(/ (- z0 (* (* (- -1/2 (* (- (* 1/4 z0) -3333333333333333/10000000000000000) z0)) z0) z0)) (+ (/ z3 (* z4 z4)) (/ z1 (* z2 z2))))

Percentage Accurate: 66.3% → 76.0%

Time: 4.3s

Alternatives: 16

Speedup: 0.9×

• 71.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]

FPCore

(FPCore (z0 z3 z4 z1 z2)
  :precision binary64
  (/
 (-
  z0
  (* (* (- -0.5 (* (- (* 0.25 z0) -0.3333333333333333) z0)) z0) z0))
 (+ (/ z3 (* z4 z4)) (/ z1 (* z2 z2)))))

C

double code(double z0, double z3, double z4, double z1, double z2) {
	return (z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)) / ((z3 / (z4 * z4)) + (z1 / (z2 * z2)));
}

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(z0, z3, z4, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    code = (z0 - ((((-0.5d0) - (((0.25d0 * z0) - (-0.3333333333333333d0)) * z0)) * z0) * z0)) / ((z3 / (z4 * z4)) + (z1 / (z2 * z2)))
end function

Java

public static double code(double z0, double z3, double z4, double z1, double z2) {
	return (z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)) / ((z3 / (z4 * z4)) + (z1 / (z2 * z2)));
}

Python

def code(z0, z3, z4, z1, z2):
	return (z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)) / ((z3 / (z4 * z4)) + (z1 / (z2 * z2)))

Julia

function code(z0, z3, z4, z1, z2)
	return Float64(Float64(z0 - Float64(Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)) / Float64(Float64(z3 / Float64(z4 * z4)) + Float64(z1 / Float64(z2 * z2))))
end

MATLAB

function tmp = code(z0, z3, z4, z1, z2)
	tmp = (z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)) / ((z3 / (z4 * z4)) + (z1 / (z2 * z2)));
end

Wolfram

code[z0_, z3_, z4_, z1_, z2_] := N[(N[(z0 - N[(N[(N[(-0.5 - N[(N[(N[(0.25 * z0), $MachinePrecision] - -0.3333333333333333), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision] / N[(N[(z3 / N[(z4 * z4), $MachinePrecision]), $MachinePrecision] + N[(z1 / N[(z2 * z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 66.3% accurate, 1.0× speedup

Math

\[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]

FPCore

(FPCore (z0 z3 z4 z1 z2)
  :precision binary64
  (/
 (-
  z0
  (* (* (- -0.5 (* (- (* 0.25 z0) -0.3333333333333333) z0)) z0) z0))
 (+ (/ z3 (* z4 z4)) (/ z1 (* z2 z2)))))

C

double code(double z0, double z3, double z4, double z1, double z2) {
	return (z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)) / ((z3 / (z4 * z4)) + (z1 / (z2 * z2)));
}

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(z0, z3, z4, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    code = (z0 - ((((-0.5d0) - (((0.25d0 * z0) - (-0.3333333333333333d0)) * z0)) * z0) * z0)) / ((z3 / (z4 * z4)) + (z1 / (z2 * z2)))
end function

Java

public static double code(double z0, double z3, double z4, double z1, double z2) {
	return (z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)) / ((z3 / (z4 * z4)) + (z1 / (z2 * z2)));
}

Python

def code(z0, z3, z4, z1, z2):
	return (z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)) / ((z3 / (z4 * z4)) + (z1 / (z2 * z2)))

Julia

function code(z0, z3, z4, z1, z2)
	return Float64(Float64(z0 - Float64(Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)) / Float64(Float64(z3 / Float64(z4 * z4)) + Float64(z1 / Float64(z2 * z2))))
end

MATLAB

function tmp = code(z0, z3, z4, z1, z2)
	tmp = (z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)) / ((z3 / (z4 * z4)) + (z1 / (z2 * z2)));
end

Wolfram

code[z0_, z3_, z4_, z1_, z2_] := N[(N[(z0 - N[(N[(N[(-0.5 - N[(N[(N[(0.25 * z0), $MachinePrecision] - -0.3333333333333333), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision] / N[(N[(z3 / N[(z4 * z4), $MachinePrecision]), $MachinePrecision] + N[(z1 / N[(z2 * z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 76.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{z1}{z2} + \frac{z2 \cdot z3}{{\left(\left|z4\right|\right)}^{2}}\\ t_1 := \frac{z2}{t\_0}\\ t_2 := z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0\\ \mathbf{if}\;\left|z4\right| \leq 3.7 \cdot 10^{-75}:\\ \;\;\;\;\frac{t\_2}{\frac{z1}{z2 \cdot z2} \cdot \left|z4\right| + \frac{z3}{\left|z4\right|}} \cdot \left|z4\right|\\ \mathbf{elif}\;\left|z4\right| \leq 10^{+108}:\\ \;\;\;\;z0 \cdot \left(z0 \cdot \left(0.5 \cdot t\_1 + z0 \cdot \left(0.25 \cdot \frac{z0 \cdot z2}{t\_0} + 0.3333333333333333 \cdot t\_1\right)\right) + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{z2 \cdot \frac{z3}{\left|z4\right| \cdot \left|z4\right|} + \frac{z1}{z2}} \cdot z2\\ \end{array} \]

FPCore

(FPCore (z0 z3 z4 z1 z2)
  :precision binary64
  (let* ((t_0 (+ (/ z1 z2) (/ (* z2 z3) (pow (fabs z4) 2.0))))
       (t_1 (/ z2 t_0))
       (t_2
        (-
         z0
         (*
          (* (- -0.5 (* (- (* 0.25 z0) -0.3333333333333333) z0)) z0)
          z0))))
  (if (<= (fabs z4) 3.7e-75)
    (*
     (/ t_2 (+ (* (/ z1 (* z2 z2)) (fabs z4)) (/ z3 (fabs z4))))
     (fabs z4))
    (if (<= (fabs z4) 1e+108)
      (*
       z0
       (+
        (*
         z0
         (+
          (* 0.5 t_1)
          (*
           z0
           (+
            (* 0.25 (/ (* z0 z2) t_0))
            (* 0.3333333333333333 t_1)))))
        t_1))
      (*
       (/ t_2 (+ (* z2 (/ z3 (* (fabs z4) (fabs z4)))) (/ z1 z2)))
       z2)))))

C

double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = (z1 / z2) + ((z2 * z3) / pow(fabs(z4), 2.0));
	double t_1 = z2 / t_0;
	double t_2 = z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0);
	double tmp;
	if (fabs(z4) <= 3.7e-75) {
		tmp = (t_2 / (((z1 / (z2 * z2)) * fabs(z4)) + (z3 / fabs(z4)))) * fabs(z4);
	} else if (fabs(z4) <= 1e+108) {
		tmp = z0 * ((z0 * ((0.5 * t_1) + (z0 * ((0.25 * ((z0 * z2) / t_0)) + (0.3333333333333333 * t_1))))) + t_1);
	} else {
		tmp = (t_2 / ((z2 * (z3 / (fabs(z4) * fabs(z4)))) + (z1 / z2))) * z2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0, z3, z4, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (z1 / z2) + ((z2 * z3) / (abs(z4) ** 2.0d0))
    t_1 = z2 / t_0
    t_2 = z0 - ((((-0.5d0) - (((0.25d0 * z0) - (-0.3333333333333333d0)) * z0)) * z0) * z0)
    if (abs(z4) <= 3.7d-75) then
        tmp = (t_2 / (((z1 / (z2 * z2)) * abs(z4)) + (z3 / abs(z4)))) * abs(z4)
    else if (abs(z4) <= 1d+108) then
        tmp = z0 * ((z0 * ((0.5d0 * t_1) + (z0 * ((0.25d0 * ((z0 * z2) / t_0)) + (0.3333333333333333d0 * t_1))))) + t_1)
    else
        tmp = (t_2 / ((z2 * (z3 / (abs(z4) * abs(z4)))) + (z1 / z2))) * z2
    end if
    code = tmp
end function

Java

public static double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = (z1 / z2) + ((z2 * z3) / Math.pow(Math.abs(z4), 2.0));
	double t_1 = z2 / t_0;
	double t_2 = z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0);
	double tmp;
	if (Math.abs(z4) <= 3.7e-75) {
		tmp = (t_2 / (((z1 / (z2 * z2)) * Math.abs(z4)) + (z3 / Math.abs(z4)))) * Math.abs(z4);
	} else if (Math.abs(z4) <= 1e+108) {
		tmp = z0 * ((z0 * ((0.5 * t_1) + (z0 * ((0.25 * ((z0 * z2) / t_0)) + (0.3333333333333333 * t_1))))) + t_1);
	} else {
		tmp = (t_2 / ((z2 * (z3 / (Math.abs(z4) * Math.abs(z4)))) + (z1 / z2))) * z2;
	}
	return tmp;
}

Python

def code(z0, z3, z4, z1, z2):
	t_0 = (z1 / z2) + ((z2 * z3) / math.pow(math.fabs(z4), 2.0))
	t_1 = z2 / t_0
	t_2 = z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)
	tmp = 0
	if math.fabs(z4) <= 3.7e-75:
		tmp = (t_2 / (((z1 / (z2 * z2)) * math.fabs(z4)) + (z3 / math.fabs(z4)))) * math.fabs(z4)
	elif math.fabs(z4) <= 1e+108:
		tmp = z0 * ((z0 * ((0.5 * t_1) + (z0 * ((0.25 * ((z0 * z2) / t_0)) + (0.3333333333333333 * t_1))))) + t_1)
	else:
		tmp = (t_2 / ((z2 * (z3 / (math.fabs(z4) * math.fabs(z4)))) + (z1 / z2))) * z2
	return tmp

Julia

function code(z0, z3, z4, z1, z2)
	t_0 = Float64(Float64(z1 / z2) + Float64(Float64(z2 * z3) / (abs(z4) ^ 2.0)))
	t_1 = Float64(z2 / t_0)
	t_2 = Float64(z0 - Float64(Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0))
	tmp = 0.0
	if (abs(z4) <= 3.7e-75)
		tmp = Float64(Float64(t_2 / Float64(Float64(Float64(z1 / Float64(z2 * z2)) * abs(z4)) + Float64(z3 / abs(z4)))) * abs(z4));
	elseif (abs(z4) <= 1e+108)
		tmp = Float64(z0 * Float64(Float64(z0 * Float64(Float64(0.5 * t_1) + Float64(z0 * Float64(Float64(0.25 * Float64(Float64(z0 * z2) / t_0)) + Float64(0.3333333333333333 * t_1))))) + t_1));
	else
		tmp = Float64(Float64(t_2 / Float64(Float64(z2 * Float64(z3 / Float64(abs(z4) * abs(z4)))) + Float64(z1 / z2))) * z2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z3, z4, z1, z2)
	t_0 = (z1 / z2) + ((z2 * z3) / (abs(z4) ^ 2.0));
	t_1 = z2 / t_0;
	t_2 = z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0);
	tmp = 0.0;
	if (abs(z4) <= 3.7e-75)
		tmp = (t_2 / (((z1 / (z2 * z2)) * abs(z4)) + (z3 / abs(z4)))) * abs(z4);
	elseif (abs(z4) <= 1e+108)
		tmp = z0 * ((z0 * ((0.5 * t_1) + (z0 * ((0.25 * ((z0 * z2) / t_0)) + (0.3333333333333333 * t_1))))) + t_1);
	else
		tmp = (t_2 / ((z2 * (z3 / (abs(z4) * abs(z4)))) + (z1 / z2))) * z2;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z3_, z4_, z1_, z2_] := Block[{t$95$0 = N[(N[(z1 / z2), $MachinePrecision] + N[(N[(z2 * z3), $MachinePrecision] / N[Power[N[Abs[z4], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z2 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(z0 - N[(N[(N[(-0.5 - N[(N[(N[(0.25 * z0), $MachinePrecision] - -0.3333333333333333), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[z4], $MachinePrecision], 3.7e-75], N[(N[(t$95$2 / N[(N[(N[(z1 / N[(z2 * z2), $MachinePrecision]), $MachinePrecision] * N[Abs[z4], $MachinePrecision]), $MachinePrecision] + N[(z3 / N[Abs[z4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[z4], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[z4], $MachinePrecision], 1e+108], N[(z0 * N[(N[(z0 * N[(N[(0.5 * t$95$1), $MachinePrecision] + N[(z0 * N[(N[(0.25 * N[(N[(z0 * z2), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 / N[(N[(z2 * N[(z3 / N[(N[Abs[z4], $MachinePrecision] * N[Abs[z4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z1 / z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z4 < 3.7000000000000002e-75

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{z3}{z4 \cdot z4}} + \frac{z1}{z2 \cdot z2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{\color{blue}{z4 \cdot z4}} + \frac{z1}{z2 \cdot z2}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{\frac{z3}{z4}}{z4}} + \frac{z1}{z2 \cdot z2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{\frac{z3}{z4}}{z4}} + \frac{z1}{z2 \cdot z2}} \]

      5. lower-/.f64 70.3%

         \[\leadsto \frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{\color{blue}{\frac{z3}{z4}}}{z4} + \frac{z1}{z2 \cdot z2}} \]

   3. Applied rewrites 70.3%

      \[\leadsto \frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{\frac{z3}{z4}}{z4}} + \frac{z1}{z2 \cdot z2}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{\frac{z3}{z4}}{z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{\frac{z3}{z4}}{z4} + \frac{z1}{z2 \cdot z2}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{\frac{z3}{z4}}{z4}} + \frac{z1}{z2 \cdot z2}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{\frac{z3}{z4}}{z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. common-denominator N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{\frac{z3}{z4} \cdot \left(z2 \cdot z2\right) + z1 \cdot z4}{z4 \cdot \left(z2 \cdot z2\right)}}} \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4} \cdot \left(z2 \cdot z2\right) + z1 \cdot z4} \cdot \left(z4 \cdot \left(z2 \cdot z2\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{z1 \cdot z4 + \frac{z3}{z4} \cdot \left(z2 \cdot z2\right)}} \cdot \left(z4 \cdot \left(z2 \cdot z2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{z1 \cdot z4 + \frac{z3}{z4} \cdot \left(z2 \cdot z2\right)} \cdot \color{blue}{\left(\left(z2 \cdot z2\right) \cdot z4\right)} \]

      9. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z1 \cdot z4 + \frac{z3}{z4} \cdot \left(z2 \cdot z2\right)}{\left(z2 \cdot z2\right) \cdot z4}}} \]

      10. common-denominator N/A

          \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{\frac{z3}{z4}}{z4}}} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{z1}{z2 \cdot z2}} + \frac{\frac{z3}{z4}}{z4}} \]

      12. add-to-fraction N/A

          \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{\frac{z1}{z2 \cdot z2} \cdot z4 + \frac{z3}{z4}}{z4}}} \]

   5. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z1}{z2 \cdot z2} \cdot z4 + \frac{z3}{z4}} \cdot z4} \]

   if 3.7000000000000002e-75 < z4 < 1e108

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{z0 \cdot \left(z0 \cdot \left(\frac{1}{2} \cdot \frac{z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}} + z0 \cdot \left(\frac{1}{4} \cdot \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}} + \frac{3333333333333333}{10000000000000000} \cdot \frac{z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}\right)\right) + \frac{z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto z0 \cdot \color{blue}{\left(z0 \cdot \left(\frac{1}{2} \cdot \frac{z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}} + z0 \cdot \left(\frac{1}{4} \cdot \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}} + \frac{3333333333333333}{10000000000000000} \cdot \frac{z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}\right)\right) + \frac{z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto z0 \cdot \left(z0 \cdot \left(\frac{1}{2} \cdot \frac{z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}} + z0 \cdot \left(\frac{1}{4} \cdot \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}} + \frac{3333333333333333}{10000000000000000} \cdot \frac{z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}\right)\right) + \color{blue}{\frac{z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}}\right) \]

   8. Applied rewrites 67.9%

      \[\leadsto \color{blue}{z0 \cdot \left(z0 \cdot \left(0.5 \cdot \frac{z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}} + z0 \cdot \left(0.25 \cdot \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}} + 0.3333333333333333 \cdot \frac{z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}\right)\right) + \frac{z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}\right)} \]

   if 1e108 < z4

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      5. lower-/.f64 70.6%

         \[\leadsto \frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

   3. Applied rewrites 70.6%

      \[\leadsto \frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{\frac{z1}{z2}}{z2}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{\frac{z1}{z2}}{z2}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      4. add-to-fraction N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      5. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2 \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2 \]

      9. *-commutative N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2 \]

      10. lower-*.f64 69.9%

          \[\leadsto \frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2 \]

   5. Applied rewrites 69.9%

      \[\leadsto \color{blue}{\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0\\ \mathbf{if}\;\left|z4\right| \leq 3.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{t\_0}{\frac{z1}{z2 \cdot z2} \cdot \left|z4\right| + \frac{z3}{\left|z4\right|}} \cdot \left|z4\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{z2 \cdot \frac{z3}{\left|z4\right| \cdot \left|z4\right|} + \frac{z1}{z2}} \cdot z2\\ \end{array} \]

FPCore

(FPCore (z0 z3 z4 z1 z2)
  :precision binary64
  (let* ((t_0
        (-
         z0
         (*
          (* (- -0.5 (* (- (* 0.25 z0) -0.3333333333333333) z0)) z0)
          z0))))
  (if (<= (fabs z4) 3.6e-16)
    (*
     (/ t_0 (+ (* (/ z1 (* z2 z2)) (fabs z4)) (/ z3 (fabs z4))))
     (fabs z4))
    (*
     (/ t_0 (+ (* z2 (/ z3 (* (fabs z4) (fabs z4)))) (/ z1 z2)))
     z2))))

C

double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0);
	double tmp;
	if (fabs(z4) <= 3.6e-16) {
		tmp = (t_0 / (((z1 / (z2 * z2)) * fabs(z4)) + (z3 / fabs(z4)))) * fabs(z4);
	} else {
		tmp = (t_0 / ((z2 * (z3 / (fabs(z4) * fabs(z4)))) + (z1 / z2))) * z2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0, z3, z4, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z0 - ((((-0.5d0) - (((0.25d0 * z0) - (-0.3333333333333333d0)) * z0)) * z0) * z0)
    if (abs(z4) <= 3.6d-16) then
        tmp = (t_0 / (((z1 / (z2 * z2)) * abs(z4)) + (z3 / abs(z4)))) * abs(z4)
    else
        tmp = (t_0 / ((z2 * (z3 / (abs(z4) * abs(z4)))) + (z1 / z2))) * z2
    end if
    code = tmp
end function

Java

public static double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0);
	double tmp;
	if (Math.abs(z4) <= 3.6e-16) {
		tmp = (t_0 / (((z1 / (z2 * z2)) * Math.abs(z4)) + (z3 / Math.abs(z4)))) * Math.abs(z4);
	} else {
		tmp = (t_0 / ((z2 * (z3 / (Math.abs(z4) * Math.abs(z4)))) + (z1 / z2))) * z2;
	}
	return tmp;
}

Python

def code(z0, z3, z4, z1, z2):
	t_0 = z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)
	tmp = 0
	if math.fabs(z4) <= 3.6e-16:
		tmp = (t_0 / (((z1 / (z2 * z2)) * math.fabs(z4)) + (z3 / math.fabs(z4)))) * math.fabs(z4)
	else:
		tmp = (t_0 / ((z2 * (z3 / (math.fabs(z4) * math.fabs(z4)))) + (z1 / z2))) * z2
	return tmp

Julia

function code(z0, z3, z4, z1, z2)
	t_0 = Float64(z0 - Float64(Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0))
	tmp = 0.0
	if (abs(z4) <= 3.6e-16)
		tmp = Float64(Float64(t_0 / Float64(Float64(Float64(z1 / Float64(z2 * z2)) * abs(z4)) + Float64(z3 / abs(z4)))) * abs(z4));
	else
		tmp = Float64(Float64(t_0 / Float64(Float64(z2 * Float64(z3 / Float64(abs(z4) * abs(z4)))) + Float64(z1 / z2))) * z2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z3, z4, z1, z2)
	t_0 = z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0);
	tmp = 0.0;
	if (abs(z4) <= 3.6e-16)
		tmp = (t_0 / (((z1 / (z2 * z2)) * abs(z4)) + (z3 / abs(z4)))) * abs(z4);
	else
		tmp = (t_0 / ((z2 * (z3 / (abs(z4) * abs(z4)))) + (z1 / z2))) * z2;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z3_, z4_, z1_, z2_] := Block[{t$95$0 = N[(z0 - N[(N[(N[(-0.5 - N[(N[(N[(0.25 * z0), $MachinePrecision] - -0.3333333333333333), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[z4], $MachinePrecision], 3.6e-16], N[(N[(t$95$0 / N[(N[(N[(z1 / N[(z2 * z2), $MachinePrecision]), $MachinePrecision] * N[Abs[z4], $MachinePrecision]), $MachinePrecision] + N[(z3 / N[Abs[z4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[z4], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(N[(z2 * N[(z3 / N[(N[Abs[z4], $MachinePrecision] * N[Abs[z4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z1 / z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z4 < 3.5999999999999998e-16

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{z3}{z4 \cdot z4}} + \frac{z1}{z2 \cdot z2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{\color{blue}{z4 \cdot z4}} + \frac{z1}{z2 \cdot z2}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{\frac{z3}{z4}}{z4}} + \frac{z1}{z2 \cdot z2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{\frac{z3}{z4}}{z4}} + \frac{z1}{z2 \cdot z2}} \]

      5. lower-/.f64 70.3%

         \[\leadsto \frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{\color{blue}{\frac{z3}{z4}}}{z4} + \frac{z1}{z2 \cdot z2}} \]

   3. Applied rewrites 70.3%

      \[\leadsto \frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{\frac{z3}{z4}}{z4}} + \frac{z1}{z2 \cdot z2}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{\frac{z3}{z4}}{z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{\frac{z3}{z4}}{z4} + \frac{z1}{z2 \cdot z2}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{\frac{z3}{z4}}{z4}} + \frac{z1}{z2 \cdot z2}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{\frac{z3}{z4}}{z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. common-denominator N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{\frac{z3}{z4} \cdot \left(z2 \cdot z2\right) + z1 \cdot z4}{z4 \cdot \left(z2 \cdot z2\right)}}} \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4} \cdot \left(z2 \cdot z2\right) + z1 \cdot z4} \cdot \left(z4 \cdot \left(z2 \cdot z2\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{z1 \cdot z4 + \frac{z3}{z4} \cdot \left(z2 \cdot z2\right)}} \cdot \left(z4 \cdot \left(z2 \cdot z2\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{z1 \cdot z4 + \frac{z3}{z4} \cdot \left(z2 \cdot z2\right)} \cdot \color{blue}{\left(\left(z2 \cdot z2\right) \cdot z4\right)} \]

      9. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z1 \cdot z4 + \frac{z3}{z4} \cdot \left(z2 \cdot z2\right)}{\left(z2 \cdot z2\right) \cdot z4}}} \]

      10. common-denominator N/A

          \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{\frac{z3}{z4}}{z4}}} \]

      11. lift-/.f64 N/A

          \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{z1}{z2 \cdot z2}} + \frac{\frac{z3}{z4}}{z4}} \]

      12. add-to-fraction N/A

          \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{\frac{z1}{z2 \cdot z2} \cdot z4 + \frac{z3}{z4}}{z4}}} \]

   5. Applied rewrites 70.4%

      \[\leadsto \color{blue}{\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z1}{z2 \cdot z2} \cdot z4 + \frac{z3}{z4}} \cdot z4} \]

   if 3.5999999999999998e-16 < z4

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      5. lower-/.f64 70.6%

         \[\leadsto \frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

   3. Applied rewrites 70.6%

      \[\leadsto \frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{\frac{z1}{z2}}{z2}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{\frac{z1}{z2}}{z2}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      4. add-to-fraction N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      5. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2 \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2 \]

      9. *-commutative N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2 \]

      10. lower-*.f64 69.9%

          \[\leadsto \frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2 \]

   5. Applied rewrites 69.9%

      \[\leadsto \color{blue}{\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\left(-1 \cdot \frac{z2}{z1}\right) \cdot z2\right)\\ \mathbf{if}\;z0 \leq -4.5 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq 1.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{z4}}\\ \mathbf{elif}\;z0 \leq 6.2 \cdot 10^{+180}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{z0 - \left(\left(-0.5 - 0.3333333333333333 \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z1}{z2 \cdot z2} \cdot z4 + \frac{z3}{z4}} \cdot z4\\ \end{array} \]

FPCore

(FPCore (z0 z3 z4 z1 z2)
  :precision binary64
  (let* ((t_0
        (*
         (-
          (*
           (* (- -0.5 (* (- (* 0.25 z0) -0.3333333333333333) z0)) z0)
           z0)
          z0)
         (* (* -1.0 (/ z2 z1)) z2))))
  (if (<= z0 -4.5e+29)
    t_0
    (if (<= z0 1.1e-9)
      (/ (* z0 z2) (+ (/ z1 z2) (/ (* (/ z3 z4) z2) z4)))
      (if (<= z0 6.2e+180)
        t_0
        (*
         (/
          (- z0 (* (* (- -0.5 (* 0.3333333333333333 z0)) z0) z0))
          (+ (* (/ z1 (* z2 z2)) z4) (/ z3 z4)))
         z4))))))

C

double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = ((((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0) - z0) * ((-1.0 * (z2 / z1)) * z2);
	double tmp;
	if (z0 <= -4.5e+29) {
		tmp = t_0;
	} else if (z0 <= 1.1e-9) {
		tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4));
	} else if (z0 <= 6.2e+180) {
		tmp = t_0;
	} else {
		tmp = ((z0 - (((-0.5 - (0.3333333333333333 * z0)) * z0) * z0)) / (((z1 / (z2 * z2)) * z4) + (z3 / z4))) * z4;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0, z3, z4, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((((-0.5d0) - (((0.25d0 * z0) - (-0.3333333333333333d0)) * z0)) * z0) * z0) - z0) * (((-1.0d0) * (z2 / z1)) * z2)
    if (z0 <= (-4.5d+29)) then
        tmp = t_0
    else if (z0 <= 1.1d-9) then
        tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4))
    else if (z0 <= 6.2d+180) then
        tmp = t_0
    else
        tmp = ((z0 - ((((-0.5d0) - (0.3333333333333333d0 * z0)) * z0) * z0)) / (((z1 / (z2 * z2)) * z4) + (z3 / z4))) * z4
    end if
    code = tmp
end function

Java

public static double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = ((((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0) - z0) * ((-1.0 * (z2 / z1)) * z2);
	double tmp;
	if (z0 <= -4.5e+29) {
		tmp = t_0;
	} else if (z0 <= 1.1e-9) {
		tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4));
	} else if (z0 <= 6.2e+180) {
		tmp = t_0;
	} else {
		tmp = ((z0 - (((-0.5 - (0.3333333333333333 * z0)) * z0) * z0)) / (((z1 / (z2 * z2)) * z4) + (z3 / z4))) * z4;
	}
	return tmp;
}

Python

def code(z0, z3, z4, z1, z2):
	t_0 = ((((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0) - z0) * ((-1.0 * (z2 / z1)) * z2)
	tmp = 0
	if z0 <= -4.5e+29:
		tmp = t_0
	elif z0 <= 1.1e-9:
		tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4))
	elif z0 <= 6.2e+180:
		tmp = t_0
	else:
		tmp = ((z0 - (((-0.5 - (0.3333333333333333 * z0)) * z0) * z0)) / (((z1 / (z2 * z2)) * z4) + (z3 / z4))) * z4
	return tmp

Julia

function code(z0, z3, z4, z1, z2)
	t_0 = Float64(Float64(Float64(Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0) - z0) * Float64(Float64(-1.0 * Float64(z2 / z1)) * z2))
	tmp = 0.0
	if (z0 <= -4.5e+29)
		tmp = t_0;
	elseif (z0 <= 1.1e-9)
		tmp = Float64(Float64(z0 * z2) / Float64(Float64(z1 / z2) + Float64(Float64(Float64(z3 / z4) * z2) / z4)));
	elseif (z0 <= 6.2e+180)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(z0 - Float64(Float64(Float64(-0.5 - Float64(0.3333333333333333 * z0)) * z0) * z0)) / Float64(Float64(Float64(z1 / Float64(z2 * z2)) * z4) + Float64(z3 / z4))) * z4);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z3, z4, z1, z2)
	t_0 = ((((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0) - z0) * ((-1.0 * (z2 / z1)) * z2);
	tmp = 0.0;
	if (z0 <= -4.5e+29)
		tmp = t_0;
	elseif (z0 <= 1.1e-9)
		tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4));
	elseif (z0 <= 6.2e+180)
		tmp = t_0;
	else
		tmp = ((z0 - (((-0.5 - (0.3333333333333333 * z0)) * z0) * z0)) / (((z1 / (z2 * z2)) * z4) + (z3 / z4))) * z4;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z3_, z4_, z1_, z2_] := Block[{t$95$0 = N[(N[(N[(N[(N[(-0.5 - N[(N[(N[(0.25 * z0), $MachinePrecision] - -0.3333333333333333), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision] - z0), $MachinePrecision] * N[(N[(-1.0 * N[(z2 / z1), $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -4.5e+29], t$95$0, If[LessEqual[z0, 1.1e-9], N[(N[(z0 * z2), $MachinePrecision] / N[(N[(z1 / z2), $MachinePrecision] + N[(N[(N[(z3 / z4), $MachinePrecision] * z2), $MachinePrecision] / z4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 6.2e+180], t$95$0, N[(N[(N[(z0 - N[(N[(N[(-0.5 - N[(0.3333333333333333 * z0), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z1 / N[(z2 * z2), $MachinePrecision]), $MachinePrecision] * z4), $MachinePrecision] + N[(z3 / z4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z4), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -4.5000000000000002e29 or 1.0999999999999999e-9 < z0 < 6.2e180

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z3 around 0

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\left(-1 \cdot \frac{z2}{z1}\right)} \cdot z2\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\left(-1 \cdot \color{blue}{\frac{z2}{z1}}\right) \cdot z2\right) \]

      2. lower-/.f64 51.0%

         \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\left(-1 \cdot \frac{z2}{\color{blue}{z1}}\right) \cdot z2\right) \]

   8. Applied rewrites 51.0%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\left(-1 \cdot \frac{z2}{z1}\right)} \cdot z2\right) \]

   if -4.5000000000000002e29 < z0 < 1.0999999999999999e-9

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{\frac{z2 \cdot z3}{{z4}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\color{blue}{z2 \cdot z3}}{{z4}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

      7. lower-pow.f64 50.0%

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

      3. pow2 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{z4 \cdot \color{blue}{z4}}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{z4 \cdot \color{blue}{z4}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{z4} \cdot z4}} \]

      6. *-commutative N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3 \cdot z2}{\color{blue}{z4} \cdot z4}} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3}{z4 \cdot z4} \cdot \color{blue}{z2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3}{z4 \cdot z4} \cdot z2} \]

      9. associate-/l/ N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4}}{z4} \cdot z2} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4}}{z4} \cdot z2} \]

      11. associate-*l/ N/A

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{\color{blue}{z4}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{\color{blue}{z4}}} \]

      13. lower-*.f64 54.8%

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{z4}} \]

   10. Applied rewrites 54.8%

       \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{\color{blue}{z4}}} \]

   if 6.2e180 < z0

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      5. lower-/.f64 70.6%

         \[\leadsto \frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

   3. Applied rewrites 70.6%

      \[\leadsto \frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \frac{z0 - \left(\left(-0.5 - \color{blue}{\frac{3333333333333333}{10000000000000000}} \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{\frac{z1}{z2}}{z2}} \]
   5. Step-by-step derivation

   6. Applied rewrites 57.0%

      \[\leadsto \frac{z0 - \left(\left(-0.5 - \color{blue}{0.3333333333333333} \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{\frac{z1}{z2}}{z2}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{z0 - \color{blue}{\left(\left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot z0\right) \cdot z0}}{\frac{z3}{z4 \cdot z4} + \frac{\frac{z1}{z2}}{z2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z0 - \color{blue}{\left(\left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot z0\right)} \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{\frac{z1}{z2}}{z2}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{z0 - \color{blue}{\left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot \left(z0 \cdot z0\right)}}{\frac{z3}{z4 \cdot z4} + \frac{\frac{z1}{z2}}{z2}} \]

      4. lower-*.f32 N/A

         \[\leadsto \frac{z0 - \left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot \color{blue}{\left(z0 \cdot z0\right)}}{\frac{z3}{z4 \cdot z4} + \frac{\frac{z1}{z2}}{z2}} \]

      5. lower-unsound-*.f32 N/A

         \[\leadsto \frac{z0 - \left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot \color{blue}{\left(z0 \cdot z0\right)}}{\frac{z3}{z4 \cdot z4} + \frac{\frac{z1}{z2}}{z2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{z0 - \color{blue}{\left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot \left(z0 \cdot z0\right)}}{\frac{z3}{z4 \cdot z4} + \frac{\frac{z1}{z2}}{z2}} \]

      7. lower-unsound-*.f64 57.0%

         \[\leadsto \frac{z0 - \left(-0.5 - 0.3333333333333333 \cdot z0\right) \cdot \color{blue}{\left(z0 \cdot z0\right)}}{\frac{z3}{z4 \cdot z4} + \frac{\frac{z1}{z2}}{z2}} \]

   8. Applied rewrites 57.0%

      \[\leadsto \frac{z0 - \color{blue}{\left(-0.5 - 0.3333333333333333 \cdot z0\right) \cdot \left(z0 \cdot z0\right)}}{\frac{z3}{z4 \cdot z4} + \frac{\frac{z1}{z2}}{z2}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot \left(z0 \cdot z0\right)}{\frac{z3}{z4 \cdot z4} + \frac{\frac{z1}{z2}}{z2}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot \left(z0 \cdot z0\right)}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot \left(z0 \cdot z0\right)}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{z0 - \left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot \left(z0 \cdot z0\right)}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{z0 - \left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot \left(z0 \cdot z0\right)}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot \left(z0 \cdot z0\right)}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{z0 - \left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot \left(z0 \cdot z0\right)}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{z0 - \left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot \left(z0 \cdot z0\right)}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot \left(z0 \cdot z0\right)}{\frac{z1}{z2 \cdot z2} + \color{blue}{\frac{z3}{z4 \cdot z4}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{z0 - \left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot \left(z0 \cdot z0\right)}{\frac{z1}{z2 \cdot z2} + \frac{z3}{\color{blue}{z4 \cdot z4}}} \]

      11. associate-/l/ N/A

          \[\leadsto \frac{z0 - \left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot \left(z0 \cdot z0\right)}{\frac{z1}{z2 \cdot z2} + \color{blue}{\frac{\frac{z3}{z4}}{z4}}} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{z0 - \left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot \left(z0 \cdot z0\right)}{\frac{z1}{z2 \cdot z2} + \frac{\color{blue}{\frac{z3}{z4}}}{z4}} \]

      13. add-to-fraction N/A

          \[\leadsto \frac{z0 - \left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot \left(z0 \cdot z0\right)}{\color{blue}{\frac{\frac{z1}{z2 \cdot z2} \cdot z4 + \frac{z3}{z4}}{z4}}} \]

      14. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{z0 - \left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot \left(z0 \cdot z0\right)}{\frac{z1}{z2 \cdot z2} \cdot z4 + \frac{z3}{z4}} \cdot z4} \]

      15. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{z0 - \left(\frac{-1}{2} - \frac{3333333333333333}{10000000000000000} \cdot z0\right) \cdot \left(z0 \cdot z0\right)}{\frac{z1}{z2 \cdot z2} \cdot z4 + \frac{z3}{z4}} \cdot z4} \]

   10. Applied rewrites 56.3%

       \[\leadsto \color{blue}{\frac{z0 - \left(\left(-0.5 - 0.3333333333333333 \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z1}{z2 \cdot z2} \cdot z4 + \frac{z3}{z4}} \cdot z4} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 69.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0\\ t_1 := \left|z4\right| \cdot \left|z4\right|\\ \mathbf{if}\;\left|z4\right| \leq 3.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{t\_0}{\left(\frac{z1}{z2 \cdot z2} \cdot \left|z4\right|\right) \cdot \left|z4\right| + z3} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{z2 \cdot \frac{z3}{t\_1} + \frac{z1}{z2}} \cdot z2\\ \end{array} \]

FPCore

(FPCore (z0 z3 z4 z1 z2)
  :precision binary64
  (let* ((t_0
        (-
         z0
         (*
          (* (- -0.5 (* (- (* 0.25 z0) -0.3333333333333333) z0)) z0)
          z0)))
       (t_1 (* (fabs z4) (fabs z4))))
  (if (<= (fabs z4) 3.6e-16)
    (*
     (/ t_0 (+ (* (* (/ z1 (* z2 z2)) (fabs z4)) (fabs z4)) z3))
     t_1)
    (* (/ t_0 (+ (* z2 (/ z3 t_1)) (/ z1 z2))) z2))))

C

double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0);
	double t_1 = fabs(z4) * fabs(z4);
	double tmp;
	if (fabs(z4) <= 3.6e-16) {
		tmp = (t_0 / ((((z1 / (z2 * z2)) * fabs(z4)) * fabs(z4)) + z3)) * t_1;
	} else {
		tmp = (t_0 / ((z2 * (z3 / t_1)) + (z1 / z2))) * z2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0, z3, z4, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z0 - ((((-0.5d0) - (((0.25d0 * z0) - (-0.3333333333333333d0)) * z0)) * z0) * z0)
    t_1 = abs(z4) * abs(z4)
    if (abs(z4) <= 3.6d-16) then
        tmp = (t_0 / ((((z1 / (z2 * z2)) * abs(z4)) * abs(z4)) + z3)) * t_1
    else
        tmp = (t_0 / ((z2 * (z3 / t_1)) + (z1 / z2))) * z2
    end if
    code = tmp
end function

Java

public static double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0);
	double t_1 = Math.abs(z4) * Math.abs(z4);
	double tmp;
	if (Math.abs(z4) <= 3.6e-16) {
		tmp = (t_0 / ((((z1 / (z2 * z2)) * Math.abs(z4)) * Math.abs(z4)) + z3)) * t_1;
	} else {
		tmp = (t_0 / ((z2 * (z3 / t_1)) + (z1 / z2))) * z2;
	}
	return tmp;
}

Python

def code(z0, z3, z4, z1, z2):
	t_0 = z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)
	t_1 = math.fabs(z4) * math.fabs(z4)
	tmp = 0
	if math.fabs(z4) <= 3.6e-16:
		tmp = (t_0 / ((((z1 / (z2 * z2)) * math.fabs(z4)) * math.fabs(z4)) + z3)) * t_1
	else:
		tmp = (t_0 / ((z2 * (z3 / t_1)) + (z1 / z2))) * z2
	return tmp

Julia

function code(z0, z3, z4, z1, z2)
	t_0 = Float64(z0 - Float64(Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0))
	t_1 = Float64(abs(z4) * abs(z4))
	tmp = 0.0
	if (abs(z4) <= 3.6e-16)
		tmp = Float64(Float64(t_0 / Float64(Float64(Float64(Float64(z1 / Float64(z2 * z2)) * abs(z4)) * abs(z4)) + z3)) * t_1);
	else
		tmp = Float64(Float64(t_0 / Float64(Float64(z2 * Float64(z3 / t_1)) + Float64(z1 / z2))) * z2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z3, z4, z1, z2)
	t_0 = z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0);
	t_1 = abs(z4) * abs(z4);
	tmp = 0.0;
	if (abs(z4) <= 3.6e-16)
		tmp = (t_0 / ((((z1 / (z2 * z2)) * abs(z4)) * abs(z4)) + z3)) * t_1;
	else
		tmp = (t_0 / ((z2 * (z3 / t_1)) + (z1 / z2))) * z2;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z3_, z4_, z1_, z2_] := Block[{t$95$0 = N[(z0 - N[(N[(N[(-0.5 - N[(N[(N[(0.25 * z0), $MachinePrecision] - -0.3333333333333333), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[z4], $MachinePrecision] * N[Abs[z4], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[z4], $MachinePrecision], 3.6e-16], N[(N[(t$95$0 / N[(N[(N[(N[(z1 / N[(z2 * z2), $MachinePrecision]), $MachinePrecision] * N[Abs[z4], $MachinePrecision]), $MachinePrecision] * N[Abs[z4], $MachinePrecision]), $MachinePrecision] + z3), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(t$95$0 / N[(N[(z2 * N[(z3 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(z1 / z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z4 < 3.5999999999999998e-16

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      5. lower-/.f64 70.6%

         \[\leadsto \frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

   3. Applied rewrites 70.6%

      \[\leadsto \frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{\frac{z1}{z2}}{z2}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{\frac{z1}{z2}}{z2}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      5. associate-/l/ N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      8. +-commutative N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z1}{z2 \cdot z2} + \color{blue}{\frac{z3}{z4 \cdot z4}}} \]

      10. add-to-fraction N/A

          \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{\frac{z1}{z2 \cdot z2} \cdot \left(z4 \cdot z4\right) + z3}{z4 \cdot z4}}} \]

      11. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z1}{z2 \cdot z2} \cdot \left(z4 \cdot z4\right) + z3} \cdot \left(z4 \cdot z4\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z1}{z2 \cdot z2} \cdot \left(z4 \cdot z4\right) + z3} \cdot \left(z4 \cdot z4\right)} \]

   5. Applied rewrites 55.1%

      \[\leadsto \color{blue}{\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\left(\frac{z1}{z2 \cdot z2} \cdot z4\right) \cdot z4 + z3} \cdot \left(z4 \cdot z4\right)} \]

   if 3.5999999999999998e-16 < z4

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      5. lower-/.f64 70.6%

         \[\leadsto \frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

   3. Applied rewrites 70.6%

      \[\leadsto \frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{\frac{z1}{z2}}{z2}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{\frac{z1}{z2}}{z2}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      4. add-to-fraction N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      5. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2 \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2 \]

      9. *-commutative N/A

         \[\leadsto \frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2 \]

      10. lower-*.f64 69.9%

          \[\leadsto \frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2 \]

   5. Applied rewrites 69.9%

      \[\leadsto \color{blue}{\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 69.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\\ \mathbf{if}\;z0 \leq -4.5 \cdot 10^{+29}:\\ \;\;\;\;t\_0 \cdot \left(\left(-1 \cdot \frac{z2}{z1}\right) \cdot z2\right)\\ \mathbf{elif}\;z0 \leq 5.6 \cdot 10^{-49}:\\ \;\;\;\;\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{z4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\frac{z1}{\left(-z2\right) \cdot z2} - \frac{z3}{z4 \cdot z4}}\\ \end{array} \]

FPCore

(FPCore (z0 z3 z4 z1 z2)
  :precision binary64
  (let* ((t_0
        (-
         (*
          (* (- -0.5 (* (- (* 0.25 z0) -0.3333333333333333) z0)) z0)
          z0)
         z0)))
  (if (<= z0 -4.5e+29)
    (* t_0 (* (* -1.0 (/ z2 z1)) z2))
    (if (<= z0 5.6e-49)
      (/ (* z0 z2) (+ (/ z1 z2) (/ (* (/ z3 z4) z2) z4)))
      (/ t_0 (- (/ z1 (* (- z2) z2)) (/ z3 (* z4 z4))))))))

C

double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0) - z0;
	double tmp;
	if (z0 <= -4.5e+29) {
		tmp = t_0 * ((-1.0 * (z2 / z1)) * z2);
	} else if (z0 <= 5.6e-49) {
		tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4));
	} else {
		tmp = t_0 / ((z1 / (-z2 * z2)) - (z3 / (z4 * z4)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0, z3, z4, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((-0.5d0) - (((0.25d0 * z0) - (-0.3333333333333333d0)) * z0)) * z0) * z0) - z0
    if (z0 <= (-4.5d+29)) then
        tmp = t_0 * (((-1.0d0) * (z2 / z1)) * z2)
    else if (z0 <= 5.6d-49) then
        tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4))
    else
        tmp = t_0 / ((z1 / (-z2 * z2)) - (z3 / (z4 * z4)))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0) - z0;
	double tmp;
	if (z0 <= -4.5e+29) {
		tmp = t_0 * ((-1.0 * (z2 / z1)) * z2);
	} else if (z0 <= 5.6e-49) {
		tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4));
	} else {
		tmp = t_0 / ((z1 / (-z2 * z2)) - (z3 / (z4 * z4)));
	}
	return tmp;
}

Python

def code(z0, z3, z4, z1, z2):
	t_0 = (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0) - z0
	tmp = 0
	if z0 <= -4.5e+29:
		tmp = t_0 * ((-1.0 * (z2 / z1)) * z2)
	elif z0 <= 5.6e-49:
		tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4))
	else:
		tmp = t_0 / ((z1 / (-z2 * z2)) - (z3 / (z4 * z4)))
	return tmp

Julia

function code(z0, z3, z4, z1, z2)
	t_0 = Float64(Float64(Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0) - z0)
	tmp = 0.0
	if (z0 <= -4.5e+29)
		tmp = Float64(t_0 * Float64(Float64(-1.0 * Float64(z2 / z1)) * z2));
	elseif (z0 <= 5.6e-49)
		tmp = Float64(Float64(z0 * z2) / Float64(Float64(z1 / z2) + Float64(Float64(Float64(z3 / z4) * z2) / z4)));
	else
		tmp = Float64(t_0 / Float64(Float64(z1 / Float64(Float64(-z2) * z2)) - Float64(z3 / Float64(z4 * z4))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z3, z4, z1, z2)
	t_0 = (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0) - z0;
	tmp = 0.0;
	if (z0 <= -4.5e+29)
		tmp = t_0 * ((-1.0 * (z2 / z1)) * z2);
	elseif (z0 <= 5.6e-49)
		tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4));
	else
		tmp = t_0 / ((z1 / (-z2 * z2)) - (z3 / (z4 * z4)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z3_, z4_, z1_, z2_] := Block[{t$95$0 = N[(N[(N[(N[(-0.5 - N[(N[(N[(0.25 * z0), $MachinePrecision] - -0.3333333333333333), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision] - z0), $MachinePrecision]}, If[LessEqual[z0, -4.5e+29], N[(t$95$0 * N[(N[(-1.0 * N[(z2 / z1), $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 5.6e-49], N[(N[(z0 * z2), $MachinePrecision] / N[(N[(z1 / z2), $MachinePrecision] + N[(N[(N[(z3 / z4), $MachinePrecision] * z2), $MachinePrecision] / z4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(z1 / N[((-z2) * z2), $MachinePrecision]), $MachinePrecision] - N[(z3 / N[(z4 * z4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -4.5000000000000002e29

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z3 around 0

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\left(-1 \cdot \frac{z2}{z1}\right)} \cdot z2\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\left(-1 \cdot \color{blue}{\frac{z2}{z1}}\right) \cdot z2\right) \]

      2. lower-/.f64 51.0%

         \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\left(-1 \cdot \frac{z2}{\color{blue}{z1}}\right) \cdot z2\right) \]

   8. Applied rewrites 51.0%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\left(-1 \cdot \frac{z2}{z1}\right)} \cdot z2\right) \]

   if -4.5000000000000002e29 < z0 < 5.5999999999999999e-49

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{\frac{z2 \cdot z3}{{z4}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\color{blue}{z2 \cdot z3}}{{z4}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

      7. lower-pow.f64 50.0%

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

      3. pow2 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{z4 \cdot \color{blue}{z4}}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{z4 \cdot \color{blue}{z4}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{z4} \cdot z4}} \]

      6. *-commutative N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3 \cdot z2}{\color{blue}{z4} \cdot z4}} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3}{z4 \cdot z4} \cdot \color{blue}{z2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3}{z4 \cdot z4} \cdot z2} \]

      9. associate-/l/ N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4}}{z4} \cdot z2} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4}}{z4} \cdot z2} \]

      11. associate-*l/ N/A

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{\color{blue}{z4}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{\color{blue}{z4}}} \]

      13. lower-*.f64 54.8%

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{z4}} \]

   10. Applied rewrites 54.8%

       \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{\color{blue}{z4}}} \]

   if 5.5999999999999999e-49 < z0

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      5. sub-negate-rev N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0}}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0}}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0}{\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)} \]

      8. add-flip N/A

         \[\leadsto \frac{\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0}{\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)} \]

      9. sub-negate N/A

         \[\leadsto \frac{\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0}{\color{blue}{\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}}} \]

      10. lower--.f64 N/A

          \[\leadsto \frac{\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0}{\color{blue}{\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}}} \]

   3. Applied rewrites 66.0%

      \[\leadsto \color{blue}{\frac{\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0}{\frac{z1}{\left(-z2\right) \cdot z2} - \frac{z3}{z4 \cdot z4}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 68.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0\\ \mathbf{if}\;z0 \leq -4.5 \cdot 10^{+29}:\\ \;\;\;\;\left(t\_0 - z0\right) \cdot \left(\left(-1 \cdot \frac{z2}{z1}\right) \cdot z2\right)\\ \mathbf{elif}\;z0 \leq 3.35 \cdot 10^{-49}:\\ \;\;\;\;\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{z4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z0 - t\_0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}\\ \end{array} \]

FPCore

(FPCore (z0 z3 z4 z1 z2)
  :precision binary64
  (let* ((t_0
        (*
         (* (- -0.5 (* (- (* 0.25 z0) -0.3333333333333333) z0)) z0)
         z0)))
  (if (<= z0 -4.5e+29)
    (* (- t_0 z0) (* (* -1.0 (/ z2 z1)) z2))
    (if (<= z0 3.35e-49)
      (/ (* z0 z2) (+ (/ z1 z2) (/ (* (/ z3 z4) z2) z4)))
      (/ (- z0 t_0) (+ (/ z3 (* z4 z4)) (/ z1 (* z2 z2))))))))

C

double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = ((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0;
	double tmp;
	if (z0 <= -4.5e+29) {
		tmp = (t_0 - z0) * ((-1.0 * (z2 / z1)) * z2);
	} else if (z0 <= 3.35e-49) {
		tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4));
	} else {
		tmp = (z0 - t_0) / ((z3 / (z4 * z4)) + (z1 / (z2 * z2)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0, z3, z4, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((-0.5d0) - (((0.25d0 * z0) - (-0.3333333333333333d0)) * z0)) * z0) * z0
    if (z0 <= (-4.5d+29)) then
        tmp = (t_0 - z0) * (((-1.0d0) * (z2 / z1)) * z2)
    else if (z0 <= 3.35d-49) then
        tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4))
    else
        tmp = (z0 - t_0) / ((z3 / (z4 * z4)) + (z1 / (z2 * z2)))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = ((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0;
	double tmp;
	if (z0 <= -4.5e+29) {
		tmp = (t_0 - z0) * ((-1.0 * (z2 / z1)) * z2);
	} else if (z0 <= 3.35e-49) {
		tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4));
	} else {
		tmp = (z0 - t_0) / ((z3 / (z4 * z4)) + (z1 / (z2 * z2)));
	}
	return tmp;
}

Python

def code(z0, z3, z4, z1, z2):
	t_0 = ((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0
	tmp = 0
	if z0 <= -4.5e+29:
		tmp = (t_0 - z0) * ((-1.0 * (z2 / z1)) * z2)
	elif z0 <= 3.35e-49:
		tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4))
	else:
		tmp = (z0 - t_0) / ((z3 / (z4 * z4)) + (z1 / (z2 * z2)))
	return tmp

Julia

function code(z0, z3, z4, z1, z2)
	t_0 = Float64(Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)
	tmp = 0.0
	if (z0 <= -4.5e+29)
		tmp = Float64(Float64(t_0 - z0) * Float64(Float64(-1.0 * Float64(z2 / z1)) * z2));
	elseif (z0 <= 3.35e-49)
		tmp = Float64(Float64(z0 * z2) / Float64(Float64(z1 / z2) + Float64(Float64(Float64(z3 / z4) * z2) / z4)));
	else
		tmp = Float64(Float64(z0 - t_0) / Float64(Float64(z3 / Float64(z4 * z4)) + Float64(z1 / Float64(z2 * z2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z3, z4, z1, z2)
	t_0 = ((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0;
	tmp = 0.0;
	if (z0 <= -4.5e+29)
		tmp = (t_0 - z0) * ((-1.0 * (z2 / z1)) * z2);
	elseif (z0 <= 3.35e-49)
		tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4));
	else
		tmp = (z0 - t_0) / ((z3 / (z4 * z4)) + (z1 / (z2 * z2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z3_, z4_, z1_, z2_] := Block[{t$95$0 = N[(N[(N[(-0.5 - N[(N[(N[(0.25 * z0), $MachinePrecision] - -0.3333333333333333), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision]}, If[LessEqual[z0, -4.5e+29], N[(N[(t$95$0 - z0), $MachinePrecision] * N[(N[(-1.0 * N[(z2 / z1), $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 3.35e-49], N[(N[(z0 * z2), $MachinePrecision] / N[(N[(z1 / z2), $MachinePrecision] + N[(N[(N[(z3 / z4), $MachinePrecision] * z2), $MachinePrecision] / z4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z0 - t$95$0), $MachinePrecision] / N[(N[(z3 / N[(z4 * z4), $MachinePrecision]), $MachinePrecision] + N[(z1 / N[(z2 * z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -4.5000000000000002e29

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z3 around 0

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\left(-1 \cdot \frac{z2}{z1}\right)} \cdot z2\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\left(-1 \cdot \color{blue}{\frac{z2}{z1}}\right) \cdot z2\right) \]

      2. lower-/.f64 51.0%

         \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\left(-1 \cdot \frac{z2}{\color{blue}{z1}}\right) \cdot z2\right) \]

   8. Applied rewrites 51.0%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\left(-1 \cdot \frac{z2}{z1}\right)} \cdot z2\right) \]

   if -4.5000000000000002e29 < z0 < 3.3499999999999999e-49

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{\frac{z2 \cdot z3}{{z4}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\color{blue}{z2 \cdot z3}}{{z4}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

      7. lower-pow.f64 50.0%

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

      3. pow2 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{z4 \cdot \color{blue}{z4}}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{z4 \cdot \color{blue}{z4}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{z4} \cdot z4}} \]

      6. *-commutative N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3 \cdot z2}{\color{blue}{z4} \cdot z4}} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3}{z4 \cdot z4} \cdot \color{blue}{z2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3}{z4 \cdot z4} \cdot z2} \]

      9. associate-/l/ N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4}}{z4} \cdot z2} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4}}{z4} \cdot z2} \]

      11. associate-*l/ N/A

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{\color{blue}{z4}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{\color{blue}{z4}}} \]

      13. lower-*.f64 54.8%

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{z4}} \]

   10. Applied rewrites 54.8%

       \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{\color{blue}{z4}}} \]

   if 3.3499999999999999e-49 < z0

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0\\ \mathbf{if}\;z0 - t\_0 \leq 10^{-15}:\\ \;\;\;\;\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{z4}}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - z0\right) \cdot \left(\left(-1 \cdot \frac{z2}{z1}\right) \cdot z2\right)\\ \end{array} \]

FPCore

(FPCore (z0 z3 z4 z1 z2)
  :precision binary64
  (let* ((t_0
        (*
         (* (- -0.5 (* (- (* 0.25 z0) -0.3333333333333333) z0)) z0)
         z0)))
  (if (<= (- z0 t_0) 1e-15)
    (/ (* z0 z2) (+ (/ z1 z2) (/ (* (/ z3 z4) z2) z4)))
    (* (- t_0 z0) (* (* -1.0 (/ z2 z1)) z2)))))

C

double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = ((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0;
	double tmp;
	if ((z0 - t_0) <= 1e-15) {
		tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4));
	} else {
		tmp = (t_0 - z0) * ((-1.0 * (z2 / z1)) * z2);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0, z3, z4, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((-0.5d0) - (((0.25d0 * z0) - (-0.3333333333333333d0)) * z0)) * z0) * z0
    if ((z0 - t_0) <= 1d-15) then
        tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4))
    else
        tmp = (t_0 - z0) * (((-1.0d0) * (z2 / z1)) * z2)
    end if
    code = tmp
end function

Java

public static double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = ((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0;
	double tmp;
	if ((z0 - t_0) <= 1e-15) {
		tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4));
	} else {
		tmp = (t_0 - z0) * ((-1.0 * (z2 / z1)) * z2);
	}
	return tmp;
}

Python

def code(z0, z3, z4, z1, z2):
	t_0 = ((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0
	tmp = 0
	if (z0 - t_0) <= 1e-15:
		tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4))
	else:
		tmp = (t_0 - z0) * ((-1.0 * (z2 / z1)) * z2)
	return tmp

Julia

function code(z0, z3, z4, z1, z2)
	t_0 = Float64(Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)
	tmp = 0.0
	if (Float64(z0 - t_0) <= 1e-15)
		tmp = Float64(Float64(z0 * z2) / Float64(Float64(z1 / z2) + Float64(Float64(Float64(z3 / z4) * z2) / z4)));
	else
		tmp = Float64(Float64(t_0 - z0) * Float64(Float64(-1.0 * Float64(z2 / z1)) * z2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z3, z4, z1, z2)
	t_0 = ((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0;
	tmp = 0.0;
	if ((z0 - t_0) <= 1e-15)
		tmp = (z0 * z2) / ((z1 / z2) + (((z3 / z4) * z2) / z4));
	else
		tmp = (t_0 - z0) * ((-1.0 * (z2 / z1)) * z2);
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z3_, z4_, z1_, z2_] := Block[{t$95$0 = N[(N[(N[(-0.5 - N[(N[(N[(0.25 * z0), $MachinePrecision] - -0.3333333333333333), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision]}, If[LessEqual[N[(z0 - t$95$0), $MachinePrecision], 1e-15], N[(N[(z0 * z2), $MachinePrecision] / N[(N[(z1 / z2), $MachinePrecision] + N[(N[(N[(z3 / z4), $MachinePrecision] * z2), $MachinePrecision] / z4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - z0), $MachinePrecision] * N[(N[(-1.0 * N[(z2 / z1), $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z0 (*.f64 (*.f64 (-.f64 #s(literal -1/2 binary64) (*.f64 (-.f64 (*.f64 #s(literal 1/4 binary64) z0) #s(literal -3333333333333333/10000000000000000 binary64)) z0)) z0) z0)) < 1.0000000000000001e-15

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{\frac{z2 \cdot z3}{{z4}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\color{blue}{z2 \cdot z3}}{{z4}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

      7. lower-pow.f64 50.0%

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

      3. pow2 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{z4 \cdot \color{blue}{z4}}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{z4 \cdot \color{blue}{z4}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{z4} \cdot z4}} \]

      6. *-commutative N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3 \cdot z2}{\color{blue}{z4} \cdot z4}} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3}{z4 \cdot z4} \cdot \color{blue}{z2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3}{z4 \cdot z4} \cdot z2} \]

      9. associate-/l/ N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4}}{z4} \cdot z2} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4}}{z4} \cdot z2} \]

      11. associate-*l/ N/A

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{\color{blue}{z4}}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{\color{blue}{z4}}} \]

      13. lower-*.f64 54.8%

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{z4}} \]

   10. Applied rewrites 54.8%

       \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\frac{z3}{z4} \cdot z2}{\color{blue}{z4}}} \]

   if 1.0000000000000001e-15 < (-.f64 z0 (*.f64 (*.f64 (-.f64 #s(literal -1/2 binary64) (*.f64 (-.f64 (*.f64 #s(literal 1/4 binary64) z0) #s(literal -3333333333333333/10000000000000000 binary64)) z0)) z0) z0))

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z3 around 0

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\left(-1 \cdot \frac{z2}{z1}\right)} \cdot z2\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\left(-1 \cdot \color{blue}{\frac{z2}{z1}}\right) \cdot z2\right) \]

      2. lower-/.f64 51.0%

         \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\left(-1 \cdot \frac{z2}{\color{blue}{z1}}\right) \cdot z2\right) \]

   8. Applied rewrites 51.0%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\left(-1 \cdot \frac{z2}{z1}\right)} \cdot z2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 64.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0\\ \mathbf{if}\;z0 - t\_0 \leq 10^{-15}:\\ \;\;\;\;\left(\frac{z2}{\left(\frac{z3}{z4 \cdot z4} \cdot z2\right) \cdot z2 + z1} \cdot z2\right) \cdot z0\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 - z0\right) \cdot \left(\left(-1 \cdot \frac{z2}{z1}\right) \cdot z2\right)\\ \end{array} \]

FPCore

(FPCore (z0 z3 z4 z1 z2)
  :precision binary64
  (let* ((t_0
        (*
         (* (- -0.5 (* (- (* 0.25 z0) -0.3333333333333333) z0)) z0)
         z0)))
  (if (<= (- z0 t_0) 1e-15)
    (* (* (/ z2 (+ (* (* (/ z3 (* z4 z4)) z2) z2) z1)) z2) z0)
    (* (- t_0 z0) (* (* -1.0 (/ z2 z1)) z2)))))

C

double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = ((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0;
	double tmp;
	if ((z0 - t_0) <= 1e-15) {
		tmp = ((z2 / ((((z3 / (z4 * z4)) * z2) * z2) + z1)) * z2) * z0;
	} else {
		tmp = (t_0 - z0) * ((-1.0 * (z2 / z1)) * z2);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0, z3, z4, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((-0.5d0) - (((0.25d0 * z0) - (-0.3333333333333333d0)) * z0)) * z0) * z0
    if ((z0 - t_0) <= 1d-15) then
        tmp = ((z2 / ((((z3 / (z4 * z4)) * z2) * z2) + z1)) * z2) * z0
    else
        tmp = (t_0 - z0) * (((-1.0d0) * (z2 / z1)) * z2)
    end if
    code = tmp
end function

Java

public static double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = ((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0;
	double tmp;
	if ((z0 - t_0) <= 1e-15) {
		tmp = ((z2 / ((((z3 / (z4 * z4)) * z2) * z2) + z1)) * z2) * z0;
	} else {
		tmp = (t_0 - z0) * ((-1.0 * (z2 / z1)) * z2);
	}
	return tmp;
}

Python

def code(z0, z3, z4, z1, z2):
	t_0 = ((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0
	tmp = 0
	if (z0 - t_0) <= 1e-15:
		tmp = ((z2 / ((((z3 / (z4 * z4)) * z2) * z2) + z1)) * z2) * z0
	else:
		tmp = (t_0 - z0) * ((-1.0 * (z2 / z1)) * z2)
	return tmp

Julia

function code(z0, z3, z4, z1, z2)
	t_0 = Float64(Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)
	tmp = 0.0
	if (Float64(z0 - t_0) <= 1e-15)
		tmp = Float64(Float64(Float64(z2 / Float64(Float64(Float64(Float64(z3 / Float64(z4 * z4)) * z2) * z2) + z1)) * z2) * z0);
	else
		tmp = Float64(Float64(t_0 - z0) * Float64(Float64(-1.0 * Float64(z2 / z1)) * z2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z3, z4, z1, z2)
	t_0 = ((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0;
	tmp = 0.0;
	if ((z0 - t_0) <= 1e-15)
		tmp = ((z2 / ((((z3 / (z4 * z4)) * z2) * z2) + z1)) * z2) * z0;
	else
		tmp = (t_0 - z0) * ((-1.0 * (z2 / z1)) * z2);
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z3_, z4_, z1_, z2_] := Block[{t$95$0 = N[(N[(N[(-0.5 - N[(N[(N[(0.25 * z0), $MachinePrecision] - -0.3333333333333333), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision]}, If[LessEqual[N[(z0 - t$95$0), $MachinePrecision], 1e-15], N[(N[(N[(z2 / N[(N[(N[(N[(z3 / N[(z4 * z4), $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision] * z2), $MachinePrecision] + z1), $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision] * z0), $MachinePrecision], N[(N[(t$95$0 - z0), $MachinePrecision] * N[(N[(-1.0 * N[(z2 / z1), $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z0 (*.f64 (*.f64 (-.f64 #s(literal -1/2 binary64) (*.f64 (-.f64 (*.f64 #s(literal 1/4 binary64) z0) #s(literal -3333333333333333/10000000000000000 binary64)) z0)) z0) z0)) < 1.0000000000000001e-15

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{\frac{z2 \cdot z3}{{z4}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\color{blue}{z2 \cdot z3}}{{z4}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

      7. lower-pow.f64 50.0%

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

   10. Applied rewrites 50.2%

       \[\leadsto \left(\frac{z2}{\left(\frac{z3}{z4 \cdot z4} \cdot z2\right) \cdot z2 + z1} \cdot z2\right) \cdot \color{blue}{z0} \]

   if 1.0000000000000001e-15 < (-.f64 z0 (*.f64 (*.f64 (-.f64 #s(literal -1/2 binary64) (*.f64 (-.f64 (*.f64 #s(literal 1/4 binary64) z0) #s(literal -3333333333333333/10000000000000000 binary64)) z0)) z0) z0))

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z3 around 0

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\left(-1 \cdot \frac{z2}{z1}\right)} \cdot z2\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\left(-1 \cdot \color{blue}{\frac{z2}{z1}}\right) \cdot z2\right) \]

      2. lower-/.f64 51.0%

         \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\left(-1 \cdot \frac{z2}{\color{blue}{z1}}\right) \cdot z2\right) \]

   8. Applied rewrites 51.0%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\left(-1 \cdot \frac{z2}{z1}\right)} \cdot z2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 55.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left|z4\right| \cdot \left|z4\right|\\ \mathbf{if}\;\left|z4\right| \leq 160000000000:\\ \;\;\;\;\frac{-z0}{\left(\frac{z1}{\left(-\left|z2\right|\right) \cdot \left|z2\right|} \cdot \left|z4\right|\right) \cdot \left|z4\right| - z3} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{z0 \cdot \left|z2\right|}{\frac{z1}{\left|z2\right|} + z3 \cdot \frac{\left|z2\right|}{t\_0}}\\ \end{array} \]

FPCore

(FPCore (z0 z3 z4 z1 z2)
  :precision binary64
  (let* ((t_0 (* (fabs z4) (fabs z4))))
  (if (<= (fabs z4) 160000000000.0)
    (*
     (/
      (- z0)
      (-
       (* (* (/ z1 (* (- (fabs z2)) (fabs z2))) (fabs z4)) (fabs z4))
       z3))
     t_0)
    (/
     (* z0 (fabs z2))
     (+ (/ z1 (fabs z2)) (* z3 (/ (fabs z2) t_0)))))))

C

double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = fabs(z4) * fabs(z4);
	double tmp;
	if (fabs(z4) <= 160000000000.0) {
		tmp = (-z0 / ((((z1 / (-fabs(z2) * fabs(z2))) * fabs(z4)) * fabs(z4)) - z3)) * t_0;
	} else {
		tmp = (z0 * fabs(z2)) / ((z1 / fabs(z2)) + (z3 * (fabs(z2) / t_0)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0, z3, z4, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(z4) * abs(z4)
    if (abs(z4) <= 160000000000.0d0) then
        tmp = (-z0 / ((((z1 / (-abs(z2) * abs(z2))) * abs(z4)) * abs(z4)) - z3)) * t_0
    else
        tmp = (z0 * abs(z2)) / ((z1 / abs(z2)) + (z3 * (abs(z2) / t_0)))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = Math.abs(z4) * Math.abs(z4);
	double tmp;
	if (Math.abs(z4) <= 160000000000.0) {
		tmp = (-z0 / ((((z1 / (-Math.abs(z2) * Math.abs(z2))) * Math.abs(z4)) * Math.abs(z4)) - z3)) * t_0;
	} else {
		tmp = (z0 * Math.abs(z2)) / ((z1 / Math.abs(z2)) + (z3 * (Math.abs(z2) / t_0)));
	}
	return tmp;
}

Python

def code(z0, z3, z4, z1, z2):
	t_0 = math.fabs(z4) * math.fabs(z4)
	tmp = 0
	if math.fabs(z4) <= 160000000000.0:
		tmp = (-z0 / ((((z1 / (-math.fabs(z2) * math.fabs(z2))) * math.fabs(z4)) * math.fabs(z4)) - z3)) * t_0
	else:
		tmp = (z0 * math.fabs(z2)) / ((z1 / math.fabs(z2)) + (z3 * (math.fabs(z2) / t_0)))
	return tmp

Julia

function code(z0, z3, z4, z1, z2)
	t_0 = Float64(abs(z4) * abs(z4))
	tmp = 0.0
	if (abs(z4) <= 160000000000.0)
		tmp = Float64(Float64(Float64(-z0) / Float64(Float64(Float64(Float64(z1 / Float64(Float64(-abs(z2)) * abs(z2))) * abs(z4)) * abs(z4)) - z3)) * t_0);
	else
		tmp = Float64(Float64(z0 * abs(z2)) / Float64(Float64(z1 / abs(z2)) + Float64(z3 * Float64(abs(z2) / t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z3, z4, z1, z2)
	t_0 = abs(z4) * abs(z4);
	tmp = 0.0;
	if (abs(z4) <= 160000000000.0)
		tmp = (-z0 / ((((z1 / (-abs(z2) * abs(z2))) * abs(z4)) * abs(z4)) - z3)) * t_0;
	else
		tmp = (z0 * abs(z2)) / ((z1 / abs(z2)) + (z3 * (abs(z2) / t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z3_, z4_, z1_, z2_] := Block[{t$95$0 = N[(N[Abs[z4], $MachinePrecision] * N[Abs[z4], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[z4], $MachinePrecision], 160000000000.0], N[(N[((-z0) / N[(N[(N[(N[(z1 / N[((-N[Abs[z2], $MachinePrecision]) * N[Abs[z2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[z4], $MachinePrecision]), $MachinePrecision] * N[Abs[z4], $MachinePrecision]), $MachinePrecision] - z3), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(z0 * N[Abs[z2], $MachinePrecision]), $MachinePrecision] / N[(N[(z1 / N[Abs[z2], $MachinePrecision]), $MachinePrecision] + N[(z3 * N[(N[Abs[z2], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z4 < 1.6e11

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\left(-1 \cdot z0\right)} \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}} \]
   5. Step-by-step derivation

      1. lower-*.f64 49.1%

         \[\leadsto \left(-1 \cdot \color{blue}{z0}\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}} \]

   6. Applied rewrites 49.1%

      \[\leadsto \color{blue}{\left(-1 \cdot z0\right)} \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-1 \cdot z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. lift-+.f64 N/A

         \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2}} + \frac{z3}{z4 \cdot z4}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\frac{z1}{\color{blue}{z2 \cdot z2}} + \frac{z3}{z4 \cdot z4}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z1}{z2}}{z2}} + \frac{z3}{z4 \cdot z4}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\frac{\color{blue}{\frac{z1}{z2}}}{z2} + \frac{z3}{z4 \cdot z4}} \]

      8. mult-flip-rev N/A

         \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2} \cdot \frac{1}{z2}} + \frac{z3}{z4 \cdot z4}} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\frac{z1}{z2} \cdot \color{blue}{\frac{1}{z2}} + \frac{z3}{z4 \cdot z4}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2} \cdot \frac{1}{z2}} + \frac{z3}{z4 \cdot z4}} \]

      11. +-commutative N/A

          \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2} \cdot \frac{1}{z2}}} \]

      12. lift-+.f64 N/A

          \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2} \cdot \frac{1}{z2}}} \]

      13. frac-2neg-rev N/A

          \[\leadsto \left(-1 \cdot z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2} \cdot \frac{1}{z2}\right)\right)}} \]

      14. metadata-eval N/A

          \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{\color{blue}{1}}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2} \cdot \frac{1}{z2}\right)\right)} \]

   8. Applied rewrites 49.3%

      \[\leadsto \color{blue}{\frac{-z0}{\frac{z1}{\left(-z2\right) \cdot z2} - \frac{z3}{z4 \cdot z4}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-z0}{\frac{z1}{\left(-z2\right) \cdot z2} - \frac{z3}{z4 \cdot z4}}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{-z0}{\color{blue}{\frac{z1}{\left(-z2\right) \cdot z2} - \frac{z3}{z4 \cdot z4}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{-z0}{\frac{z1}{\left(-z2\right) \cdot z2} - \color{blue}{\frac{z3}{z4 \cdot z4}}} \]

      4. sub-to-fraction N/A

         \[\leadsto \frac{-z0}{\color{blue}{\frac{\frac{z1}{\left(-z2\right) \cdot z2} \cdot \left(z4 \cdot z4\right) - z3}{z4 \cdot z4}}} \]

      5. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{-z0}{\frac{z1}{\left(-z2\right) \cdot z2} \cdot \left(z4 \cdot z4\right) - z3} \cdot \left(z4 \cdot z4\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-z0}{\frac{z1}{\left(-z2\right) \cdot z2} \cdot \left(z4 \cdot z4\right) - z3} \cdot \left(z4 \cdot z4\right)} \]

   10. Applied rewrites 42.2%

       \[\leadsto \color{blue}{\frac{-z0}{\left(\frac{z1}{\left(-z2\right) \cdot z2} \cdot z4\right) \cdot z4 - z3} \cdot \left(z4 \cdot z4\right)} \]

   if 1.6e11 < z4

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{\frac{z2 \cdot z3}{{z4}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\color{blue}{z2 \cdot z3}}{{z4}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

      7. lower-pow.f64 50.0%

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3 \cdot z2}{{\color{blue}{z4}}^{2}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3 \cdot z2}{{z4}^{\color{blue}{2}}}} \]

      5. pow2 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3 \cdot z2}{z4 \cdot \color{blue}{z4}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3 \cdot z2}{z4 \cdot \color{blue}{z4}}} \]

      7. associate-/l* N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + z3 \cdot \color{blue}{\frac{z2}{z4 \cdot z4}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + z3 \cdot \color{blue}{\frac{z2}{z4 \cdot z4}}} \]

      9. lower-/.f64 52.2%

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + z3 \cdot \frac{z2}{\color{blue}{z4 \cdot z4}}} \]

   10. Applied rewrites 52.2%

       \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + z3 \cdot \color{blue}{\frac{z2}{z4 \cdot z4}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 55.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|z4\right| \leq 1.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{z0}{\frac{z1}{\left|z2\right| \cdot \left|z2\right|} \cdot \left|z4\right| + \frac{z3}{\left|z4\right|}} \cdot \left|z4\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{z0 \cdot \left|z2\right|}{\frac{z1}{\left|z2\right|} + z3 \cdot \frac{\left|z2\right|}{\left|z4\right| \cdot \left|z4\right|}}\\ \end{array} \]

FPCore

(FPCore (z0 z3 z4 z1 z2)
  :precision binary64
  (if (<= (fabs z4) 1.2e-9)
  (*
   (/
    z0
    (+ (* (/ z1 (* (fabs z2) (fabs z2))) (fabs z4)) (/ z3 (fabs z4))))
   (fabs z4))
  (/
   (* z0 (fabs z2))
   (+ (/ z1 (fabs z2)) (* z3 (/ (fabs z2) (* (fabs z4) (fabs z4))))))))

C

double code(double z0, double z3, double z4, double z1, double z2) {
	double tmp;
	if (fabs(z4) <= 1.2e-9) {
		tmp = (z0 / (((z1 / (fabs(z2) * fabs(z2))) * fabs(z4)) + (z3 / fabs(z4)))) * fabs(z4);
	} else {
		tmp = (z0 * fabs(z2)) / ((z1 / fabs(z2)) + (z3 * (fabs(z2) / (fabs(z4) * fabs(z4)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0, z3, z4, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8) :: tmp
    if (abs(z4) <= 1.2d-9) then
        tmp = (z0 / (((z1 / (abs(z2) * abs(z2))) * abs(z4)) + (z3 / abs(z4)))) * abs(z4)
    else
        tmp = (z0 * abs(z2)) / ((z1 / abs(z2)) + (z3 * (abs(z2) / (abs(z4) * abs(z4)))))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z3, double z4, double z1, double z2) {
	double tmp;
	if (Math.abs(z4) <= 1.2e-9) {
		tmp = (z0 / (((z1 / (Math.abs(z2) * Math.abs(z2))) * Math.abs(z4)) + (z3 / Math.abs(z4)))) * Math.abs(z4);
	} else {
		tmp = (z0 * Math.abs(z2)) / ((z1 / Math.abs(z2)) + (z3 * (Math.abs(z2) / (Math.abs(z4) * Math.abs(z4)))));
	}
	return tmp;
}

Python

def code(z0, z3, z4, z1, z2):
	tmp = 0
	if math.fabs(z4) <= 1.2e-9:
		tmp = (z0 / (((z1 / (math.fabs(z2) * math.fabs(z2))) * math.fabs(z4)) + (z3 / math.fabs(z4)))) * math.fabs(z4)
	else:
		tmp = (z0 * math.fabs(z2)) / ((z1 / math.fabs(z2)) + (z3 * (math.fabs(z2) / (math.fabs(z4) * math.fabs(z4)))))
	return tmp

Julia

function code(z0, z3, z4, z1, z2)
	tmp = 0.0
	if (abs(z4) <= 1.2e-9)
		tmp = Float64(Float64(z0 / Float64(Float64(Float64(z1 / Float64(abs(z2) * abs(z2))) * abs(z4)) + Float64(z3 / abs(z4)))) * abs(z4));
	else
		tmp = Float64(Float64(z0 * abs(z2)) / Float64(Float64(z1 / abs(z2)) + Float64(z3 * Float64(abs(z2) / Float64(abs(z4) * abs(z4))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z3, z4, z1, z2)
	tmp = 0.0;
	if (abs(z4) <= 1.2e-9)
		tmp = (z0 / (((z1 / (abs(z2) * abs(z2))) * abs(z4)) + (z3 / abs(z4)))) * abs(z4);
	else
		tmp = (z0 * abs(z2)) / ((z1 / abs(z2)) + (z3 * (abs(z2) / (abs(z4) * abs(z4)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z3_, z4_, z1_, z2_] := If[LessEqual[N[Abs[z4], $MachinePrecision], 1.2e-9], N[(N[(z0 / N[(N[(N[(z1 / N[(N[Abs[z2], $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[z4], $MachinePrecision]), $MachinePrecision] + N[(z3 / N[Abs[z4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[z4], $MachinePrecision]), $MachinePrecision], N[(N[(z0 * N[Abs[z2], $MachinePrecision]), $MachinePrecision] / N[(N[(z1 / N[Abs[z2], $MachinePrecision]), $MachinePrecision] + N[(z3 * N[(N[Abs[z2], $MachinePrecision] / N[(N[Abs[z4], $MachinePrecision] * N[Abs[z4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z4 < 1.2e-9

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{\frac{z2 \cdot z3}{{z4}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\color{blue}{z2 \cdot z3}}{{z4}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

      7. lower-pow.f64 50.0%

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

   10. Applied rewrites 50.6%

       \[\leadsto \frac{z0}{\frac{z1}{z2 \cdot z2} \cdot z4 + \frac{z3}{z4}} \cdot \color{blue}{z4} \]

   if 1.2e-9 < z4

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{\frac{z2 \cdot z3}{{z4}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\color{blue}{z2 \cdot z3}}{{z4}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

      7. lower-pow.f64 50.0%

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3 \cdot z2}{{\color{blue}{z4}}^{2}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3 \cdot z2}{{z4}^{\color{blue}{2}}}} \]

      5. pow2 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3 \cdot z2}{z4 \cdot \color{blue}{z4}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z3 \cdot z2}{z4 \cdot \color{blue}{z4}}} \]

      7. associate-/l* N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + z3 \cdot \color{blue}{\frac{z2}{z4 \cdot z4}}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + z3 \cdot \color{blue}{\frac{z2}{z4 \cdot z4}}} \]

      9. lower-/.f64 52.2%

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + z3 \cdot \frac{z2}{\color{blue}{z4 \cdot z4}}} \]

   10. Applied rewrites 52.2%

       \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + z3 \cdot \color{blue}{\frac{z2}{z4 \cdot z4}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 54.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|z2\right| \leq 5.2 \cdot 10^{+212}:\\ \;\;\;\;\frac{\left|z2\right| \cdot z0}{\left(\frac{z3}{z4 \cdot z4} \cdot \left|z2\right|\right) \cdot \left|z2\right| + z1} \cdot \left|z2\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{z0}{\frac{z1}{\left|z2\right| \cdot \left|z2\right|} \cdot z4 + \frac{z3}{z4}} \cdot z4\\ \end{array} \]

FPCore

(FPCore (z0 z3 z4 z1 z2)
  :precision binary64
  (if (<= (fabs z2) 5.2e+212)
  (*
   (/
    (* (fabs z2) z0)
    (+ (* (* (/ z3 (* z4 z4)) (fabs z2)) (fabs z2)) z1))
   (fabs z2))
  (* (/ z0 (+ (* (/ z1 (* (fabs z2) (fabs z2))) z4) (/ z3 z4))) z4)))

C

double code(double z0, double z3, double z4, double z1, double z2) {
	double tmp;
	if (fabs(z2) <= 5.2e+212) {
		tmp = ((fabs(z2) * z0) / ((((z3 / (z4 * z4)) * fabs(z2)) * fabs(z2)) + z1)) * fabs(z2);
	} else {
		tmp = (z0 / (((z1 / (fabs(z2) * fabs(z2))) * z4) + (z3 / z4))) * z4;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0, z3, z4, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8) :: tmp
    if (abs(z2) <= 5.2d+212) then
        tmp = ((abs(z2) * z0) / ((((z3 / (z4 * z4)) * abs(z2)) * abs(z2)) + z1)) * abs(z2)
    else
        tmp = (z0 / (((z1 / (abs(z2) * abs(z2))) * z4) + (z3 / z4))) * z4
    end if
    code = tmp
end function

Java

public static double code(double z0, double z3, double z4, double z1, double z2) {
	double tmp;
	if (Math.abs(z2) <= 5.2e+212) {
		tmp = ((Math.abs(z2) * z0) / ((((z3 / (z4 * z4)) * Math.abs(z2)) * Math.abs(z2)) + z1)) * Math.abs(z2);
	} else {
		tmp = (z0 / (((z1 / (Math.abs(z2) * Math.abs(z2))) * z4) + (z3 / z4))) * z4;
	}
	return tmp;
}

Python

def code(z0, z3, z4, z1, z2):
	tmp = 0
	if math.fabs(z2) <= 5.2e+212:
		tmp = ((math.fabs(z2) * z0) / ((((z3 / (z4 * z4)) * math.fabs(z2)) * math.fabs(z2)) + z1)) * math.fabs(z2)
	else:
		tmp = (z0 / (((z1 / (math.fabs(z2) * math.fabs(z2))) * z4) + (z3 / z4))) * z4
	return tmp

Julia

function code(z0, z3, z4, z1, z2)
	tmp = 0.0
	if (abs(z2) <= 5.2e+212)
		tmp = Float64(Float64(Float64(abs(z2) * z0) / Float64(Float64(Float64(Float64(z3 / Float64(z4 * z4)) * abs(z2)) * abs(z2)) + z1)) * abs(z2));
	else
		tmp = Float64(Float64(z0 / Float64(Float64(Float64(z1 / Float64(abs(z2) * abs(z2))) * z4) + Float64(z3 / z4))) * z4);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z3, z4, z1, z2)
	tmp = 0.0;
	if (abs(z2) <= 5.2e+212)
		tmp = ((abs(z2) * z0) / ((((z3 / (z4 * z4)) * abs(z2)) * abs(z2)) + z1)) * abs(z2);
	else
		tmp = (z0 / (((z1 / (abs(z2) * abs(z2))) * z4) + (z3 / z4))) * z4;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z3_, z4_, z1_, z2_] := If[LessEqual[N[Abs[z2], $MachinePrecision], 5.2e+212], N[(N[(N[(N[Abs[z2], $MachinePrecision] * z0), $MachinePrecision] / N[(N[(N[(N[(z3 / N[(z4 * z4), $MachinePrecision]), $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision] + z1), $MachinePrecision]), $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision], N[(N[(z0 / N[(N[(N[(z1 / N[(N[Abs[z2], $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z4), $MachinePrecision] + N[(z3 / z4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z4), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z2 < 5.1999999999999997e212

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{\frac{z2 \cdot z3}{{z4}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\color{blue}{z2 \cdot z3}}{{z4}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

      7. lower-pow.f64 50.0%

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

   10. Applied rewrites 50.7%

       \[\leadsto \frac{z2 \cdot z0}{\left(\frac{z3}{z4 \cdot z4} \cdot z2\right) \cdot z2 + z1} \cdot \color{blue}{z2} \]

   if 5.1999999999999997e212 < z2

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{\frac{z2 \cdot z3}{{z4}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\color{blue}{z2 \cdot z3}}{{z4}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

      7. lower-pow.f64 50.0%

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

   10. Applied rewrites 50.6%

       \[\leadsto \frac{z0}{\frac{z1}{z2 \cdot z2} \cdot z4 + \frac{z3}{z4}} \cdot \color{blue}{z4} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 53.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{z3}{z4 \cdot z4}\\ \mathbf{if}\;\left|z2\right| \leq 5.2 \cdot 10^{+200}:\\ \;\;\;\;\frac{\left|z2\right| \cdot z0}{\left(t\_0 \cdot \left|z2\right|\right) \cdot \left|z2\right| + z1} \cdot \left|z2\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{-z0}{\frac{z1}{\left(-\left|z2\right|\right) \cdot \left|z2\right|} - t\_0}\\ \end{array} \]

FPCore

(FPCore (z0 z3 z4 z1 z2)
  :precision binary64
  (let* ((t_0 (/ z3 (* z4 z4))))
  (if (<= (fabs z2) 5.2e+200)
    (*
     (/ (* (fabs z2) z0) (+ (* (* t_0 (fabs z2)) (fabs z2)) z1))
     (fabs z2))
    (/ (- z0) (- (/ z1 (* (- (fabs z2)) (fabs z2))) t_0)))))

C

double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = z3 / (z4 * z4);
	double tmp;
	if (fabs(z2) <= 5.2e+200) {
		tmp = ((fabs(z2) * z0) / (((t_0 * fabs(z2)) * fabs(z2)) + z1)) * fabs(z2);
	} else {
		tmp = -z0 / ((z1 / (-fabs(z2) * fabs(z2))) - t_0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0, z3, z4, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z3 / (z4 * z4)
    if (abs(z2) <= 5.2d+200) then
        tmp = ((abs(z2) * z0) / (((t_0 * abs(z2)) * abs(z2)) + z1)) * abs(z2)
    else
        tmp = -z0 / ((z1 / (-abs(z2) * abs(z2))) - t_0)
    end if
    code = tmp
end function

Java

public static double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = z3 / (z4 * z4);
	double tmp;
	if (Math.abs(z2) <= 5.2e+200) {
		tmp = ((Math.abs(z2) * z0) / (((t_0 * Math.abs(z2)) * Math.abs(z2)) + z1)) * Math.abs(z2);
	} else {
		tmp = -z0 / ((z1 / (-Math.abs(z2) * Math.abs(z2))) - t_0);
	}
	return tmp;
}

Python

def code(z0, z3, z4, z1, z2):
	t_0 = z3 / (z4 * z4)
	tmp = 0
	if math.fabs(z2) <= 5.2e+200:
		tmp = ((math.fabs(z2) * z0) / (((t_0 * math.fabs(z2)) * math.fabs(z2)) + z1)) * math.fabs(z2)
	else:
		tmp = -z0 / ((z1 / (-math.fabs(z2) * math.fabs(z2))) - t_0)
	return tmp

Julia

function code(z0, z3, z4, z1, z2)
	t_0 = Float64(z3 / Float64(z4 * z4))
	tmp = 0.0
	if (abs(z2) <= 5.2e+200)
		tmp = Float64(Float64(Float64(abs(z2) * z0) / Float64(Float64(Float64(t_0 * abs(z2)) * abs(z2)) + z1)) * abs(z2));
	else
		tmp = Float64(Float64(-z0) / Float64(Float64(z1 / Float64(Float64(-abs(z2)) * abs(z2))) - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z3, z4, z1, z2)
	t_0 = z3 / (z4 * z4);
	tmp = 0.0;
	if (abs(z2) <= 5.2e+200)
		tmp = ((abs(z2) * z0) / (((t_0 * abs(z2)) * abs(z2)) + z1)) * abs(z2);
	else
		tmp = -z0 / ((z1 / (-abs(z2) * abs(z2))) - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z3_, z4_, z1_, z2_] := Block[{t$95$0 = N[(z3 / N[(z4 * z4), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[z2], $MachinePrecision], 5.2e+200], N[(N[(N[(N[Abs[z2], $MachinePrecision] * z0), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Abs[z2], $MachinePrecision]), $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision] + z1), $MachinePrecision]), $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision], N[((-z0) / N[(N[(z1 / N[((-N[Abs[z2], $MachinePrecision]) * N[Abs[z2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z2 < 5.2000000000000003e200

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{\frac{z2 \cdot z3}{{z4}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\color{blue}{z2 \cdot z3}}{{z4}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

      7. lower-pow.f64 50.0%

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

   10. Applied rewrites 50.7%

       \[\leadsto \frac{z2 \cdot z0}{\left(\frac{z3}{z4 \cdot z4} \cdot z2\right) \cdot z2 + z1} \cdot \color{blue}{z2} \]

   if 5.2000000000000003e200 < z2

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

   4. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\left(-1 \cdot z0\right)} \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}} \]
   5. Step-by-step derivation

      1. lower-*.f64 49.1%

         \[\leadsto \left(-1 \cdot \color{blue}{z0}\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}} \]

   6. Applied rewrites 49.1%

      \[\leadsto \color{blue}{\left(-1 \cdot z0\right)} \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-1 \cdot z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. lift-+.f64 N/A

         \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2}} + \frac{z3}{z4 \cdot z4}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\frac{z1}{\color{blue}{z2 \cdot z2}} + \frac{z3}{z4 \cdot z4}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z1}{z2}}{z2}} + \frac{z3}{z4 \cdot z4}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\frac{\color{blue}{\frac{z1}{z2}}}{z2} + \frac{z3}{z4 \cdot z4}} \]

      8. mult-flip-rev N/A

         \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2} \cdot \frac{1}{z2}} + \frac{z3}{z4 \cdot z4}} \]

      9. lift-/.f64 N/A

         \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\frac{z1}{z2} \cdot \color{blue}{\frac{1}{z2}} + \frac{z3}{z4 \cdot z4}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2} \cdot \frac{1}{z2}} + \frac{z3}{z4 \cdot z4}} \]

      11. +-commutative N/A

          \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2} \cdot \frac{1}{z2}}} \]

      12. lift-+.f64 N/A

          \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2} \cdot \frac{1}{z2}}} \]

      13. frac-2neg-rev N/A

          \[\leadsto \left(-1 \cdot z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2} \cdot \frac{1}{z2}\right)\right)}} \]

      14. metadata-eval N/A

          \[\leadsto \left(-1 \cdot z0\right) \cdot \frac{\color{blue}{1}}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2} \cdot \frac{1}{z2}\right)\right)} \]

   8. Applied rewrites 49.3%

      \[\leadsto \color{blue}{\frac{-z0}{\frac{z1}{\left(-z2\right) \cdot z2} - \frac{z3}{z4 \cdot z4}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 52.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{z3}{z4 \cdot z4}\\ \mathbf{if}\;\left|z2\right| \leq 3 \cdot 10^{+212}:\\ \;\;\;\;\frac{\left|z2\right| \cdot z0}{\left(t\_0 \cdot \left|z2\right|\right) \cdot \left|z2\right| + z1} \cdot \left|z2\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{z0}{t\_0 + \frac{z1}{\left|z2\right| \cdot \left|z2\right|}}\\ \end{array} \]

FPCore

(FPCore (z0 z3 z4 z1 z2)
  :precision binary64
  (let* ((t_0 (/ z3 (* z4 z4))))
  (if (<= (fabs z2) 3e+212)
    (*
     (/ (* (fabs z2) z0) (+ (* (* t_0 (fabs z2)) (fabs z2)) z1))
     (fabs z2))
    (/ z0 (+ t_0 (/ z1 (* (fabs z2) (fabs z2))))))))

C

double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = z3 / (z4 * z4);
	double tmp;
	if (fabs(z2) <= 3e+212) {
		tmp = ((fabs(z2) * z0) / (((t_0 * fabs(z2)) * fabs(z2)) + z1)) * fabs(z2);
	} else {
		tmp = z0 / (t_0 + (z1 / (fabs(z2) * fabs(z2))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0, z3, z4, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z3 / (z4 * z4)
    if (abs(z2) <= 3d+212) then
        tmp = ((abs(z2) * z0) / (((t_0 * abs(z2)) * abs(z2)) + z1)) * abs(z2)
    else
        tmp = z0 / (t_0 + (z1 / (abs(z2) * abs(z2))))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = z3 / (z4 * z4);
	double tmp;
	if (Math.abs(z2) <= 3e+212) {
		tmp = ((Math.abs(z2) * z0) / (((t_0 * Math.abs(z2)) * Math.abs(z2)) + z1)) * Math.abs(z2);
	} else {
		tmp = z0 / (t_0 + (z1 / (Math.abs(z2) * Math.abs(z2))));
	}
	return tmp;
}

Python

def code(z0, z3, z4, z1, z2):
	t_0 = z3 / (z4 * z4)
	tmp = 0
	if math.fabs(z2) <= 3e+212:
		tmp = ((math.fabs(z2) * z0) / (((t_0 * math.fabs(z2)) * math.fabs(z2)) + z1)) * math.fabs(z2)
	else:
		tmp = z0 / (t_0 + (z1 / (math.fabs(z2) * math.fabs(z2))))
	return tmp

Julia

function code(z0, z3, z4, z1, z2)
	t_0 = Float64(z3 / Float64(z4 * z4))
	tmp = 0.0
	if (abs(z2) <= 3e+212)
		tmp = Float64(Float64(Float64(abs(z2) * z0) / Float64(Float64(Float64(t_0 * abs(z2)) * abs(z2)) + z1)) * abs(z2));
	else
		tmp = Float64(z0 / Float64(t_0 + Float64(z1 / Float64(abs(z2) * abs(z2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z3, z4, z1, z2)
	t_0 = z3 / (z4 * z4);
	tmp = 0.0;
	if (abs(z2) <= 3e+212)
		tmp = ((abs(z2) * z0) / (((t_0 * abs(z2)) * abs(z2)) + z1)) * abs(z2);
	else
		tmp = z0 / (t_0 + (z1 / (abs(z2) * abs(z2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z3_, z4_, z1_, z2_] := Block[{t$95$0 = N[(z3 / N[(z4 * z4), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[z2], $MachinePrecision], 3e+212], N[(N[(N[(N[Abs[z2], $MachinePrecision] * z0), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Abs[z2], $MachinePrecision]), $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision] + z1), $MachinePrecision]), $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision], N[(z0 / N[(t$95$0 + N[(z1 / N[(N[Abs[z2], $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z2 < 3.0000000000000001e212

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{\frac{z2 \cdot z3}{{z4}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\color{blue}{z2 \cdot z3}}{{z4}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

      7. lower-pow.f64 50.0%

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

   10. Applied rewrites 50.7%

       \[\leadsto \frac{z2 \cdot z0}{\left(\frac{z3}{z4 \cdot z4} \cdot z2\right) \cdot z2 + z1} \cdot \color{blue}{z2} \]

   if 3.0000000000000001e212 < z2

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{\frac{z2 \cdot z3}{{z4}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\color{blue}{z2 \cdot z3}}{{z4}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

      7. lower-pow.f64 50.0%

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

      5. pow2 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{z4 \cdot \color{blue}{z4}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{z4 \cdot \color{blue}{z4}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{z4} \cdot z4}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + z2 \cdot \color{blue}{\frac{z3}{z4 \cdot z4}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + z2 \cdot \frac{z3}{z4 \cdot \color{blue}{z4}}} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{z0 \cdot z2}{z2 \cdot \frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2}}} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{z0 \cdot z2}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2}}} \]

      13. associate-/l* N/A

          \[\leadsto z0 \cdot \color{blue}{\frac{z2}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}}} \]

      14. div-flip-rev N/A

          \[\leadsto z0 \cdot \frac{1}{\color{blue}{\frac{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}}{z2}}} \]

   10. Applied rewrites 49.2%

       \[\leadsto \frac{z0}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 52.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{z3}{z4 \cdot z4}\\ \mathbf{if}\;\left|z2\right| \leq 3.1 \cdot 10^{+212}:\\ \;\;\;\;\left|z2\right| \cdot \left(z0 \cdot \frac{\left|z2\right|}{\left(t\_0 \cdot \left|z2\right|\right) \cdot \left|z2\right| + z1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z0}{t\_0 + \frac{z1}{\left|z2\right| \cdot \left|z2\right|}}\\ \end{array} \]

FPCore

(FPCore (z0 z3 z4 z1 z2)
  :precision binary64
  (let* ((t_0 (/ z3 (* z4 z4))))
  (if (<= (fabs z2) 3.1e+212)
    (*
     (fabs z2)
     (* z0 (/ (fabs z2) (+ (* (* t_0 (fabs z2)) (fabs z2)) z1))))
    (/ z0 (+ t_0 (/ z1 (* (fabs z2) (fabs z2))))))))

C

double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = z3 / (z4 * z4);
	double tmp;
	if (fabs(z2) <= 3.1e+212) {
		tmp = fabs(z2) * (z0 * (fabs(z2) / (((t_0 * fabs(z2)) * fabs(z2)) + z1)));
	} else {
		tmp = z0 / (t_0 + (z1 / (fabs(z2) * fabs(z2))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0, z3, z4, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z3 / (z4 * z4)
    if (abs(z2) <= 3.1d+212) then
        tmp = abs(z2) * (z0 * (abs(z2) / (((t_0 * abs(z2)) * abs(z2)) + z1)))
    else
        tmp = z0 / (t_0 + (z1 / (abs(z2) * abs(z2))))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = z3 / (z4 * z4);
	double tmp;
	if (Math.abs(z2) <= 3.1e+212) {
		tmp = Math.abs(z2) * (z0 * (Math.abs(z2) / (((t_0 * Math.abs(z2)) * Math.abs(z2)) + z1)));
	} else {
		tmp = z0 / (t_0 + (z1 / (Math.abs(z2) * Math.abs(z2))));
	}
	return tmp;
}

Python

def code(z0, z3, z4, z1, z2):
	t_0 = z3 / (z4 * z4)
	tmp = 0
	if math.fabs(z2) <= 3.1e+212:
		tmp = math.fabs(z2) * (z0 * (math.fabs(z2) / (((t_0 * math.fabs(z2)) * math.fabs(z2)) + z1)))
	else:
		tmp = z0 / (t_0 + (z1 / (math.fabs(z2) * math.fabs(z2))))
	return tmp

Julia

function code(z0, z3, z4, z1, z2)
	t_0 = Float64(z3 / Float64(z4 * z4))
	tmp = 0.0
	if (abs(z2) <= 3.1e+212)
		tmp = Float64(abs(z2) * Float64(z0 * Float64(abs(z2) / Float64(Float64(Float64(t_0 * abs(z2)) * abs(z2)) + z1))));
	else
		tmp = Float64(z0 / Float64(t_0 + Float64(z1 / Float64(abs(z2) * abs(z2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z3, z4, z1, z2)
	t_0 = z3 / (z4 * z4);
	tmp = 0.0;
	if (abs(z2) <= 3.1e+212)
		tmp = abs(z2) * (z0 * (abs(z2) / (((t_0 * abs(z2)) * abs(z2)) + z1)));
	else
		tmp = z0 / (t_0 + (z1 / (abs(z2) * abs(z2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z3_, z4_, z1_, z2_] := Block[{t$95$0 = N[(z3 / N[(z4 * z4), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[z2], $MachinePrecision], 3.1e+212], N[(N[Abs[z2], $MachinePrecision] * N[(z0 * N[(N[Abs[z2], $MachinePrecision] / N[(N[(N[(t$95$0 * N[Abs[z2], $MachinePrecision]), $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision] + z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z0 / N[(t$95$0 + N[(z1 / N[(N[Abs[z2], $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z2 < 3.1e212

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{\frac{z2 \cdot z3}{{z4}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\color{blue}{z2 \cdot z3}}{{z4}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

      7. lower-pow.f64 50.0%

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

   10. Applied rewrites 50.6%

       \[\leadsto z2 \cdot \color{blue}{\left(z0 \cdot \frac{z2}{\left(\frac{z3}{z4 \cdot z4} \cdot z2\right) \cdot z2 + z1}\right)} \]

   if 3.1e212 < z2

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{\frac{z2 \cdot z3}{{z4}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\color{blue}{z2 \cdot z3}}{{z4}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

      7. lower-pow.f64 50.0%

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

      5. pow2 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{z4 \cdot \color{blue}{z4}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{z4 \cdot \color{blue}{z4}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{z4} \cdot z4}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + z2 \cdot \color{blue}{\frac{z3}{z4 \cdot z4}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + z2 \cdot \frac{z3}{z4 \cdot \color{blue}{z4}}} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{z0 \cdot z2}{z2 \cdot \frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2}}} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{z0 \cdot z2}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2}}} \]

      13. associate-/l* N/A

          \[\leadsto z0 \cdot \color{blue}{\frac{z2}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}}} \]

      14. div-flip-rev N/A

          \[\leadsto z0 \cdot \frac{1}{\color{blue}{\frac{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}}{z2}}} \]

   10. Applied rewrites 49.2%

       \[\leadsto \frac{z0}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 52.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\\ \mathbf{if}\;\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{t\_0} \leq 4 \cdot 10^{+202}:\\ \;\;\;\;\frac{z0}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{z0 \cdot z2}{\frac{z1}{z2}}\\ \end{array} \]

FPCore

(FPCore (z0 z3 z4 z1 z2)
  :precision binary64
  (let* ((t_0 (+ (/ z3 (* z4 z4)) (/ z1 (* z2 z2)))))
  (if (<=
       (/
        (-
         z0
         (*
          (* (- -0.5 (* (- (* 0.25 z0) -0.3333333333333333) z0)) z0)
          z0))
        t_0)
       4e+202)
    (/ z0 t_0)
    (/ (* z0 z2) (/ z1 z2)))))

C

double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = (z3 / (z4 * z4)) + (z1 / (z2 * z2));
	double tmp;
	if (((z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)) / t_0) <= 4e+202) {
		tmp = z0 / t_0;
	} else {
		tmp = (z0 * z2) / (z1 / z2);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0, z3, z4, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z3 / (z4 * z4)) + (z1 / (z2 * z2))
    if (((z0 - ((((-0.5d0) - (((0.25d0 * z0) - (-0.3333333333333333d0)) * z0)) * z0) * z0)) / t_0) <= 4d+202) then
        tmp = z0 / t_0
    else
        tmp = (z0 * z2) / (z1 / z2)
    end if
    code = tmp
end function

Java

public static double code(double z0, double z3, double z4, double z1, double z2) {
	double t_0 = (z3 / (z4 * z4)) + (z1 / (z2 * z2));
	double tmp;
	if (((z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)) / t_0) <= 4e+202) {
		tmp = z0 / t_0;
	} else {
		tmp = (z0 * z2) / (z1 / z2);
	}
	return tmp;
}

Python

def code(z0, z3, z4, z1, z2):
	t_0 = (z3 / (z4 * z4)) + (z1 / (z2 * z2))
	tmp = 0
	if ((z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)) / t_0) <= 4e+202:
		tmp = z0 / t_0
	else:
		tmp = (z0 * z2) / (z1 / z2)
	return tmp

Julia

function code(z0, z3, z4, z1, z2)
	t_0 = Float64(Float64(z3 / Float64(z4 * z4)) + Float64(z1 / Float64(z2 * z2)))
	tmp = 0.0
	if (Float64(Float64(z0 - Float64(Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)) / t_0) <= 4e+202)
		tmp = Float64(z0 / t_0);
	else
		tmp = Float64(Float64(z0 * z2) / Float64(z1 / z2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z3, z4, z1, z2)
	t_0 = (z3 / (z4 * z4)) + (z1 / (z2 * z2));
	tmp = 0.0;
	if (((z0 - (((-0.5 - (((0.25 * z0) - -0.3333333333333333) * z0)) * z0) * z0)) / t_0) <= 4e+202)
		tmp = z0 / t_0;
	else
		tmp = (z0 * z2) / (z1 / z2);
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z3_, z4_, z1_, z2_] := Block[{t$95$0 = N[(N[(z3 / N[(z4 * z4), $MachinePrecision]), $MachinePrecision] + N[(z1 / N[(z2 * z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(z0 - N[(N[(N[(-0.5 - N[(N[(N[(0.25 * z0), $MachinePrecision] - -0.3333333333333333), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], 4e+202], N[(z0 / t$95$0), $MachinePrecision], N[(N[(z0 * z2), $MachinePrecision] / N[(z1 / z2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z0 (*.f64 (*.f64 (-.f64 #s(literal -1/2 binary64) (*.f64 (-.f64 (*.f64 #s(literal 1/4 binary64) z0) #s(literal -3333333333333333/10000000000000000 binary64)) z0)) z0) z0)) (+.f64 (/.f64 z3 (*.f64 z4 z4)) (/.f64 z1 (*.f64 z2 z2)))) < 3.9999999999999996e202

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{\frac{z2 \cdot z3}{{z4}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\color{blue}{z2 \cdot z3}}{{z4}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

      7. lower-pow.f64 50.0%

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

      5. pow2 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{z4 \cdot \color{blue}{z4}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{z4 \cdot \color{blue}{z4}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{z4} \cdot z4}} \]

      8. associate-/l* N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + z2 \cdot \color{blue}{\frac{z3}{z4 \cdot z4}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + z2 \cdot \frac{z3}{z4 \cdot \color{blue}{z4}}} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{z0 \cdot z2}{z2 \cdot \frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2}}} \]

      12. lift-/.f64 N/A

          \[\leadsto \frac{z0 \cdot z2}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2}}} \]

      13. associate-/l* N/A

          \[\leadsto z0 \cdot \color{blue}{\frac{z2}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}}} \]

      14. div-flip-rev N/A

          \[\leadsto z0 \cdot \frac{1}{\color{blue}{\frac{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}}{z2}}} \]

   10. Applied rewrites 49.2%

       \[\leadsto \frac{z0}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

   if 3.9999999999999996e202 < (/.f64 (-.f64 z0 (*.f64 (*.f64 (-.f64 #s(literal -1/2 binary64) (*.f64 (-.f64 (*.f64 #s(literal 1/4 binary64) z0) #s(literal -3333333333333333/10000000000000000 binary64)) z0)) z0) z0)) (+.f64 (/.f64 z3 (*.f64 z4 z4)) (/.f64 z1 (*.f64 z2 z2))))

   1. Initial program 66.3%

      \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      3. mult-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

      5. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      6. sub-negate-rev N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

      8. frac-2neg N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

      11. add-flip N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

      12. sub-negate N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

      13. sub-negate-rev N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

      6. associate-/l/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

      8. add-to-fraction N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

      9. associate-/r/ N/A

         \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

      14. lower-*.f64 69.1%

          \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   5. Applied rewrites 69.1%

      \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

   6. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{\frac{z2 \cdot z3}{{z4}^{2}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\color{blue}{z2 \cdot z3}}{{z4}^{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

      7. lower-pow.f64 50.0%

         \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

   8. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

   9. Taylor expanded in z3 around 0

      \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{\color{blue}{z2}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 39.8%

          \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2}} \]

   11. Applied rewrites 39.8%

       \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{\color{blue}{z2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 39.8% accurate, 2.7× speedup

Math

\[\frac{z0 \cdot z2}{\frac{z1}{z2}} \]

FPCore

(FPCore (z0 z3 z4 z1 z2)
  :precision binary64
  (/ (* z0 z2) (/ z1 z2)))

C

double code(double z0, double z3, double z4, double z1, double z2) {
	return (z0 * z2) / (z1 / z2);
}

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(z0, z3, z4, z1, z2)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z3
    real(8), intent (in) :: z4
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    code = (z0 * z2) / (z1 / z2)
end function

Java

public static double code(double z0, double z3, double z4, double z1, double z2) {
	return (z0 * z2) / (z1 / z2);
}

Python

def code(z0, z3, z4, z1, z2):
	return (z0 * z2) / (z1 / z2)

Julia

function code(z0, z3, z4, z1, z2)
	return Float64(Float64(z0 * z2) / Float64(z1 / z2))
end

MATLAB

function tmp = code(z0, z3, z4, z1, z2)
	tmp = (z0 * z2) / (z1 / z2);
end

Wolfram

code[z0_, z3_, z4_, z1_, z2_] := N[(N[(z0 * z2), $MachinePrecision] / N[(z1 / z2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.3%

   \[\frac{z0 - \left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

   3. mult-flip N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)}} \]

   5. lift--.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z0 - \left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0\right)}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

   6. sub-negate-rev N/A

      \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

   7. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)} \]

   8. frac-2neg N/A

      \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)}} \]

   9. metadata-eval N/A

      \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)\right)\right)\right)} \]

   10. lift-+.f64 N/A

       \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}\right)}\right)\right)\right)} \]

   11. add-flip N/A

       \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)\right)}\right)\right)\right)} \]

   12. sub-negate N/A

       \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right) - \frac{z3}{z4 \cdot z4}\right)}\right)} \]

   13. sub-negate-rev N/A

       \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} - \left(\mathsf{neg}\left(\frac{z1}{z2 \cdot z2}\right)\right)}} \]

3. Applied rewrites 66.2%

   \[\leadsto \color{blue}{\left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\frac{-1}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

   2. lift-+.f64 N/A

      \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z1}{z2 \cdot z2} + \frac{z3}{z4 \cdot z4}}} \]

   3. +-commutative N/A

      \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} + \frac{z1}{z2 \cdot z2}}} \]

   4. lift-/.f64 N/A

      \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{z1}{z2 \cdot z2}}} \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{z1}{\color{blue}{z2 \cdot z2}}} \]

   6. associate-/l/ N/A

      \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \color{blue}{\frac{\frac{z1}{z2}}{z2}}} \]

   7. lift-/.f64 N/A

      \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\frac{z3}{z4 \cdot z4} + \frac{\color{blue}{\frac{z1}{z2}}}{z2}} \]

   8. add-to-fraction N/A

      \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \frac{-1}{\color{blue}{\frac{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}{z2}}} \]

   9. associate-/r/ N/A

      \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}} \cdot z2\right)} \]

   11. lower-/.f64 N/A

       \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\color{blue}{\frac{-1}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

   12. lower-+.f64 N/A

       \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{\frac{z3}{z4 \cdot z4} \cdot z2 + \frac{z1}{z2}}} \cdot z2\right) \]

   13. *-commutative N/A

       \[\leadsto \left(\left(\left(\frac{-1}{2} - \left(\frac{1}{4} \cdot z0 - \frac{-3333333333333333}{10000000000000000}\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

   14. lower-*.f64 69.1%

       \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \left(\frac{-1}{\color{blue}{z2 \cdot \frac{z3}{z4 \cdot z4}} + \frac{z1}{z2}} \cdot z2\right) \]

5. Applied rewrites 69.1%

   \[\leadsto \left(\left(\left(-0.5 - \left(0.25 \cdot z0 - -0.3333333333333333\right) \cdot z0\right) \cdot z0\right) \cdot z0 - z0\right) \cdot \color{blue}{\left(\frac{-1}{z2 \cdot \frac{z3}{z4 \cdot z4} + \frac{z1}{z2}} \cdot z2\right)} \]

6. Taylor expanded in z0 around 0

   \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{z0 \cdot z2}{\color{blue}{\frac{z1}{z2}} + \frac{z2 \cdot z3}{{z4}^{2}}} \]

   3. lower-+.f64 N/A

      \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \color{blue}{\frac{z2 \cdot z3}{{z4}^{2}}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{\color{blue}{z2 \cdot z3}}{{z4}^{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{\color{blue}{{z4}^{2}}}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{\color{blue}{z4}}^{2}}} \]

   7. lower-pow.f64 50.0%

      \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{\color{blue}{2}}}} \]

8. Applied rewrites 50.0%

   \[\leadsto \color{blue}{\frac{z0 \cdot z2}{\frac{z1}{z2} + \frac{z2 \cdot z3}{{z4}^{2}}}} \]

9. Taylor expanded in z3 around 0

   \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{\color{blue}{z2}}} \]
10. Step-by-step derivation

    1. lower-/.f64 39.8%

       \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{z2}} \]

11. Applied rewrites 39.8%

    \[\leadsto \frac{z0 \cdot z2}{\frac{z1}{\color{blue}{z2}}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z0 z3 z4 z1 z2)
  :name "(/ (- z0 (* (* (- -1/2 (* (- (* 1/4 z0) -3333333333333333/10000000000000000) z0)) z0) z0)) (+ (/ z3 (* z4 z4)) (/ z1 (* z2 z2))))"
  :precision binary64
  (/ (- z0 (* (* (- -0.5 (* (- (* 0.25 z0) -0.3333333333333333) z0)) z0) z0)) (+ (/ z3 (* z4 z4)) (/ z1 (* z2 z2)))))