma}\left(\frac{-1}{e^{wj}} \cdot \left(wj - 1\right), x, wj \cdot \left(wj - 1\right)\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 1e-8)
     (fma
      (* (exp (- wj)) x)
      (/ (- wj 1.0) (fma wj wj -1.0))
      (* (* wj wj) (fma (- (* (- 1.0 wj) wj) 1.0) wj 1.0)))
     (-
      wj
      (/
       (fma (* (/ -1.0 (exp wj)) (- wj 1.0)) x (* wj (- wj 1.0)))
       (fma wj wj -1.0))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 1e-8) {
		tmp = fma((exp(-wj) * x), ((wj - 1.0) / fma(wj, wj, -1.0)), ((wj * wj) * fma((((1.0 - wj) * wj) - 1.0), wj, 1.0)));
	} else {
		tmp = wj - (fma(((-1.0 / exp(wj)) * (wj - 1.0)), x, (wj * (wj - 1.0))) / fma(wj, wj, -1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 1e-8)
		tmp = fma(Float64(exp(Float64(-wj)) * x), Float64(Float64(wj - 1.0) / fma(wj, wj, -1.0)), Float64(Float64(wj * wj) * fma(Float64(Float64(Float64(1.0 - wj) * wj) - 1.0), wj, 1.0)));
	else
		tmp = Float64(wj - Float64(fma(Float64(Float64(-1.0 / exp(wj)) * Float64(wj - 1.0)), x, Float64(wj * Float64(wj - 1.0))) / fma(wj, wj, -1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-8], N[(N[(N[Exp[(-wj)], $MachinePrecision] * x), $MachinePrecision] * N[(N[(wj - 1.0), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(wj * wj), $MachinePrecision] * N[(N[(N[(N[(1.0 - wj), $MachinePrecision] * wj), $MachinePrecision] - 1.0), $MachinePrecision] * wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(N[(N[(-1.0 / N[Exp[wj], $MachinePrecision]), $MachinePrecision] * N[(wj - 1.0), $MachinePrecision]), $MachinePrecision] * x + N[(wj * N[(wj - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 1e-8

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. distribute-rgt1-in N/A

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

      6. remove-double-neg N/A

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

      7. flip3-+ N/A

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

      8. lift-exp.f64 N/A

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

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

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\left(\cosh wj + \sinh wj\right)}} \]

      10. flip-+ N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\frac{\cosh wj \cdot \cosh wj - \sinh wj \cdot \sinh wj}{\cosh wj - \sinh wj}}} \]

      11. sinh-cosh N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{\color{blue}{1}}{\cosh wj - \sinh wj}} \]

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

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{1}{\color{blue}{e^{\mathsf{neg}\left(wj\right)}}}} \]

      13. flip3-+ N/A

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

      14. flip-+ N/A

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

      15. frac-times N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 87.0%

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

   8. Taylor expanded in wj around 0

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

   10. Applied rewrites 100.0%

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

   if 1e-8 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 90.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. distribute-rgt1-in N/A

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

      6. remove-double-neg N/A

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

      7. flip3-+ N/A

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

      8. lift-exp.f64 N/A

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

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

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\left(\cosh wj + \sinh wj\right)}} \]

      10. flip-+ N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\frac{\cosh wj \cdot \cosh wj - \sinh wj \cdot \sinh wj}{\cosh wj - \sinh wj}}} \]

      11. sinh-cosh N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{\color{blue}{1}}{\cosh wj - \sinh wj}} \]

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

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{1}{\color{blue}{e^{\mathsf{neg}\left(wj\right)}}}} \]

      13. flip3-+ N/A

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

      14. flip-+ N/A

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

      15. frac-times N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 92.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.6%

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

4. Final simplification 99.8%

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

Alternative 3: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, wj, 0.5\right) \cdot wj - 1, wj, 1\right) \cdot x, \frac{wj - 1}{\mathsf{fma}\left(wj, wj, -1\right)}, \left(wj \cdot wj\right) \cdot \mathsf{fma}\left(\left(1 - wj\right) \cdot wj - 1, wj, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\mathsf{fma}\left(\frac{-1}{e^{wj}} \cdot \left(wj - 1\right), x, wj \cdot \left(wj - 1\right)\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 1e-8)
     (fma
      (* (fma (- (* (fma -0.16666666666666666 wj 0.5) wj) 1.0) wj 1.0) x)
      (/ (- wj 1.0) (fma wj wj -1.0))
      (* (* wj wj) (fma (- (* (- 1.0 wj) wj) 1.0) wj 1.0)))
     (-
      wj
      (/
       (fma (* (/ -1.0 (exp wj)) (- wj 1.0)) x (* wj (- wj 1.0)))
       (fma wj wj -1.0))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 1e-8) {
		tmp = fma((fma(((fma(-0.16666666666666666, wj, 0.5) * wj) - 1.0), wj, 1.0) * x), ((wj - 1.0) / fma(wj, wj, -1.0)), ((wj * wj) * fma((((1.0 - wj) * wj) - 1.0), wj, 1.0)));
	} else {
		tmp = wj - (fma(((-1.0 / exp(wj)) * (wj - 1.0)), x, (wj * (wj - 1.0))) / fma(wj, wj, -1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 1e-8)
		tmp = fma(Float64(fma(Float64(Float64(fma(-0.16666666666666666, wj, 0.5) * wj) - 1.0), wj, 1.0) * x), Float64(Float64(wj - 1.0) / fma(wj, wj, -1.0)), Float64(Float64(wj * wj) * fma(Float64(Float64(Float64(1.0 - wj) * wj) - 1.0), wj, 1.0)));
	else
		tmp = Float64(wj - Float64(fma(Float64(Float64(-1.0 / exp(wj)) * Float64(wj - 1.0)), x, Float64(wj * Float64(wj - 1.0))) / fma(wj, wj, -1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-8], N[(N[(N[(N[(N[(N[(-0.16666666666666666 * wj + 0.5), $MachinePrecision] * wj), $MachinePrecision] - 1.0), $MachinePrecision] * wj + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(N[(wj - 1.0), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(wj * wj), $MachinePrecision] * N[(N[(N[(N[(1.0 - wj), $MachinePrecision] * wj), $MachinePrecision] - 1.0), $MachinePrecision] * wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(N[(N[(-1.0 / N[Exp[wj], $MachinePrecision]), $MachinePrecision] * N[(wj - 1.0), $MachinePrecision]), $MachinePrecision] * x + N[(wj * N[(wj - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 1e-8

   1. Initial program 73.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. distribute-rgt1-in N/A

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

      6. remove-double-neg N/A

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

      7. flip3-+ N/A

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

      8. lift-exp.f64 N/A

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

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

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\left(\cosh wj + \sinh wj\right)}} \]

      10. flip-+ N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\frac{\cosh wj \cdot \cosh wj - \sinh wj \cdot \sinh wj}{\cosh wj - \sinh wj}}} \]

      11. sinh-cosh N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{\color{blue}{1}}{\cosh wj - \sinh wj}} \]

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

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{1}{\color{blue}{e^{\mathsf{neg}\left(wj\right)}}}} \]

      13. flip3-+ N/A

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

      14. flip-+ N/A

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

      15. frac-times N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 73.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 87.0%

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

   8. Taylor expanded in wj around 0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in wj around 0

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

   13. Applied rewrites 99.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, wj, 0.5\right) \cdot wj - 1, wj, 1\right) \cdot x, \frac{\color{blue}{wj} - 1}{\mathsf{fma}\left(wj, wj, -1\right)}, \left(wj \cdot wj\right) \cdot \mathsf{fma}\left(\left(1 - wj\right) \cdot wj - 1, wj, 1\right)\right) \]

   if 1e-8 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 90.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. distribute-rgt1-in N/A

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

      6. remove-double-neg N/A

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

      7. flip3-+ N/A

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

      8. lift-exp.f64 N/A

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

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

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\left(\cosh wj + \sinh wj\right)}} \]

      10. flip-+ N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\frac{\cosh wj \cdot \cosh wj - \sinh wj \cdot \sinh wj}{\cosh wj - \sinh wj}}} \]

      11. sinh-cosh N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{\color{blue}{1}}{\cosh wj - \sinh wj}} \]

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

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{1}{\color{blue}{e^{\mathsf{neg}\left(wj\right)}}}} \]

      13. flip3-+ N/A

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

      14. flip-+ N/A

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

      15. frac-times N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 92.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, wj, 0.5\right) \cdot wj - 1, wj, 1\right) \cdot x, \frac{wj - 1}{\mathsf{fma}\left(wj, wj, -1\right)}, \left(wj \cdot wj\right) \cdot \mathsf{fma}\left(\left(1 - wj\right) \cdot wj - 1, wj, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\mathsf{fma}\left(\frac{-1}{e^{wj}} \cdot \left(wj - 1\right), x, wj \cdot \left(wj - 1\right)\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.31:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, wj, 0.5\right) \cdot wj - 1, wj, 1\right) \cdot x, \frac{wj - 1}{\mathsf{fma}\left(wj, wj, -1\right)}, \left(wj \cdot wj\right) \cdot \mathsf{fma}\left(\left(1 - wj\right) \cdot wj - 1, wj, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj \cdot \left(wj - 1\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.31)
   (fma
    (* (fma (- (* (fma -0.16666666666666666 wj 0.5) wj) 1.0) wj 1.0) x)
    (/ (- wj 1.0) (fma wj wj -1.0))
    (* (* wj wj) (fma (- (* (- 1.0 wj) wj) 1.0) wj 1.0)))
   (- wj (/ (* wj (- wj 1.0)) (fma wj wj -1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.31) {
		tmp = fma((fma(((fma(-0.16666666666666666, wj, 0.5) * wj) - 1.0), wj, 1.0) * x), ((wj - 1.0) / fma(wj, wj, -1.0)), ((wj * wj) * fma((((1.0 - wj) * wj) - 1.0), wj, 1.0)));
	} else {
		tmp = wj - ((wj * (wj - 1.0)) / fma(wj, wj, -1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.31)
		tmp = fma(Float64(fma(Float64(Float64(fma(-0.16666666666666666, wj, 0.5) * wj) - 1.0), wj, 1.0) * x), Float64(Float64(wj - 1.0) / fma(wj, wj, -1.0)), Float64(Float64(wj * wj) * fma(Float64(Float64(Float64(1.0 - wj) * wj) - 1.0), wj, 1.0)));
	else
		tmp = Float64(wj - Float64(Float64(wj * Float64(wj - 1.0)) / fma(wj, wj, -1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.31], N[(N[(N[(N[(N[(N[(-0.16666666666666666 * wj + 0.5), $MachinePrecision] * wj), $MachinePrecision] - 1.0), $MachinePrecision] * wj + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(N[(wj - 1.0), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(wj * wj), $MachinePrecision] * N[(N[(N[(N[(1.0 - wj), $MachinePrecision] * wj), $MachinePrecision] - 1.0), $MachinePrecision] * wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj * N[(wj - 1.0), $MachinePrecision]), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.309999999999999998

   1. Initial program 79.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. distribute-rgt1-in N/A

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

      6. remove-double-neg N/A

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

      7. flip3-+ N/A

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

      8. lift-exp.f64 N/A

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

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

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\left(\cosh wj + \sinh wj\right)}} \]

      10. flip-+ N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\frac{\cosh wj \cdot \cosh wj - \sinh wj \cdot \sinh wj}{\cosh wj - \sinh wj}}} \]

      11. sinh-cosh N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{\color{blue}{1}}{\cosh wj - \sinh wj}} \]

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

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{1}{\color{blue}{e^{\mathsf{neg}\left(wj\right)}}}} \]

      13. flip3-+ N/A

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

      14. flip-+ N/A

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

      15. frac-times N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 90.5%

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

   8. Taylor expanded in wj around 0

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

   10. Applied rewrites 99.7%

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

   11. Taylor expanded in wj around 0

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

   13. Applied rewrites 98.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, wj, 0.5\right) \cdot wj - 1, wj, 1\right) \cdot x, \frac{\color{blue}{wj} - 1}{\mathsf{fma}\left(wj, wj, -1\right)}, \left(wj \cdot wj\right) \cdot \mathsf{fma}\left(\left(1 - wj\right) \cdot wj - 1, wj, 1\right)\right) \]

   if 0.309999999999999998 < wj

   1. Initial program 16.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. distribute-rgt1-in N/A

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

      6. remove-double-neg N/A

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

      7. flip3-+ N/A

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

      8. lift-exp.f64 N/A

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

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

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\left(\cosh wj + \sinh wj\right)}} \]

      10. flip-+ N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\frac{\cosh wj \cdot \cosh wj - \sinh wj \cdot \sinh wj}{\cosh wj - \sinh wj}}} \]

      11. sinh-cosh N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{\color{blue}{1}}{\cosh wj - \sinh wj}} \]

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

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{1}{\color{blue}{e^{\mathsf{neg}\left(wj\right)}}}} \]

      13. flip3-+ N/A

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

      14. flip-+ N/A

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

      15. frac-times N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 16.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.31:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, wj, 0.5\right) \cdot wj - 1, wj, 1\right) \cdot x, \frac{wj - 1}{\mathsf{fma}\left(wj, wj, -1\right)}, \left(wj \cdot wj\right) \cdot \mathsf{fma}\left(\left(1 - wj\right) \cdot wj - 1, wj, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj \cdot \left(wj - 1\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.072:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(0.5, wj, -1\right), wj, x\right), \frac{wj - 1}{\mathsf{fma}\left(wj, wj, -1\right)}, \left(wj \cdot wj\right) \cdot \mathsf{fma}\left(\left(1 - wj\right) \cdot wj - 1, wj, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj \cdot \left(wj - 1\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.072)
   (fma
    (fma (* x (fma 0.5 wj -1.0)) wj x)
    (/ (- wj 1.0) (fma wj wj -1.0))
    (* (* wj wj) (fma (- (* (- 1.0 wj) wj) 1.0) wj 1.0)))
   (- wj (/ (* wj (- wj 1.0)) (fma wj wj -1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.072) {
		tmp = fma(fma((x * fma(0.5, wj, -1.0)), wj, x), ((wj - 1.0) / fma(wj, wj, -1.0)), ((wj * wj) * fma((((1.0 - wj) * wj) - 1.0), wj, 1.0)));
	} else {
		tmp = wj - ((wj * (wj - 1.0)) / fma(wj, wj, -1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.072)
		tmp = fma(fma(Float64(x * fma(0.5, wj, -1.0)), wj, x), Float64(Float64(wj - 1.0) / fma(wj, wj, -1.0)), Float64(Float64(wj * wj) * fma(Float64(Float64(Float64(1.0 - wj) * wj) - 1.0), wj, 1.0)));
	else
		tmp = Float64(wj - Float64(Float64(wj * Float64(wj - 1.0)) / fma(wj, wj, -1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.072], N[(N[(N[(x * N[(0.5 * wj + -1.0), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision] * N[(N[(wj - 1.0), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(wj * wj), $MachinePrecision] * N[(N[(N[(N[(1.0 - wj), $MachinePrecision] * wj), $MachinePrecision] - 1.0), $MachinePrecision] * wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj * N[(wj - 1.0), $MachinePrecision]), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.0719999999999999946

   1. Initial program 79.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. distribute-rgt1-in N/A

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

      6. remove-double-neg N/A

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

      7. flip3-+ N/A

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

      8. lift-exp.f64 N/A

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

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

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\left(\cosh wj + \sinh wj\right)}} \]

      10. flip-+ N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\frac{\cosh wj \cdot \cosh wj - \sinh wj \cdot \sinh wj}{\cosh wj - \sinh wj}}} \]

      11. sinh-cosh N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{\color{blue}{1}}{\cosh wj - \sinh wj}} \]

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

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{1}{\color{blue}{e^{\mathsf{neg}\left(wj\right)}}}} \]

      13. flip3-+ N/A

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

      14. flip-+ N/A

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

      15. frac-times N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 90.5%

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

   8. Taylor expanded in wj around 0

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

   10. Applied rewrites 99.7%

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

   11. Taylor expanded in wj around 0

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

   13. Applied rewrites 98.4%

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

   if 0.0719999999999999946 < wj

   1. Initial program 16.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. distribute-rgt1-in N/A

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

      6. remove-double-neg N/A

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

      7. flip3-+ N/A

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

      8. lift-exp.f64 N/A

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

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

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\left(\cosh wj + \sinh wj\right)}} \]

      10. flip-+ N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\frac{\cosh wj \cdot \cosh wj - \sinh wj \cdot \sinh wj}{\cosh wj - \sinh wj}}} \]

      11. sinh-cosh N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{\color{blue}{1}}{\cosh wj - \sinh wj}} \]

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

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{1}{\color{blue}{e^{\mathsf{neg}\left(wj\right)}}}} \]

      13. flip3-+ N/A

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

      14. flip-+ N/A

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

      15. frac-times N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 16.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.072:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot \mathsf{fma}\left(0.5, wj, -1\right), wj, x\right), \frac{wj - 1}{\mathsf{fma}\left(wj, wj, -1\right)}, \left(wj \cdot wj\right) \cdot \mathsf{fma}\left(\left(1 - wj\right) \cdot wj - 1, wj, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj \cdot \left(wj - 1\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.049:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - wj\right) \cdot x, \frac{wj - 1}{\mathsf{fma}\left(wj, wj, -1\right)}, \left(wj \cdot wj\right) \cdot \mathsf{fma}\left(\left(1 - wj\right) \cdot wj - 1, wj, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj \cdot \left(wj - 1\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.049)
   (fma
    (* (- 1.0 wj) x)
    (/ (- wj 1.0) (fma wj wj -1.0))
    (* (* wj wj) (fma (- (* (- 1.0 wj) wj) 1.0) wj 1.0)))
   (- wj (/ (* wj (- wj 1.0)) (fma wj wj -1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.049) {
		tmp = fma(((1.0 - wj) * x), ((wj - 1.0) / fma(wj, wj, -1.0)), ((wj * wj) * fma((((1.0 - wj) * wj) - 1.0), wj, 1.0)));
	} else {
		tmp = wj - ((wj * (wj - 1.0)) / fma(wj, wj, -1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.049)
		tmp = fma(Float64(Float64(1.0 - wj) * x), Float64(Float64(wj - 1.0) / fma(wj, wj, -1.0)), Float64(Float64(wj * wj) * fma(Float64(Float64(Float64(1.0 - wj) * wj) - 1.0), wj, 1.0)));
	else
		tmp = Float64(wj - Float64(Float64(wj * Float64(wj - 1.0)) / fma(wj, wj, -1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.049], N[(N[(N[(1.0 - wj), $MachinePrecision] * x), $MachinePrecision] * N[(N[(wj - 1.0), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(wj * wj), $MachinePrecision] * N[(N[(N[(N[(1.0 - wj), $MachinePrecision] * wj), $MachinePrecision] - 1.0), $MachinePrecision] * wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(wj - N[(N[(wj * N[(wj - 1.0), $MachinePrecision]), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.049000000000000002

   1. Initial program 79.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. distribute-rgt1-in N/A

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

      6. remove-double-neg N/A

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

      7. flip3-+ N/A

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

      8. lift-exp.f64 N/A

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

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

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\left(\cosh wj + \sinh wj\right)}} \]

      10. flip-+ N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\frac{\cosh wj \cdot \cosh wj - \sinh wj \cdot \sinh wj}{\cosh wj - \sinh wj}}} \]

      11. sinh-cosh N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{\color{blue}{1}}{\cosh wj - \sinh wj}} \]

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

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{1}{\color{blue}{e^{\mathsf{neg}\left(wj\right)}}}} \]

      13. flip3-+ N/A

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

      14. flip-+ N/A

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

      15. frac-times N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 90.5%

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

   8. Taylor expanded in wj around 0

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

   10. Applied rewrites 99.7%

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

   11. Taylor expanded in wj around 0

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

   13. Applied rewrites 98.2%

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

   if 0.049000000000000002 < wj

   1. Initial program 16.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. distribute-rgt1-in N/A

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

      6. remove-double-neg N/A

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

      7. flip3-+ N/A

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

      8. lift-exp.f64 N/A

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

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

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\left(\cosh wj + \sinh wj\right)}} \]

      10. flip-+ N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\frac{\cosh wj \cdot \cosh wj - \sinh wj \cdot \sinh wj}{\cosh wj - \sinh wj}}} \]

      11. sinh-cosh N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{\color{blue}{1}}{\cosh wj - \sinh wj}} \]

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

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{1}{\color{blue}{e^{\mathsf{neg}\left(wj\right)}}}} \]

      13. flip3-+ N/A

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

      14. flip-+ N/A

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

      15. frac-times N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 16.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.049:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - wj\right) \cdot x, \frac{wj - 1}{\mathsf{fma}\left(wj, wj, -1\right)}, \left(wj \cdot wj\right) \cdot \mathsf{fma}\left(\left(1 - wj\right) \cdot wj - 1, wj, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj \cdot \left(wj - 1\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00082:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(1 - \mathsf{fma}\left(\mathsf{fma}\left(-3, x, \mathsf{fma}\left(0.6666666666666666, x, x \cdot 5\right)\right), wj, wj\right)\right) - -2.5 \cdot x, wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj \cdot \left(wj - 1\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00082)
   (fma
    (fma
     (-
      (- 1.0 (fma (fma -3.0 x (fma 0.6666666666666666 x (* x 5.0))) wj wj))
      (* -2.5 x))
     wj
     (* -2.0 x))
    wj
    x)
   (- wj (/ (* wj (- wj 1.0)) (fma wj wj -1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00082) {
		tmp = fma(fma(((1.0 - fma(fma(-3.0, x, fma(0.6666666666666666, x, (x * 5.0))), wj, wj)) - (-2.5 * x)), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - ((wj * (wj - 1.0)) / fma(wj, wj, -1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00082)
		tmp = fma(fma(Float64(Float64(1.0 - fma(fma(-3.0, x, fma(0.6666666666666666, x, Float64(x * 5.0))), wj, wj)) - Float64(-2.5 * x)), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - Float64(Float64(wj * Float64(wj - 1.0)) / fma(wj, wj, -1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00082], N[(N[(N[(N[(1.0 - N[(N[(-3.0 * x + N[(0.6666666666666666 * x + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + wj), $MachinePrecision]), $MachinePrecision] - N[(-2.5 * x), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(wj * N[(wj - 1.0), $MachinePrecision]), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 8.1999999999999998e-4

   1. Initial program 79.7%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 98.3%

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

   if 8.1999999999999998e-4 < wj

   1. Initial program 35.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. distribute-rgt1-in N/A

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

      6. remove-double-neg N/A

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

      7. flip3-+ N/A

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

      8. lift-exp.f64 N/A

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

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

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\left(\cosh wj + \sinh wj\right)}} \]

      10. flip-+ N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\frac{\cosh wj \cdot \cosh wj - \sinh wj \cdot \sinh wj}{\cosh wj - \sinh wj}}} \]

      11. sinh-cosh N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{\color{blue}{1}}{\cosh wj - \sinh wj}} \]

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

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{1}{\color{blue}{e^{\mathsf{neg}\left(wj\right)}}}} \]

      13. flip3-+ N/A

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

      14. flip-+ N/A

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

      15. frac-times N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 86.9%

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

Alternative 8: 97.6% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0008:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1 - wj, wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj \cdot \left(wj - 1\right)}{\mathsf{fma}\left(wj, wj, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0008)
   (fma (fma (- 1.0 wj) wj (* -2.0 x)) wj x)
   (- wj (/ (* wj (- wj 1.0)) (fma wj wj -1.0)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0008) {
		tmp = fma(fma((1.0 - wj), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = wj - ((wj * (wj - 1.0)) / fma(wj, wj, -1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0008)
		tmp = fma(fma(Float64(1.0 - wj), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = Float64(wj - Float64(Float64(wj * Float64(wj - 1.0)) / fma(wj, wj, -1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0008], N[(N[(N[(1.0 - wj), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(wj - N[(N[(wj * N[(wj - 1.0), $MachinePrecision]), $MachinePrecision] / N[(wj * wj + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 8.00000000000000038e-4

   1. Initial program 79.7%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 98.3%

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

   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 98.2%

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

   if 8.00000000000000038e-4 < wj

   1. Initial program 35.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

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

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

      5. distribute-rgt1-in N/A

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

      6. remove-double-neg N/A

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

      7. flip3-+ N/A

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

      8. lift-exp.f64 N/A

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

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

         \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\left(\cosh wj + \sinh wj\right)}} \]

      10. flip-+ N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\frac{\cosh wj \cdot \cosh wj - \sinh wj \cdot \sinh wj}{\cosh wj - \sinh wj}}} \]

      11. sinh-cosh N/A

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{\color{blue}{1}}{\cosh wj - \sinh wj}} \]

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

          \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{1}{\color{blue}{e^{\mathsf{neg}\left(wj\right)}}}} \]

      13. flip3-+ N/A

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

      14. flip-+ N/A

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

      15. frac-times N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 35.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 86.9%

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

Alternative 9: 96.3% accurate, 13.8× speedup

Math

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

FPCore

(FPCore (wj x)
 :precision binary64
 (fma (fma (fma 2.5 x 1.0) wj (* -2.0 x)) wj x))

C

double code(double wj, double x) {
	return fma(fma(fma(2.5, x, 1.0), wj, (-2.0 * x)), wj, x);
}

Julia

function code(wj, x)
	return fma(fma(fma(2.5, x, 1.0), wj, Float64(-2.0 * x)), wj, x)
end

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

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

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

   5. distribute-rgt1-in N/A

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

   6. remove-double-neg N/A

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

   7. flip3-+ N/A

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

   8. lift-exp.f64 N/A

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

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

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\left(\cosh wj + \sinh wj\right)}} \]

   10. flip-+ N/A

       \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \color{blue}{\frac{\cosh wj \cdot \cosh wj - \sinh wj \cdot \sinh wj}{\cosh wj - \sinh wj}}} \]

   11. sinh-cosh N/A

       \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{\color{blue}{1}}{\cosh wj - \sinh wj}} \]

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

       \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\frac{{wj}^{3} + {1}^{3}}{wj \cdot wj + \left(1 \cdot 1 - wj \cdot 1\right)} \cdot \frac{1}{\color{blue}{e^{\mathsf{neg}\left(wj\right)}}}} \]

   13. flip3-+ N/A

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

   14. flip-+ N/A

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

   15. frac-times N/A

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

   16. lower-/.f64 N/A

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

4. Applied rewrites 79.1%

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

5. Taylor expanded in wj around 0

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

7. Applied rewrites 95.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot x - \left(1 - \mathsf{fma}\left(\left(-x\right) - 1, 0.5, \mathsf{fma}\left(-0.16666666666666666, x, 0.5\right)\right)\right), wj, x + \mathsf{fma}\left(0.5, x, -\left(\left(-x\right) - 1\right)\right)\right), wj, 1 + \left(-2 \cdot x - 1\right)\right), wj, x\right)} \]

8. Taylor expanded in wj around 0

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

10. Applied rewrites 95.5%

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

11. Final simplification 95.5%

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

Alternative 10: 96.6% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 95.7%

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

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 95.4%

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

Alternative 11: 96.1% accurate, 19.5× speedup

Math

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

FPCore

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

C

double code(double wj, double x) {
	return (((1.0 - wj) * wj) * wj) + 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 = (((1.0d0 - wj) * wj) * wj) + x
end function

Java

public static double code(double wj, double x) {
	return (((1.0 - wj) * wj) * wj) + x;
}

Python

def code(wj, x):
	return (((1.0 - wj) * wj) * wj) + x

Julia

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

MATLAB

function tmp = code(wj, x)
	tmp = (((1.0 - wj) * wj) * wj) + x;
end

Wolfram

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

Derivation

1. Initial program 78.4%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 95.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 94.8%

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

10. Applied rewrites 94.8%

    \[\leadsto \left(\left(1 - wj\right) \cdot wj\right) \cdot wj + \color{blue}{x} \]
11. Add Preprocessing

Alternative 12: 96.1% accurate, 22.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.4%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 95.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 94.8%

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

Alternative 13: 95.7% accurate, 47.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 95.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 94.8%

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

9. Taylor expanded in wj around 0

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

11. Applied rewrites 94.7%

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

Alternative 14: 84.6% 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 78.4%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 85.1%

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

Alternative 15: 4.4% accurate, 331.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 wj)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 78.4%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. 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: 79.0% 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 2025018 
(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))))))