(sqrt (- (/ (* (* (* z1 z0) z0) (* (* z1 z0) z0)) (* (* (* (* z4 z3) z2) z2) (* (* (* z4 z3) z2) z2))) (* z5 z5)))

Percentage Accurate: 43.7% → 94.6%

Time: 11.7s

Alternatives: 24

Speedup: 1.4×

• 69.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} t_0 := \left(z1 \cdot z0\right) \cdot z0\\ t_1 := \left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\\ \sqrt{\frac{t\_0 \cdot t\_0}{t\_1 \cdot t\_1} - z5 \cdot z5} \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0 (* (* z1 z0) z0)) (t_1 (* (* (* z4 z3) z2) z2)))
  (sqrt (- (/ (* t_0 t_0) (* t_1 t_1)) (* z5 z5)))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = (z1 * z0) * z0;
	double t_1 = ((z4 * z3) * z2) * z2;
	return sqrt((((t_0 * t_0) / (t_1 * t_1)) - (z5 * z5)));
}

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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    real(8) :: t_0
    real(8) :: t_1
    t_0 = (z1 * z0) * z0
    t_1 = ((z4 * z3) * z2) * z2
    code = sqrt((((t_0 * t_0) / (t_1 * t_1)) - (z5 * z5)))
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = (z1 * z0) * z0;
	double t_1 = ((z4 * z3) * z2) * z2;
	return Math.sqrt((((t_0 * t_0) / (t_1 * t_1)) - (z5 * z5)));
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = (z1 * z0) * z0
	t_1 = ((z4 * z3) * z2) * z2
	return math.sqrt((((t_0 * t_0) / (t_1 * t_1)) - (z5 * z5)))

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = Float64(Float64(z1 * z0) * z0)
	t_1 = Float64(Float64(Float64(z4 * z3) * z2) * z2)
	return sqrt(Float64(Float64(Float64(t_0 * t_0) / Float64(t_1 * t_1)) - Float64(z5 * z5)))
end

MATLAB

function tmp = code(z1, z0, z4, z3, z2, z5)
	t_0 = (z1 * z0) * z0;
	t_1 = ((z4 * z3) * z2) * z2;
	tmp = sqrt((((t_0 * t_0) / (t_1 * t_1)) - (z5 * z5)));
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[(N[(z1 * z0), $MachinePrecision] * z0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(z4 * z3), $MachinePrecision] * z2), $MachinePrecision] * z2), $MachinePrecision]}, N[Sqrt[N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(z5 * z5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Initial Program: 43.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(z1 \cdot z0\right) \cdot z0\\ t_1 := \left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\\ \sqrt{\frac{t\_0 \cdot t\_0}{t\_1 \cdot t\_1} - z5 \cdot z5} \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0 (* (* z1 z0) z0)) (t_1 (* (* (* z4 z3) z2) z2)))
  (sqrt (- (/ (* t_0 t_0) (* t_1 t_1)) (* z5 z5)))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = (z1 * z0) * z0;
	double t_1 = ((z4 * z3) * z2) * z2;
	return sqrt((((t_0 * t_0) / (t_1 * t_1)) - (z5 * z5)));
}

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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    real(8) :: t_0
    real(8) :: t_1
    t_0 = (z1 * z0) * z0
    t_1 = ((z4 * z3) * z2) * z2
    code = sqrt((((t_0 * t_0) / (t_1 * t_1)) - (z5 * z5)))
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = (z1 * z0) * z0;
	double t_1 = ((z4 * z3) * z2) * z2;
	return Math.sqrt((((t_0 * t_0) / (t_1 * t_1)) - (z5 * z5)));
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = (z1 * z0) * z0
	t_1 = ((z4 * z3) * z2) * z2
	return math.sqrt((((t_0 * t_0) / (t_1 * t_1)) - (z5 * z5)))

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = Float64(Float64(z1 * z0) * z0)
	t_1 = Float64(Float64(Float64(z4 * z3) * z2) * z2)
	return sqrt(Float64(Float64(Float64(t_0 * t_0) / Float64(t_1 * t_1)) - Float64(z5 * z5)))
end

MATLAB

function tmp = code(z1, z0, z4, z3, z2, z5)
	t_0 = (z1 * z0) * z0;
	t_1 = ((z4 * z3) * z2) * z2;
	tmp = sqrt((((t_0 * t_0) / (t_1 * t_1)) - (z5 * z5)));
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[(N[(z1 * z0), $MachinePrecision] * z0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(z4 * z3), $MachinePrecision] * z2), $MachinePrecision] * z2), $MachinePrecision]}, N[Sqrt[N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(z5 * z5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Alternative 1: 94.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|z4\right|, \left|z3\right|\right)\\ t_1 := \mathsf{max}\left(\left|z4\right|, \left|z3\right|\right)\\ t_2 := \frac{\frac{\left|z1\right| \cdot z0}{t\_1 \cdot z2} \cdot z0}{t\_0 \cdot z2}\\ t_3 := \sqrt{t\_2 - z5} \cdot \sqrt{t\_2 + z5}\\ t_4 := \frac{z0}{z2} \cdot \frac{\left|z1\right| \cdot \frac{z0}{z2}}{t\_1 \cdot t\_0}\\ t_5 := t\_0 \cdot t\_1\\ \mathbf{if}\;t\_5 \leq 10^{-312}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_5 \leq 10^{+279}:\\ \;\;\;\;\sqrt{t\_4 - z5} \cdot \sqrt{t\_4 + z5}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0 (fmin (fabs z4) (fabs z3)))
       (t_1 (fmax (fabs z4) (fabs z3)))
       (t_2 (/ (* (/ (* (fabs z1) z0) (* t_1 z2)) z0) (* t_0 z2)))
       (t_3 (* (sqrt (- t_2 z5)) (sqrt (+ t_2 z5))))
       (t_4 (* (/ z0 z2) (/ (* (fabs z1) (/ z0 z2)) (* t_1 t_0))))
       (t_5 (* t_0 t_1)))
  (if (<= t_5 1e-312)
    t_3
    (if (<= t_5 1e+279) (* (sqrt (- t_4 z5)) (sqrt (+ t_4 z5))) t_3))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmin(fabs(z4), fabs(z3));
	double t_1 = fmax(fabs(z4), fabs(z3));
	double t_2 = (((fabs(z1) * z0) / (t_1 * z2)) * z0) / (t_0 * z2);
	double t_3 = sqrt((t_2 - z5)) * sqrt((t_2 + z5));
	double t_4 = (z0 / z2) * ((fabs(z1) * (z0 / z2)) / (t_1 * t_0));
	double t_5 = t_0 * t_1;
	double tmp;
	if (t_5 <= 1e-312) {
		tmp = t_3;
	} else if (t_5 <= 1e+279) {
		tmp = sqrt((t_4 - z5)) * sqrt((t_4 + z5));
	} else {
		tmp = t_3;
	}
	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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = fmin(abs(z4), abs(z3))
    t_1 = fmax(abs(z4), abs(z3))
    t_2 = (((abs(z1) * z0) / (t_1 * z2)) * z0) / (t_0 * z2)
    t_3 = sqrt((t_2 - z5)) * sqrt((t_2 + z5))
    t_4 = (z0 / z2) * ((abs(z1) * (z0 / z2)) / (t_1 * t_0))
    t_5 = t_0 * t_1
    if (t_5 <= 1d-312) then
        tmp = t_3
    else if (t_5 <= 1d+279) then
        tmp = sqrt((t_4 - z5)) * sqrt((t_4 + z5))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmin(Math.abs(z4), Math.abs(z3));
	double t_1 = fmax(Math.abs(z4), Math.abs(z3));
	double t_2 = (((Math.abs(z1) * z0) / (t_1 * z2)) * z0) / (t_0 * z2);
	double t_3 = Math.sqrt((t_2 - z5)) * Math.sqrt((t_2 + z5));
	double t_4 = (z0 / z2) * ((Math.abs(z1) * (z0 / z2)) / (t_1 * t_0));
	double t_5 = t_0 * t_1;
	double tmp;
	if (t_5 <= 1e-312) {
		tmp = t_3;
	} else if (t_5 <= 1e+279) {
		tmp = Math.sqrt((t_4 - z5)) * Math.sqrt((t_4 + z5));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = fmin(math.fabs(z4), math.fabs(z3))
	t_1 = fmax(math.fabs(z4), math.fabs(z3))
	t_2 = (((math.fabs(z1) * z0) / (t_1 * z2)) * z0) / (t_0 * z2)
	t_3 = math.sqrt((t_2 - z5)) * math.sqrt((t_2 + z5))
	t_4 = (z0 / z2) * ((math.fabs(z1) * (z0 / z2)) / (t_1 * t_0))
	t_5 = t_0 * t_1
	tmp = 0
	if t_5 <= 1e-312:
		tmp = t_3
	elif t_5 <= 1e+279:
		tmp = math.sqrt((t_4 - z5)) * math.sqrt((t_4 + z5))
	else:
		tmp = t_3
	return tmp

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = fmin(abs(z4), abs(z3))
	t_1 = fmax(abs(z4), abs(z3))
	t_2 = Float64(Float64(Float64(Float64(abs(z1) * z0) / Float64(t_1 * z2)) * z0) / Float64(t_0 * z2))
	t_3 = Float64(sqrt(Float64(t_2 - z5)) * sqrt(Float64(t_2 + z5)))
	t_4 = Float64(Float64(z0 / z2) * Float64(Float64(abs(z1) * Float64(z0 / z2)) / Float64(t_1 * t_0)))
	t_5 = Float64(t_0 * t_1)
	tmp = 0.0
	if (t_5 <= 1e-312)
		tmp = t_3;
	elseif (t_5 <= 1e+279)
		tmp = Float64(sqrt(Float64(t_4 - z5)) * sqrt(Float64(t_4 + z5)));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z4, z3, z2, z5)
	t_0 = min(abs(z4), abs(z3));
	t_1 = max(abs(z4), abs(z3));
	t_2 = (((abs(z1) * z0) / (t_1 * z2)) * z0) / (t_0 * z2);
	t_3 = sqrt((t_2 - z5)) * sqrt((t_2 + z5));
	t_4 = (z0 / z2) * ((abs(z1) * (z0 / z2)) / (t_1 * t_0));
	t_5 = t_0 * t_1;
	tmp = 0.0;
	if (t_5 <= 1e-312)
		tmp = t_3;
	elseif (t_5 <= 1e+279)
		tmp = sqrt((t_4 - z5)) * sqrt((t_4 + z5));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[Min[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Abs[z1], $MachinePrecision] * z0), $MachinePrecision] / N[(t$95$1 * z2), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / N[(t$95$0 * z2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t$95$2 - z5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$2 + z5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(z0 / z2), $MachinePrecision] * N[(N[(N[Abs[z1], $MachinePrecision] * N[(z0 / z2), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$5, 1e-312], t$95$3, If[LessEqual[t$95$5, 1e+279], N[(N[Sqrt[N[(t$95$4 - z5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$4 + z5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z4 z3) < 9.9999999999846534e-313 or 1.0000000000000001e279 < (*.f64 z4 z3)

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      13. lower-*.f64 34.0%

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      16. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      19. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      22. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

   5. Applied rewrites 34.0%

      \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      13. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      16. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      19. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

      22. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

   7. Applied rewrites 40.7%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      3. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      9. lower-/.f64 39.1%

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      14. lower-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      17. lift-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

   9. Applied rewrites 40.4%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) + z5} \]

       3. associate-*l/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} + z5} \]

       5. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       6. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} + z5} \]

       7. associate-/r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       8. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       9. lower-/.f64 41.8%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       10. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} + z5} \]

       12. associate-/l* N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       13. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       14. lower-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} + z5} \]

       16. *-commutative N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

       17. lift-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

   11. Applied rewrites 43.9%

       \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       2. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} \cdot \frac{z0}{z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       3. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} \cdot \frac{z0}{z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} \cdot \frac{z0}{z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       5. associate-/r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3}}{z4}} \cdot \frac{z0}{z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       6. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3}}{z4} \cdot \color{blue}{\frac{z0}{z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       7. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3} \cdot z0}{z4 \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       8. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3} \cdot z0}{z4 \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       9. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3} \cdot z0}}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       10. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z3} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       11. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z3} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       12. associate-*r/ N/A

           \[\leadsto \sqrt{\frac{\frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z3} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       13. associate-/l/ N/A

           \[\leadsto \sqrt{\frac{\color{blue}{\frac{z1 \cdot z0}{z2 \cdot z3}} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       14. *-commutative N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{\color{blue}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{\color{blue}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       16. lower-/.f64 N/A

           \[\leadsto \sqrt{\frac{\color{blue}{\frac{z1 \cdot z0}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       17. lower-*.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{\color{blue}{z1 \cdot z0}}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       18. lower-*.f64 37.2%

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{\color{blue}{z4 \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

   13. Applied rewrites 37.2%

       \[\leadsto \sqrt{\color{blue}{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]

       2. *-commutative N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} \cdot \frac{z0}{z2}} + z5} \]

       3. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} \cdot \frac{z0}{z2} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} \cdot \frac{z0}{z2} + z5} \]

       5. associate-/r* N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3}}{z4}} \cdot \frac{z0}{z2} + z5} \]

       6. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3}}{z4} \cdot \color{blue}{\frac{z0}{z2}} + z5} \]

       7. frac-times N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3} \cdot z0}{z4 \cdot z2}} + z5} \]

       8. lower-/.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3} \cdot z0}{z4 \cdot z2}} + z5} \]

       9. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3} \cdot z0}}{z4 \cdot z2} + z5} \]

       10. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z3} \cdot z0}{z4 \cdot z2} + z5} \]

       11. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z3} \cdot z0}{z4 \cdot z2} + z5} \]

       12. associate-*r/ N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z3} \cdot z0}{z4 \cdot z2} + z5} \]

       13. associate-/l/ N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\color{blue}{\frac{z1 \cdot z0}{z2 \cdot z3}} \cdot z0}{z4 \cdot z2} + z5} \]

       14. *-commutative N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{z1 \cdot z0}{\color{blue}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{z1 \cdot z0}{\color{blue}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} + z5} \]

       16. lower-/.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\color{blue}{\frac{z1 \cdot z0}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} + z5} \]

       17. lower-*.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{\color{blue}{z1 \cdot z0}}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} + z5} \]

       18. lower-*.f64 41.3%

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{\color{blue}{z4 \cdot z2}} + z5} \]

   15. Applied rewrites 41.3%

       \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2}} + z5} \]

   if 9.9999999999846534e-313 < (*.f64 z4 z3) < 1.0000000000000001e279

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      13. lower-*.f64 34.0%

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      16. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      19. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      22. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

   5. Applied rewrites 34.0%

      \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      13. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      16. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      19. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

      22. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

   7. Applied rewrites 40.7%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      3. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      9. lower-/.f64 39.1%

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      14. lower-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      17. lift-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

   9. Applied rewrites 40.4%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) + z5} \]

       3. associate-*l/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} + z5} \]

       5. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       6. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} + z5} \]

       7. associate-/r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       8. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       9. lower-/.f64 41.8%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       10. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} + z5} \]

       12. associate-/l* N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       13. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       14. lower-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} + z5} \]

       16. *-commutative N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

       17. lift-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

   11. Applied rewrites 43.9%

       \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|z4\right|, \left|z3\right|\right)\\ t_1 := \mathsf{max}\left(\left|z4\right|, \left|z3\right|\right)\\ t_2 := \frac{\frac{\left|z1\right| \cdot z0}{t\_1 \cdot z2} \cdot z0}{t\_0 \cdot z2}\\ t_3 := \sqrt{t\_2 - \left|z5\right|} \cdot \sqrt{t\_2 + \left|z5\right|}\\ t_4 := t\_0 \cdot t\_1\\ \mathbf{if}\;t\_4 \leq 10^{-238}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 10^{+279}:\\ \;\;\;\;\sqrt{\frac{z0}{z2} \cdot \frac{\left|z1\right| \cdot \frac{z0}{z2}}{t\_1 \cdot t\_0} - \left|z5\right|} \cdot \sqrt{\left|z5\right| - \left(-\frac{z0}{z2}\right) \cdot \left(\frac{z0}{t\_4 \cdot z2} \cdot \left|z1\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0 (fmin (fabs z4) (fabs z3)))
       (t_1 (fmax (fabs z4) (fabs z3)))
       (t_2 (/ (* (/ (* (fabs z1) z0) (* t_1 z2)) z0) (* t_0 z2)))
       (t_3 (* (sqrt (- t_2 (fabs z5))) (sqrt (+ t_2 (fabs z5)))))
       (t_4 (* t_0 t_1)))
  (if (<= t_4 1e-238)
    t_3
    (if (<= t_4 1e+279)
      (*
       (sqrt
        (-
         (* (/ z0 z2) (/ (* (fabs z1) (/ z0 z2)) (* t_1 t_0)))
         (fabs z5)))
       (sqrt
        (-
         (fabs z5)
         (* (- (/ z0 z2)) (* (/ z0 (* t_4 z2)) (fabs z1))))))
      t_3))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmin(fabs(z4), fabs(z3));
	double t_1 = fmax(fabs(z4), fabs(z3));
	double t_2 = (((fabs(z1) * z0) / (t_1 * z2)) * z0) / (t_0 * z2);
	double t_3 = sqrt((t_2 - fabs(z5))) * sqrt((t_2 + fabs(z5)));
	double t_4 = t_0 * t_1;
	double tmp;
	if (t_4 <= 1e-238) {
		tmp = t_3;
	} else if (t_4 <= 1e+279) {
		tmp = sqrt((((z0 / z2) * ((fabs(z1) * (z0 / z2)) / (t_1 * t_0))) - fabs(z5))) * sqrt((fabs(z5) - (-(z0 / z2) * ((z0 / (t_4 * z2)) * fabs(z1)))));
	} else {
		tmp = t_3;
	}
	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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = fmin(abs(z4), abs(z3))
    t_1 = fmax(abs(z4), abs(z3))
    t_2 = (((abs(z1) * z0) / (t_1 * z2)) * z0) / (t_0 * z2)
    t_3 = sqrt((t_2 - abs(z5))) * sqrt((t_2 + abs(z5)))
    t_4 = t_0 * t_1
    if (t_4 <= 1d-238) then
        tmp = t_3
    else if (t_4 <= 1d+279) then
        tmp = sqrt((((z0 / z2) * ((abs(z1) * (z0 / z2)) / (t_1 * t_0))) - abs(z5))) * sqrt((abs(z5) - (-(z0 / z2) * ((z0 / (t_4 * z2)) * abs(z1)))))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmin(Math.abs(z4), Math.abs(z3));
	double t_1 = fmax(Math.abs(z4), Math.abs(z3));
	double t_2 = (((Math.abs(z1) * z0) / (t_1 * z2)) * z0) / (t_0 * z2);
	double t_3 = Math.sqrt((t_2 - Math.abs(z5))) * Math.sqrt((t_2 + Math.abs(z5)));
	double t_4 = t_0 * t_1;
	double tmp;
	if (t_4 <= 1e-238) {
		tmp = t_3;
	} else if (t_4 <= 1e+279) {
		tmp = Math.sqrt((((z0 / z2) * ((Math.abs(z1) * (z0 / z2)) / (t_1 * t_0))) - Math.abs(z5))) * Math.sqrt((Math.abs(z5) - (-(z0 / z2) * ((z0 / (t_4 * z2)) * Math.abs(z1)))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = fmin(math.fabs(z4), math.fabs(z3))
	t_1 = fmax(math.fabs(z4), math.fabs(z3))
	t_2 = (((math.fabs(z1) * z0) / (t_1 * z2)) * z0) / (t_0 * z2)
	t_3 = math.sqrt((t_2 - math.fabs(z5))) * math.sqrt((t_2 + math.fabs(z5)))
	t_4 = t_0 * t_1
	tmp = 0
	if t_4 <= 1e-238:
		tmp = t_3
	elif t_4 <= 1e+279:
		tmp = math.sqrt((((z0 / z2) * ((math.fabs(z1) * (z0 / z2)) / (t_1 * t_0))) - math.fabs(z5))) * math.sqrt((math.fabs(z5) - (-(z0 / z2) * ((z0 / (t_4 * z2)) * math.fabs(z1)))))
	else:
		tmp = t_3
	return tmp

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = fmin(abs(z4), abs(z3))
	t_1 = fmax(abs(z4), abs(z3))
	t_2 = Float64(Float64(Float64(Float64(abs(z1) * z0) / Float64(t_1 * z2)) * z0) / Float64(t_0 * z2))
	t_3 = Float64(sqrt(Float64(t_2 - abs(z5))) * sqrt(Float64(t_2 + abs(z5))))
	t_4 = Float64(t_0 * t_1)
	tmp = 0.0
	if (t_4 <= 1e-238)
		tmp = t_3;
	elseif (t_4 <= 1e+279)
		tmp = Float64(sqrt(Float64(Float64(Float64(z0 / z2) * Float64(Float64(abs(z1) * Float64(z0 / z2)) / Float64(t_1 * t_0))) - abs(z5))) * sqrt(Float64(abs(z5) - Float64(Float64(-Float64(z0 / z2)) * Float64(Float64(z0 / Float64(t_4 * z2)) * abs(z1))))));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z4, z3, z2, z5)
	t_0 = min(abs(z4), abs(z3));
	t_1 = max(abs(z4), abs(z3));
	t_2 = (((abs(z1) * z0) / (t_1 * z2)) * z0) / (t_0 * z2);
	t_3 = sqrt((t_2 - abs(z5))) * sqrt((t_2 + abs(z5)));
	t_4 = t_0 * t_1;
	tmp = 0.0;
	if (t_4 <= 1e-238)
		tmp = t_3;
	elseif (t_4 <= 1e+279)
		tmp = sqrt((((z0 / z2) * ((abs(z1) * (z0 / z2)) / (t_1 * t_0))) - abs(z5))) * sqrt((abs(z5) - (-(z0 / z2) * ((z0 / (t_4 * z2)) * abs(z1)))));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[Min[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Abs[z1], $MachinePrecision] * z0), $MachinePrecision] / N[(t$95$1 * z2), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / N[(t$95$0 * z2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t$95$2 - N[Abs[z5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$2 + N[Abs[z5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$4, 1e-238], t$95$3, If[LessEqual[t$95$4, 1e+279], N[(N[Sqrt[N[(N[(N[(z0 / z2), $MachinePrecision] * N[(N[(N[Abs[z1], $MachinePrecision] * N[(z0 / z2), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Abs[z5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[z5], $MachinePrecision] - N[((-N[(z0 / z2), $MachinePrecision]) * N[(N[(z0 / N[(t$95$4 * z2), $MachinePrecision]), $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z4 z3) < 9.9999999999999999e-239 or 1.0000000000000001e279 < (*.f64 z4 z3)

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      13. lower-*.f64 34.0%

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      16. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      19. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      22. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

   5. Applied rewrites 34.0%

      \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      13. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      16. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      19. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

      22. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

   7. Applied rewrites 40.7%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      3. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      9. lower-/.f64 39.1%

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      14. lower-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      17. lift-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

   9. Applied rewrites 40.4%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) + z5} \]

       3. associate-*l/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} + z5} \]

       5. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       6. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} + z5} \]

       7. associate-/r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       8. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       9. lower-/.f64 41.8%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       10. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} + z5} \]

       12. associate-/l* N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       13. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       14. lower-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} + z5} \]

       16. *-commutative N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

       17. lift-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

   11. Applied rewrites 43.9%

       \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       2. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} \cdot \frac{z0}{z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       3. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} \cdot \frac{z0}{z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} \cdot \frac{z0}{z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       5. associate-/r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3}}{z4}} \cdot \frac{z0}{z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       6. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3}}{z4} \cdot \color{blue}{\frac{z0}{z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       7. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3} \cdot z0}{z4 \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       8. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3} \cdot z0}{z4 \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       9. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3} \cdot z0}}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       10. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z3} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       11. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z3} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       12. associate-*r/ N/A

           \[\leadsto \sqrt{\frac{\frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z3} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       13. associate-/l/ N/A

           \[\leadsto \sqrt{\frac{\color{blue}{\frac{z1 \cdot z0}{z2 \cdot z3}} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       14. *-commutative N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{\color{blue}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{\color{blue}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       16. lower-/.f64 N/A

           \[\leadsto \sqrt{\frac{\color{blue}{\frac{z1 \cdot z0}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       17. lower-*.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{\color{blue}{z1 \cdot z0}}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       18. lower-*.f64 37.2%

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{\color{blue}{z4 \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

   13. Applied rewrites 37.2%

       \[\leadsto \sqrt{\color{blue}{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]

       2. *-commutative N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} \cdot \frac{z0}{z2}} + z5} \]

       3. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} \cdot \frac{z0}{z2} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} \cdot \frac{z0}{z2} + z5} \]

       5. associate-/r* N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3}}{z4}} \cdot \frac{z0}{z2} + z5} \]

       6. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3}}{z4} \cdot \color{blue}{\frac{z0}{z2}} + z5} \]

       7. frac-times N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3} \cdot z0}{z4 \cdot z2}} + z5} \]

       8. lower-/.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3} \cdot z0}{z4 \cdot z2}} + z5} \]

       9. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3} \cdot z0}}{z4 \cdot z2} + z5} \]

       10. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z3} \cdot z0}{z4 \cdot z2} + z5} \]

       11. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z3} \cdot z0}{z4 \cdot z2} + z5} \]

       12. associate-*r/ N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z3} \cdot z0}{z4 \cdot z2} + z5} \]

       13. associate-/l/ N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\color{blue}{\frac{z1 \cdot z0}{z2 \cdot z3}} \cdot z0}{z4 \cdot z2} + z5} \]

       14. *-commutative N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{z1 \cdot z0}{\color{blue}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{z1 \cdot z0}{\color{blue}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} + z5} \]

       16. lower-/.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\color{blue}{\frac{z1 \cdot z0}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} + z5} \]

       17. lower-*.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{\color{blue}{z1 \cdot z0}}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} + z5} \]

       18. lower-*.f64 41.3%

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{\color{blue}{z4 \cdot z2}} + z5} \]

   15. Applied rewrites 41.3%

       \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2}} + z5} \]

   if 9.9999999999999999e-239 < (*.f64 z4 z3) < 1.0000000000000001e279

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      13. lower-*.f64 34.0%

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      16. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      19. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      22. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

   5. Applied rewrites 34.0%

      \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      13. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      16. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      19. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

      22. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

   7. Applied rewrites 40.7%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      3. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      9. lower-/.f64 39.1%

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      14. lower-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      17. lift-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

   9. Applied rewrites 40.4%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) + z5} \]

       3. associate-*l/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} + z5} \]

       5. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       6. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} + z5} \]

       7. associate-/r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       8. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       9. lower-/.f64 41.8%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       10. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} + z5} \]

       12. associate-/l* N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       13. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       14. lower-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} + z5} \]

       16. *-commutative N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

       17. lift-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

   11. Applied rewrites 43.9%

       \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]
   12. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5}} \]

       2. +-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\color{blue}{z5 + \frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}}} \]

       3. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{z5 + \color{blue}{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}}} \]

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

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\color{blue}{z5 - \left(\mathsf{neg}\left(\frac{z0}{z2}\right)\right) \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}}} \]

       5. lower--.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\color{blue}{z5 - \left(\mathsf{neg}\left(\frac{z0}{z2}\right)\right) \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}}} \]

       6. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{z5 - \color{blue}{\left(\mathsf{neg}\left(\frac{z0}{z2}\right)\right) \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}}} \]

       7. lower-neg.f64 43.9%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{z5 - \color{blue}{\left(-\frac{z0}{z2}\right)} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} \]

       8. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{z5 - \left(-\frac{z0}{z2}\right) \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}}} \]

       9. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{z5 - \left(-\frac{z0}{z2}\right) \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z3 \cdot z4}} \]

       10. associate-*r/ N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{z5 - \left(-\frac{z0}{z2}\right) \cdot \color{blue}{\left(z1 \cdot \frac{\frac{z0}{z2}}{z3 \cdot z4}\right)}} \]

       11. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{z5 - \left(-\frac{z0}{z2}\right) \cdot \left(z1 \cdot \frac{\color{blue}{\frac{z0}{z2}}}{z3 \cdot z4}\right)} \]

       12. associate-/l/ N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{z5 - \left(-\frac{z0}{z2}\right) \cdot \left(z1 \cdot \color{blue}{\frac{z0}{z2 \cdot \left(z3 \cdot z4\right)}}\right)} \]

       13. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{z5 - \left(-\frac{z0}{z2}\right) \cdot \left(z1 \cdot \frac{z0}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}}\right)} \]

       14. *-commutative N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{z5 - \left(-\frac{z0}{z2}\right) \cdot \color{blue}{\left(\frac{z0}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z1\right)}} \]

       15. lower-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{z5 - \left(-\frac{z0}{z2}\right) \cdot \color{blue}{\left(\frac{z0}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z1\right)}} \]

       16. lower-/.f64 41.8%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{z5 - \left(-\frac{z0}{z2}\right) \cdot \left(\color{blue}{\frac{z0}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z1\right)} \]

       17. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{z5 - \left(-\frac{z0}{z2}\right) \cdot \left(\frac{z0}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z1\right)} \]

       18. *-commutative N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{z5 - \left(-\frac{z0}{z2}\right) \cdot \left(\frac{z0}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z1\right)} \]

       19. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{z5 - \left(-\frac{z0}{z2}\right) \cdot \left(\frac{z0}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z1\right)} \]

       20. *-commutative N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{z5 - \left(-\frac{z0}{z2}\right) \cdot \left(\frac{z0}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z1\right)} \]

       21. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{z5 - \left(-\frac{z0}{z2}\right) \cdot \left(\frac{z0}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z1\right)} \]

       22. lift-*.f64 41.8%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{z5 - \left(-\frac{z0}{z2}\right) \cdot \left(\frac{z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z1\right)} \]

   13. Applied rewrites 41.8%

       \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\color{blue}{z5 - \left(-\frac{z0}{z2}\right) \cdot \left(\frac{z0}{\left(z4 \cdot z3\right) \cdot z2} \cdot z1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|z4\right|, \left|z3\right|\right)\\ t_1 := \mathsf{max}\left(\left|z4\right|, \left|z3\right|\right)\\ t_2 := \frac{\frac{\left|z1\right| \cdot z0}{t\_1 \cdot z2} \cdot z0}{t\_0 \cdot z2}\\ t_3 := \sqrt{t\_2 - z5} \cdot \sqrt{t\_2 + z5}\\ t_4 := t\_0 \cdot t\_1\\ t_5 := \frac{z0}{z2} \cdot \left(\frac{\left|z1\right|}{t\_4 \cdot z2} \cdot z0\right)\\ \mathbf{if}\;t\_4 \leq 5 \cdot 10^{-147}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+275}:\\ \;\;\;\;\sqrt{t\_5 - z5} \cdot \sqrt{t\_5 + z5}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0 (fmin (fabs z4) (fabs z3)))
       (t_1 (fmax (fabs z4) (fabs z3)))
       (t_2 (/ (* (/ (* (fabs z1) z0) (* t_1 z2)) z0) (* t_0 z2)))
       (t_3 (* (sqrt (- t_2 z5)) (sqrt (+ t_2 z5))))
       (t_4 (* t_0 t_1))
       (t_5 (* (/ z0 z2) (* (/ (fabs z1) (* t_4 z2)) z0))))
  (if (<= t_4 5e-147)
    t_3
    (if (<= t_4 4e+275) (* (sqrt (- t_5 z5)) (sqrt (+ t_5 z5))) t_3))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmin(fabs(z4), fabs(z3));
	double t_1 = fmax(fabs(z4), fabs(z3));
	double t_2 = (((fabs(z1) * z0) / (t_1 * z2)) * z0) / (t_0 * z2);
	double t_3 = sqrt((t_2 - z5)) * sqrt((t_2 + z5));
	double t_4 = t_0 * t_1;
	double t_5 = (z0 / z2) * ((fabs(z1) / (t_4 * z2)) * z0);
	double tmp;
	if (t_4 <= 5e-147) {
		tmp = t_3;
	} else if (t_4 <= 4e+275) {
		tmp = sqrt((t_5 - z5)) * sqrt((t_5 + z5));
	} else {
		tmp = t_3;
	}
	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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = fmin(abs(z4), abs(z3))
    t_1 = fmax(abs(z4), abs(z3))
    t_2 = (((abs(z1) * z0) / (t_1 * z2)) * z0) / (t_0 * z2)
    t_3 = sqrt((t_2 - z5)) * sqrt((t_2 + z5))
    t_4 = t_0 * t_1
    t_5 = (z0 / z2) * ((abs(z1) / (t_4 * z2)) * z0)
    if (t_4 <= 5d-147) then
        tmp = t_3
    else if (t_4 <= 4d+275) then
        tmp = sqrt((t_5 - z5)) * sqrt((t_5 + z5))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmin(Math.abs(z4), Math.abs(z3));
	double t_1 = fmax(Math.abs(z4), Math.abs(z3));
	double t_2 = (((Math.abs(z1) * z0) / (t_1 * z2)) * z0) / (t_0 * z2);
	double t_3 = Math.sqrt((t_2 - z5)) * Math.sqrt((t_2 + z5));
	double t_4 = t_0 * t_1;
	double t_5 = (z0 / z2) * ((Math.abs(z1) / (t_4 * z2)) * z0);
	double tmp;
	if (t_4 <= 5e-147) {
		tmp = t_3;
	} else if (t_4 <= 4e+275) {
		tmp = Math.sqrt((t_5 - z5)) * Math.sqrt((t_5 + z5));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = fmin(math.fabs(z4), math.fabs(z3))
	t_1 = fmax(math.fabs(z4), math.fabs(z3))
	t_2 = (((math.fabs(z1) * z0) / (t_1 * z2)) * z0) / (t_0 * z2)
	t_3 = math.sqrt((t_2 - z5)) * math.sqrt((t_2 + z5))
	t_4 = t_0 * t_1
	t_5 = (z0 / z2) * ((math.fabs(z1) / (t_4 * z2)) * z0)
	tmp = 0
	if t_4 <= 5e-147:
		tmp = t_3
	elif t_4 <= 4e+275:
		tmp = math.sqrt((t_5 - z5)) * math.sqrt((t_5 + z5))
	else:
		tmp = t_3
	return tmp

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = fmin(abs(z4), abs(z3))
	t_1 = fmax(abs(z4), abs(z3))
	t_2 = Float64(Float64(Float64(Float64(abs(z1) * z0) / Float64(t_1 * z2)) * z0) / Float64(t_0 * z2))
	t_3 = Float64(sqrt(Float64(t_2 - z5)) * sqrt(Float64(t_2 + z5)))
	t_4 = Float64(t_0 * t_1)
	t_5 = Float64(Float64(z0 / z2) * Float64(Float64(abs(z1) / Float64(t_4 * z2)) * z0))
	tmp = 0.0
	if (t_4 <= 5e-147)
		tmp = t_3;
	elseif (t_4 <= 4e+275)
		tmp = Float64(sqrt(Float64(t_5 - z5)) * sqrt(Float64(t_5 + z5)));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z4, z3, z2, z5)
	t_0 = min(abs(z4), abs(z3));
	t_1 = max(abs(z4), abs(z3));
	t_2 = (((abs(z1) * z0) / (t_1 * z2)) * z0) / (t_0 * z2);
	t_3 = sqrt((t_2 - z5)) * sqrt((t_2 + z5));
	t_4 = t_0 * t_1;
	t_5 = (z0 / z2) * ((abs(z1) / (t_4 * z2)) * z0);
	tmp = 0.0;
	if (t_4 <= 5e-147)
		tmp = t_3;
	elseif (t_4 <= 4e+275)
		tmp = sqrt((t_5 - z5)) * sqrt((t_5 + z5));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[Min[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[Abs[z1], $MachinePrecision] * z0), $MachinePrecision] / N[(t$95$1 * z2), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / N[(t$95$0 * z2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t$95$2 - z5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$2 + z5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(N[(z0 / z2), $MachinePrecision] * N[(N[(N[Abs[z1], $MachinePrecision] / N[(t$95$4 * z2), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 5e-147], t$95$3, If[LessEqual[t$95$4, 4e+275], N[(N[Sqrt[N[(t$95$5 - z5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$5 + z5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z4 z3) < 5.0000000000000001e-147 or 3.9999999999999998e275 < (*.f64 z4 z3)

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      13. lower-*.f64 34.0%

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      16. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      19. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      22. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

   5. Applied rewrites 34.0%

      \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      13. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      16. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      19. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

      22. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

   7. Applied rewrites 40.7%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      3. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      9. lower-/.f64 39.1%

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      14. lower-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      17. lift-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

   9. Applied rewrites 40.4%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) + z5} \]

       3. associate-*l/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} + z5} \]

       5. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       6. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} + z5} \]

       7. associate-/r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       8. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       9. lower-/.f64 41.8%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       10. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} + z5} \]

       12. associate-/l* N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       13. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       14. lower-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} + z5} \]

       16. *-commutative N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

       17. lift-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

   11. Applied rewrites 43.9%

       \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       2. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} \cdot \frac{z0}{z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       3. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} \cdot \frac{z0}{z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} \cdot \frac{z0}{z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       5. associate-/r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3}}{z4}} \cdot \frac{z0}{z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       6. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3}}{z4} \cdot \color{blue}{\frac{z0}{z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       7. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3} \cdot z0}{z4 \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       8. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3} \cdot z0}{z4 \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       9. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3} \cdot z0}}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       10. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z3} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       11. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z3} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       12. associate-*r/ N/A

           \[\leadsto \sqrt{\frac{\frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z3} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       13. associate-/l/ N/A

           \[\leadsto \sqrt{\frac{\color{blue}{\frac{z1 \cdot z0}{z2 \cdot z3}} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       14. *-commutative N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{\color{blue}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{\color{blue}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       16. lower-/.f64 N/A

           \[\leadsto \sqrt{\frac{\color{blue}{\frac{z1 \cdot z0}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       17. lower-*.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{\color{blue}{z1 \cdot z0}}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       18. lower-*.f64 37.2%

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{\color{blue}{z4 \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

   13. Applied rewrites 37.2%

       \[\leadsto \sqrt{\color{blue}{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]

       2. *-commutative N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} \cdot \frac{z0}{z2}} + z5} \]

       3. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} \cdot \frac{z0}{z2} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} \cdot \frac{z0}{z2} + z5} \]

       5. associate-/r* N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3}}{z4}} \cdot \frac{z0}{z2} + z5} \]

       6. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3}}{z4} \cdot \color{blue}{\frac{z0}{z2}} + z5} \]

       7. frac-times N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3} \cdot z0}{z4 \cdot z2}} + z5} \]

       8. lower-/.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{\frac{z1 \cdot \frac{z0}{z2}}{z3} \cdot z0}{z4 \cdot z2}} + z5} \]

       9. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3} \cdot z0}}{z4 \cdot z2} + z5} \]

       10. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z3} \cdot z0}{z4 \cdot z2} + z5} \]

       11. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z3} \cdot z0}{z4 \cdot z2} + z5} \]

       12. associate-*r/ N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z3} \cdot z0}{z4 \cdot z2} + z5} \]

       13. associate-/l/ N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\color{blue}{\frac{z1 \cdot z0}{z2 \cdot z3}} \cdot z0}{z4 \cdot z2} + z5} \]

       14. *-commutative N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{z1 \cdot z0}{\color{blue}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{z1 \cdot z0}{\color{blue}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} + z5} \]

       16. lower-/.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\color{blue}{\frac{z1 \cdot z0}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} + z5} \]

       17. lower-*.f64 N/A

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{\color{blue}{z1 \cdot z0}}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} + z5} \]

       18. lower-*.f64 41.3%

           \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{\color{blue}{z4 \cdot z2}} + z5} \]

   15. Applied rewrites 41.3%

       \[\leadsto \sqrt{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{\frac{z1 \cdot z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2}} + z5} \]

   if 5.0000000000000001e-147 < (*.f64 z4 z3) < 3.9999999999999998e275

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      13. lower-*.f64 34.0%

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      16. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      19. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      22. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

   5. Applied rewrites 34.0%

      \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      13. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      16. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      19. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

      22. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

   7. Applied rewrites 40.7%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 84.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|z4\right|, \left|z3\right|\right)\\ t_1 := \mathsf{max}\left(\left|z4\right|, \left|z3\right|\right)\\ t_2 := \frac{z0}{t\_0} \cdot \frac{\left(-\left|z1\right|\right) \cdot z0}{z2 \cdot \left(t\_1 \cdot z2\right)}\\ t_3 := \sqrt{\left(t\_2 + z5\right) \cdot \left(t\_2 - z5\right)}\\ t_4 := t\_0 \cdot t\_1\\ t_5 := \frac{z0}{z2} \cdot \left(\frac{\left|z1\right|}{t\_4 \cdot z2} \cdot z0\right)\\ \mathbf{if}\;t\_4 \leq 10^{-312}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+275}:\\ \;\;\;\;\sqrt{t\_5 - z5} \cdot \sqrt{t\_5 + z5}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0 (fmin (fabs z4) (fabs z3)))
       (t_1 (fmax (fabs z4) (fabs z3)))
       (t_2 (* (/ z0 t_0) (/ (* (- (fabs z1)) z0) (* z2 (* t_1 z2)))))
       (t_3 (sqrt (* (+ t_2 z5) (- t_2 z5))))
       (t_4 (* t_0 t_1))
       (t_5 (* (/ z0 z2) (* (/ (fabs z1) (* t_4 z2)) z0))))
  (if (<= t_4 1e-312)
    t_3
    (if (<= t_4 4e+275) (* (sqrt (- t_5 z5)) (sqrt (+ t_5 z5))) t_3))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmin(fabs(z4), fabs(z3));
	double t_1 = fmax(fabs(z4), fabs(z3));
	double t_2 = (z0 / t_0) * ((-fabs(z1) * z0) / (z2 * (t_1 * z2)));
	double t_3 = sqrt(((t_2 + z5) * (t_2 - z5)));
	double t_4 = t_0 * t_1;
	double t_5 = (z0 / z2) * ((fabs(z1) / (t_4 * z2)) * z0);
	double tmp;
	if (t_4 <= 1e-312) {
		tmp = t_3;
	} else if (t_4 <= 4e+275) {
		tmp = sqrt((t_5 - z5)) * sqrt((t_5 + z5));
	} else {
		tmp = t_3;
	}
	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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = fmin(abs(z4), abs(z3))
    t_1 = fmax(abs(z4), abs(z3))
    t_2 = (z0 / t_0) * ((-abs(z1) * z0) / (z2 * (t_1 * z2)))
    t_3 = sqrt(((t_2 + z5) * (t_2 - z5)))
    t_4 = t_0 * t_1
    t_5 = (z0 / z2) * ((abs(z1) / (t_4 * z2)) * z0)
    if (t_4 <= 1d-312) then
        tmp = t_3
    else if (t_4 <= 4d+275) then
        tmp = sqrt((t_5 - z5)) * sqrt((t_5 + z5))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmin(Math.abs(z4), Math.abs(z3));
	double t_1 = fmax(Math.abs(z4), Math.abs(z3));
	double t_2 = (z0 / t_0) * ((-Math.abs(z1) * z0) / (z2 * (t_1 * z2)));
	double t_3 = Math.sqrt(((t_2 + z5) * (t_2 - z5)));
	double t_4 = t_0 * t_1;
	double t_5 = (z0 / z2) * ((Math.abs(z1) / (t_4 * z2)) * z0);
	double tmp;
	if (t_4 <= 1e-312) {
		tmp = t_3;
	} else if (t_4 <= 4e+275) {
		tmp = Math.sqrt((t_5 - z5)) * Math.sqrt((t_5 + z5));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = fmin(math.fabs(z4), math.fabs(z3))
	t_1 = fmax(math.fabs(z4), math.fabs(z3))
	t_2 = (z0 / t_0) * ((-math.fabs(z1) * z0) / (z2 * (t_1 * z2)))
	t_3 = math.sqrt(((t_2 + z5) * (t_2 - z5)))
	t_4 = t_0 * t_1
	t_5 = (z0 / z2) * ((math.fabs(z1) / (t_4 * z2)) * z0)
	tmp = 0
	if t_4 <= 1e-312:
		tmp = t_3
	elif t_4 <= 4e+275:
		tmp = math.sqrt((t_5 - z5)) * math.sqrt((t_5 + z5))
	else:
		tmp = t_3
	return tmp

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = fmin(abs(z4), abs(z3))
	t_1 = fmax(abs(z4), abs(z3))
	t_2 = Float64(Float64(z0 / t_0) * Float64(Float64(Float64(-abs(z1)) * z0) / Float64(z2 * Float64(t_1 * z2))))
	t_3 = sqrt(Float64(Float64(t_2 + z5) * Float64(t_2 - z5)))
	t_4 = Float64(t_0 * t_1)
	t_5 = Float64(Float64(z0 / z2) * Float64(Float64(abs(z1) / Float64(t_4 * z2)) * z0))
	tmp = 0.0
	if (t_4 <= 1e-312)
		tmp = t_3;
	elseif (t_4 <= 4e+275)
		tmp = Float64(sqrt(Float64(t_5 - z5)) * sqrt(Float64(t_5 + z5)));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z4, z3, z2, z5)
	t_0 = min(abs(z4), abs(z3));
	t_1 = max(abs(z4), abs(z3));
	t_2 = (z0 / t_0) * ((-abs(z1) * z0) / (z2 * (t_1 * z2)));
	t_3 = sqrt(((t_2 + z5) * (t_2 - z5)));
	t_4 = t_0 * t_1;
	t_5 = (z0 / z2) * ((abs(z1) / (t_4 * z2)) * z0);
	tmp = 0.0;
	if (t_4 <= 1e-312)
		tmp = t_3;
	elseif (t_4 <= 4e+275)
		tmp = sqrt((t_5 - z5)) * sqrt((t_5 + z5));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[Min[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(z0 / t$95$0), $MachinePrecision] * N[(N[((-N[Abs[z1], $MachinePrecision]) * z0), $MachinePrecision] / N[(z2 * N[(t$95$1 * z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(t$95$2 + z5), $MachinePrecision] * N[(t$95$2 - z5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(N[(z0 / z2), $MachinePrecision] * N[(N[(N[Abs[z1], $MachinePrecision] / N[(t$95$4 * z2), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1e-312], t$95$3, If[LessEqual[t$95$4, 4e+275], N[(N[Sqrt[N[(t$95$5 - z5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$5 + z5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z4 z3) < 9.9999999999846534e-313 or 3.9999999999999998e275 < (*.f64 z4 z3)

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. sub-flip N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} + \left(\mathsf{neg}\left(z5 \cdot z5\right)\right)}} \]

      3. add-flip N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      7. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      9. remove-double-neg N/A

         \[\leadsto \sqrt{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - \color{blue}{z5 \cdot z5}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - \color{blue}{z5 \cdot z5}} \]

      11. difference-of-squares N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) + z5\right) \cdot \left(\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - z5\right)}} \]

   3. Applied rewrites 67.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{z0 \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\color{blue}{z0 \cdot z1}}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{\frac{z0 \cdot z1}{z2}}{z2 \cdot \left(z3 \cdot z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      8. frac-2neg N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\color{blue}{-\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\color{blue}{\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z0 \cdot z1}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      17. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{\left(z2 \cdot z3\right) \cdot z4}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      18. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      20. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      21. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      22. lower-neg.f64 68.2%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

   5. Applied rewrites 68.2%

      \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)} - z5\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) - z5\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{z0 \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\color{blue}{z0 \cdot z1}}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5\right)} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{\frac{z0 \cdot z1}{z2}}{z2 \cdot \left(z3 \cdot z4\right)}} - z5\right)} \]

      8. frac-2neg N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\color{blue}{-\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\color{blue}{\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z0 \cdot z1}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right)} - z5\right)} \]

      17. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{\left(z2 \cdot z3\right) \cdot z4}\right)} - z5\right)} \]

      18. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} - z5\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} - z5\right)} \]

      20. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} - z5\right)} \]

      21. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} - z5\right)} \]

      22. lower-neg.f64 75.0%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(-z4\right)}} - z5\right)} \]

   7. Applied rewrites 75.0%

      \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} - z5\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{\left(-z0\right) \cdot \left(-\frac{z1 \cdot z0}{z2}\right)}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{\left(-z0\right) \cdot \left(-\frac{z1 \cdot z0}{z2}\right)}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\frac{\left(-z0\right) \cdot \left(-\frac{z1 \cdot z0}{z2}\right)}{\color{blue}{\left(-z4\right) \cdot \left(z3 \cdot z2\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{-z0}{-z4} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{-z0}{-z4} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      8. lift-neg.f64 N/A

         \[\leadsto \sqrt{\left(\frac{\color{blue}{\mathsf{neg}\left(z0\right)}}{-z4} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      9. lift-neg.f64 N/A

         \[\leadsto \sqrt{\left(\frac{\mathsf{neg}\left(z0\right)}{\color{blue}{\mathsf{neg}\left(z4\right)}} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      10. frac-2neg-rev N/A

          \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z4}} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z4}} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      12. lift-neg.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{z1 \cdot z0}{z2}\right)}}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\mathsf{neg}\left(\color{blue}{\frac{z1 \cdot z0}{z2}}\right)}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      14. distribute-neg-frac N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\color{blue}{\frac{\mathsf{neg}\left(z1 \cdot z0\right)}{z2}}}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      15. associate-/l/ N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \color{blue}{\frac{\mathsf{neg}\left(z1 \cdot z0\right)}{z2 \cdot \left(z3 \cdot z2\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      16. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \color{blue}{\frac{\mathsf{neg}\left(z1 \cdot z0\right)}{z2 \cdot \left(z3 \cdot z2\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\mathsf{neg}\left(\color{blue}{z1 \cdot z0}\right)}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      18. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\color{blue}{\left(\mathsf{neg}\left(z1\right)\right) \cdot z0}}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\color{blue}{\left(\mathsf{neg}\left(z1\right)\right) \cdot z0}}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      20. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\color{blue}{\left(-z1\right)} \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      21. lower-*.f64 69.7%

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{\color{blue}{z2 \cdot \left(z3 \cdot z2\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

   9. Applied rewrites 69.7%

      \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} - z5\right)} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} - z5\right)} \]

       3. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\frac{\left(-z0\right) \cdot \left(-\frac{z1 \cdot z0}{z2}\right)}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} - z5\right)} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{\left(-z0\right) \cdot \left(-\frac{z1 \cdot z0}{z2}\right)}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} - z5\right)} \]

       5. *-commutative N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{\left(-z0\right) \cdot \left(-\frac{z1 \cdot z0}{z2}\right)}{\color{blue}{\left(-z4\right) \cdot \left(z3 \cdot z2\right)}} - z5\right)} \]

       6. times-frac N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\frac{-z0}{-z4} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2}} - z5\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\frac{-z0}{-z4} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2}} - z5\right)} \]

       8. lift-neg.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{\color{blue}{\mathsf{neg}\left(z0\right)}}{-z4} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} - z5\right)} \]

       9. lift-neg.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{\mathsf{neg}\left(z0\right)}{\color{blue}{\mathsf{neg}\left(z4\right)}} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} - z5\right)} \]

       10. frac-2neg-rev N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\frac{z0}{z4}} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} - z5\right)} \]

       11. lower-/.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\frac{z0}{z4}} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} - z5\right)} \]

       12. lift-neg.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{z1 \cdot z0}{z2}\right)}}{z3 \cdot z2} - z5\right)} \]

       13. lift-/.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\mathsf{neg}\left(\color{blue}{\frac{z1 \cdot z0}{z2}}\right)}{z3 \cdot z2} - z5\right)} \]

       14. distribute-neg-frac N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\color{blue}{\frac{\mathsf{neg}\left(z1 \cdot z0\right)}{z2}}}{z3 \cdot z2} - z5\right)} \]

       15. associate-/l/ N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \color{blue}{\frac{\mathsf{neg}\left(z1 \cdot z0\right)}{z2 \cdot \left(z3 \cdot z2\right)}} - z5\right)} \]

       16. lower-/.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \color{blue}{\frac{\mathsf{neg}\left(z1 \cdot z0\right)}{z2 \cdot \left(z3 \cdot z2\right)}} - z5\right)} \]

       17. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\mathsf{neg}\left(\color{blue}{z1 \cdot z0}\right)}{z2 \cdot \left(z3 \cdot z2\right)} - z5\right)} \]

       18. distribute-lft-neg-in N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\color{blue}{\left(\mathsf{neg}\left(z1\right)\right) \cdot z0}}{z2 \cdot \left(z3 \cdot z2\right)} - z5\right)} \]

       19. lower-*.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\color{blue}{\left(\mathsf{neg}\left(z1\right)\right) \cdot z0}}{z2 \cdot \left(z3 \cdot z2\right)} - z5\right)} \]

       20. lower-neg.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\color{blue}{\left(-z1\right)} \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} - z5\right)} \]

       21. lower-*.f64 71.6%

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{\color{blue}{z2 \cdot \left(z3 \cdot z2\right)}} - z5\right)} \]

   11. Applied rewrites 71.6%

       \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)}} - z5\right)} \]

   if 9.9999999999846534e-313 < (*.f64 z4 z3) < 3.9999999999999998e275

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      13. lower-*.f64 34.0%

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      16. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      19. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      22. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

   5. Applied rewrites 34.0%

      \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      13. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      16. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      19. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

      22. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

   7. Applied rewrites 40.7%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 82.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|z4\right|, \left|z3\right|\right)\\ t_1 := \mathsf{max}\left(\left|z4\right|, \left|z3\right|\right) \cdot z2\\ t_2 := \frac{z0}{z2} \cdot \left(\frac{\left|z1\right|}{t\_1 \cdot t\_0} \cdot z0\right)\\ t_3 := \frac{z0}{t\_0} \cdot \frac{\left(-\left|z1\right|\right) \cdot z0}{z2 \cdot t\_1}\\ \mathbf{if}\;t\_0 \leq 3.7 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{t\_2 - z5} \cdot \sqrt{t\_2 + z5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t\_3 + z5\right) \cdot \left(t\_3 - z5\right)}\\ \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0 (fmin (fabs z4) (fabs z3)))
       (t_1 (* (fmax (fabs z4) (fabs z3)) z2))
       (t_2 (* (/ z0 z2) (* (/ (fabs z1) (* t_1 t_0)) z0)))
       (t_3 (* (/ z0 t_0) (/ (* (- (fabs z1)) z0) (* z2 t_1)))))
  (if (<= t_0 3.7e+124)
    (* (sqrt (- t_2 z5)) (sqrt (+ t_2 z5)))
    (sqrt (* (+ t_3 z5) (- t_3 z5))))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmin(fabs(z4), fabs(z3));
	double t_1 = fmax(fabs(z4), fabs(z3)) * z2;
	double t_2 = (z0 / z2) * ((fabs(z1) / (t_1 * t_0)) * z0);
	double t_3 = (z0 / t_0) * ((-fabs(z1) * z0) / (z2 * t_1));
	double tmp;
	if (t_0 <= 3.7e+124) {
		tmp = sqrt((t_2 - z5)) * sqrt((t_2 + z5));
	} else {
		tmp = sqrt(((t_3 + z5) * (t_3 - z5)));
	}
	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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(abs(z4), abs(z3))
    t_1 = fmax(abs(z4), abs(z3)) * z2
    t_2 = (z0 / z2) * ((abs(z1) / (t_1 * t_0)) * z0)
    t_3 = (z0 / t_0) * ((-abs(z1) * z0) / (z2 * t_1))
    if (t_0 <= 3.7d+124) then
        tmp = sqrt((t_2 - z5)) * sqrt((t_2 + z5))
    else
        tmp = sqrt(((t_3 + z5) * (t_3 - z5)))
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmin(Math.abs(z4), Math.abs(z3));
	double t_1 = fmax(Math.abs(z4), Math.abs(z3)) * z2;
	double t_2 = (z0 / z2) * ((Math.abs(z1) / (t_1 * t_0)) * z0);
	double t_3 = (z0 / t_0) * ((-Math.abs(z1) * z0) / (z2 * t_1));
	double tmp;
	if (t_0 <= 3.7e+124) {
		tmp = Math.sqrt((t_2 - z5)) * Math.sqrt((t_2 + z5));
	} else {
		tmp = Math.sqrt(((t_3 + z5) * (t_3 - z5)));
	}
	return tmp;
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = fmin(math.fabs(z4), math.fabs(z3))
	t_1 = fmax(math.fabs(z4), math.fabs(z3)) * z2
	t_2 = (z0 / z2) * ((math.fabs(z1) / (t_1 * t_0)) * z0)
	t_3 = (z0 / t_0) * ((-math.fabs(z1) * z0) / (z2 * t_1))
	tmp = 0
	if t_0 <= 3.7e+124:
		tmp = math.sqrt((t_2 - z5)) * math.sqrt((t_2 + z5))
	else:
		tmp = math.sqrt(((t_3 + z5) * (t_3 - z5)))
	return tmp

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = fmin(abs(z4), abs(z3))
	t_1 = Float64(fmax(abs(z4), abs(z3)) * z2)
	t_2 = Float64(Float64(z0 / z2) * Float64(Float64(abs(z1) / Float64(t_1 * t_0)) * z0))
	t_3 = Float64(Float64(z0 / t_0) * Float64(Float64(Float64(-abs(z1)) * z0) / Float64(z2 * t_1)))
	tmp = 0.0
	if (t_0 <= 3.7e+124)
		tmp = Float64(sqrt(Float64(t_2 - z5)) * sqrt(Float64(t_2 + z5)));
	else
		tmp = sqrt(Float64(Float64(t_3 + z5) * Float64(t_3 - z5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z4, z3, z2, z5)
	t_0 = min(abs(z4), abs(z3));
	t_1 = max(abs(z4), abs(z3)) * z2;
	t_2 = (z0 / z2) * ((abs(z1) / (t_1 * t_0)) * z0);
	t_3 = (z0 / t_0) * ((-abs(z1) * z0) / (z2 * t_1));
	tmp = 0.0;
	if (t_0 <= 3.7e+124)
		tmp = sqrt((t_2 - z5)) * sqrt((t_2 + z5));
	else
		tmp = sqrt(((t_3 + z5) * (t_3 - z5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[Min[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Max[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision] * z2), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z0 / z2), $MachinePrecision] * N[(N[(N[Abs[z1], $MachinePrecision] / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z0 / t$95$0), $MachinePrecision] * N[(N[((-N[Abs[z1], $MachinePrecision]) * z0), $MachinePrecision] / N[(z2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 3.7e+124], N[(N[Sqrt[N[(t$95$2 - z5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$2 + z5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(t$95$3 + z5), $MachinePrecision] * N[(t$95$3 - z5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if z4 < 3.7000000000000001e124

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      13. lower-*.f64 34.0%

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      16. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      19. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      22. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

   5. Applied rewrites 34.0%

      \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      13. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      16. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      19. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

      22. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

   7. Applied rewrites 40.7%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      3. associate-*l* N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z4 \cdot \left(z3 \cdot z2\right)}} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{z4 \cdot \color{blue}{\left(z3 \cdot z2\right)}} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z2\right) \cdot z4}} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      6. lower-*.f64 39.7%

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z2\right) \cdot z4}} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

   9. Applied rewrites 39.7%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z2\right) \cdot z4}} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z3 \cdot z2\right) \cdot z4} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) + z5} \]

       2. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z3 \cdot z2\right) \cdot z4} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

       3. associate-*l* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z3 \cdot z2\right) \cdot z4} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z4 \cdot \left(z3 \cdot z2\right)}} \cdot z0\right) + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z3 \cdot z2\right) \cdot z4} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{z4 \cdot \color{blue}{\left(z3 \cdot z2\right)}} \cdot z0\right) + z5} \]

       5. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z3 \cdot z2\right) \cdot z4} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z2\right) \cdot z4}} \cdot z0\right) + z5} \]

       6. lower-*.f64 41.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z3 \cdot z2\right) \cdot z4} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z2\right) \cdot z4}} \cdot z0\right) + z5} \]

   11. Applied rewrites 41.4%

       \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z3 \cdot z2\right) \cdot z4} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z2\right) \cdot z4}} \cdot z0\right) + z5} \]

   if 3.7000000000000001e124 < z4

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. sub-flip N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} + \left(\mathsf{neg}\left(z5 \cdot z5\right)\right)}} \]

      3. add-flip N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      7. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      9. remove-double-neg N/A

         \[\leadsto \sqrt{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - \color{blue}{z5 \cdot z5}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - \color{blue}{z5 \cdot z5}} \]

      11. difference-of-squares N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) + z5\right) \cdot \left(\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - z5\right)}} \]

   3. Applied rewrites 67.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{z0 \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\color{blue}{z0 \cdot z1}}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{\frac{z0 \cdot z1}{z2}}{z2 \cdot \left(z3 \cdot z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      8. frac-2neg N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\color{blue}{-\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\color{blue}{\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z0 \cdot z1}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      17. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{\left(z2 \cdot z3\right) \cdot z4}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      18. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      20. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      21. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      22. lower-neg.f64 68.2%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

   5. Applied rewrites 68.2%

      \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)} - z5\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) - z5\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{z0 \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\color{blue}{z0 \cdot z1}}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5\right)} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{\frac{z0 \cdot z1}{z2}}{z2 \cdot \left(z3 \cdot z4\right)}} - z5\right)} \]

      8. frac-2neg N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\color{blue}{-\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\color{blue}{\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z0 \cdot z1}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right)} - z5\right)} \]

      17. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{\left(z2 \cdot z3\right) \cdot z4}\right)} - z5\right)} \]

      18. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} - z5\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} - z5\right)} \]

      20. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} - z5\right)} \]

      21. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} - z5\right)} \]

      22. lower-neg.f64 75.0%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(-z4\right)}} - z5\right)} \]

   7. Applied rewrites 75.0%

      \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} - z5\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{\left(-z0\right) \cdot \left(-\frac{z1 \cdot z0}{z2}\right)}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{\left(-z0\right) \cdot \left(-\frac{z1 \cdot z0}{z2}\right)}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\frac{\left(-z0\right) \cdot \left(-\frac{z1 \cdot z0}{z2}\right)}{\color{blue}{\left(-z4\right) \cdot \left(z3 \cdot z2\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{-z0}{-z4} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{-z0}{-z4} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      8. lift-neg.f64 N/A

         \[\leadsto \sqrt{\left(\frac{\color{blue}{\mathsf{neg}\left(z0\right)}}{-z4} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      9. lift-neg.f64 N/A

         \[\leadsto \sqrt{\left(\frac{\mathsf{neg}\left(z0\right)}{\color{blue}{\mathsf{neg}\left(z4\right)}} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      10. frac-2neg-rev N/A

          \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z4}} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z4}} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      12. lift-neg.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{z1 \cdot z0}{z2}\right)}}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\mathsf{neg}\left(\color{blue}{\frac{z1 \cdot z0}{z2}}\right)}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      14. distribute-neg-frac N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\color{blue}{\frac{\mathsf{neg}\left(z1 \cdot z0\right)}{z2}}}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      15. associate-/l/ N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \color{blue}{\frac{\mathsf{neg}\left(z1 \cdot z0\right)}{z2 \cdot \left(z3 \cdot z2\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      16. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \color{blue}{\frac{\mathsf{neg}\left(z1 \cdot z0\right)}{z2 \cdot \left(z3 \cdot z2\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\mathsf{neg}\left(\color{blue}{z1 \cdot z0}\right)}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      18. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\color{blue}{\left(\mathsf{neg}\left(z1\right)\right) \cdot z0}}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\color{blue}{\left(\mathsf{neg}\left(z1\right)\right) \cdot z0}}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      20. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\color{blue}{\left(-z1\right)} \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      21. lower-*.f64 69.7%

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{\color{blue}{z2 \cdot \left(z3 \cdot z2\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

   9. Applied rewrites 69.7%

      \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} - z5\right)} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} - z5\right)} \]

       3. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\frac{\left(-z0\right) \cdot \left(-\frac{z1 \cdot z0}{z2}\right)}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} - z5\right)} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{\left(-z0\right) \cdot \left(-\frac{z1 \cdot z0}{z2}\right)}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} - z5\right)} \]

       5. *-commutative N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{\left(-z0\right) \cdot \left(-\frac{z1 \cdot z0}{z2}\right)}{\color{blue}{\left(-z4\right) \cdot \left(z3 \cdot z2\right)}} - z5\right)} \]

       6. times-frac N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\frac{-z0}{-z4} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2}} - z5\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\frac{-z0}{-z4} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2}} - z5\right)} \]

       8. lift-neg.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{\color{blue}{\mathsf{neg}\left(z0\right)}}{-z4} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} - z5\right)} \]

       9. lift-neg.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{\mathsf{neg}\left(z0\right)}{\color{blue}{\mathsf{neg}\left(z4\right)}} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} - z5\right)} \]

       10. frac-2neg-rev N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\frac{z0}{z4}} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} - z5\right)} \]

       11. lower-/.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\frac{z0}{z4}} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} - z5\right)} \]

       12. lift-neg.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{z1 \cdot z0}{z2}\right)}}{z3 \cdot z2} - z5\right)} \]

       13. lift-/.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\mathsf{neg}\left(\color{blue}{\frac{z1 \cdot z0}{z2}}\right)}{z3 \cdot z2} - z5\right)} \]

       14. distribute-neg-frac N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\color{blue}{\frac{\mathsf{neg}\left(z1 \cdot z0\right)}{z2}}}{z3 \cdot z2} - z5\right)} \]

       15. associate-/l/ N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \color{blue}{\frac{\mathsf{neg}\left(z1 \cdot z0\right)}{z2 \cdot \left(z3 \cdot z2\right)}} - z5\right)} \]

       16. lower-/.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \color{blue}{\frac{\mathsf{neg}\left(z1 \cdot z0\right)}{z2 \cdot \left(z3 \cdot z2\right)}} - z5\right)} \]

       17. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\mathsf{neg}\left(\color{blue}{z1 \cdot z0}\right)}{z2 \cdot \left(z3 \cdot z2\right)} - z5\right)} \]

       18. distribute-lft-neg-in N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\color{blue}{\left(\mathsf{neg}\left(z1\right)\right) \cdot z0}}{z2 \cdot \left(z3 \cdot z2\right)} - z5\right)} \]

       19. lower-*.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\color{blue}{\left(\mathsf{neg}\left(z1\right)\right) \cdot z0}}{z2 \cdot \left(z3 \cdot z2\right)} - z5\right)} \]

       20. lower-neg.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\color{blue}{\left(-z1\right)} \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} - z5\right)} \]

       21. lower-*.f64 71.6%

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{\color{blue}{z2 \cdot \left(z3 \cdot z2\right)}} - z5\right)} \]

   11. Applied rewrites 71.6%

       \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)}} - z5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{z0}{\left|z4\right|} \cdot \frac{\left(-\left|z1\right|\right) \cdot z0}{z2 \cdot \left(\left|z3\right| \cdot z2\right)}\\ t_1 := \sqrt{\left(t\_0 + z5\right) \cdot \left(t\_0 - z5\right)}\\ t_2 := \left(\left(\left|z4\right| \cdot \left|z3\right|\right) \cdot z2\right) \cdot z2\\ t_3 := \left(\left|z1\right| \cdot z0\right) \cdot \frac{z0}{t\_2}\\ \mathbf{if}\;t\_2 \leq 10^{-322}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+287}:\\ \;\;\;\;\sqrt{t\_3 - z5} \cdot \sqrt{t\_3 + z5}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0
        (*
         (/ z0 (fabs z4))
         (/ (* (- (fabs z1)) z0) (* z2 (* (fabs z3) z2)))))
       (t_1 (sqrt (* (+ t_0 z5) (- t_0 z5))))
       (t_2 (* (* (* (fabs z4) (fabs z3)) z2) z2))
       (t_3 (* (* (fabs z1) z0) (/ z0 t_2))))
  (if (<= t_2 1e-322)
    t_1
    (if (<= t_2 1e+287) (* (sqrt (- t_3 z5)) (sqrt (+ t_3 z5))) t_1))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = (z0 / fabs(z4)) * ((-fabs(z1) * z0) / (z2 * (fabs(z3) * z2)));
	double t_1 = sqrt(((t_0 + z5) * (t_0 - z5)));
	double t_2 = ((fabs(z4) * fabs(z3)) * z2) * z2;
	double t_3 = (fabs(z1) * z0) * (z0 / t_2);
	double tmp;
	if (t_2 <= 1e-322) {
		tmp = t_1;
	} else if (t_2 <= 1e+287) {
		tmp = sqrt((t_3 - z5)) * sqrt((t_3 + z5));
	} else {
		tmp = t_1;
	}
	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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (z0 / abs(z4)) * ((-abs(z1) * z0) / (z2 * (abs(z3) * z2)))
    t_1 = sqrt(((t_0 + z5) * (t_0 - z5)))
    t_2 = ((abs(z4) * abs(z3)) * z2) * z2
    t_3 = (abs(z1) * z0) * (z0 / t_2)
    if (t_2 <= 1d-322) then
        tmp = t_1
    else if (t_2 <= 1d+287) then
        tmp = sqrt((t_3 - z5)) * sqrt((t_3 + z5))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = (z0 / Math.abs(z4)) * ((-Math.abs(z1) * z0) / (z2 * (Math.abs(z3) * z2)));
	double t_1 = Math.sqrt(((t_0 + z5) * (t_0 - z5)));
	double t_2 = ((Math.abs(z4) * Math.abs(z3)) * z2) * z2;
	double t_3 = (Math.abs(z1) * z0) * (z0 / t_2);
	double tmp;
	if (t_2 <= 1e-322) {
		tmp = t_1;
	} else if (t_2 <= 1e+287) {
		tmp = Math.sqrt((t_3 - z5)) * Math.sqrt((t_3 + z5));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = (z0 / math.fabs(z4)) * ((-math.fabs(z1) * z0) / (z2 * (math.fabs(z3) * z2)))
	t_1 = math.sqrt(((t_0 + z5) * (t_0 - z5)))
	t_2 = ((math.fabs(z4) * math.fabs(z3)) * z2) * z2
	t_3 = (math.fabs(z1) * z0) * (z0 / t_2)
	tmp = 0
	if t_2 <= 1e-322:
		tmp = t_1
	elif t_2 <= 1e+287:
		tmp = math.sqrt((t_3 - z5)) * math.sqrt((t_3 + z5))
	else:
		tmp = t_1
	return tmp

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = Float64(Float64(z0 / abs(z4)) * Float64(Float64(Float64(-abs(z1)) * z0) / Float64(z2 * Float64(abs(z3) * z2))))
	t_1 = sqrt(Float64(Float64(t_0 + z5) * Float64(t_0 - z5)))
	t_2 = Float64(Float64(Float64(abs(z4) * abs(z3)) * z2) * z2)
	t_3 = Float64(Float64(abs(z1) * z0) * Float64(z0 / t_2))
	tmp = 0.0
	if (t_2 <= 1e-322)
		tmp = t_1;
	elseif (t_2 <= 1e+287)
		tmp = Float64(sqrt(Float64(t_3 - z5)) * sqrt(Float64(t_3 + z5)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z4, z3, z2, z5)
	t_0 = (z0 / abs(z4)) * ((-abs(z1) * z0) / (z2 * (abs(z3) * z2)));
	t_1 = sqrt(((t_0 + z5) * (t_0 - z5)));
	t_2 = ((abs(z4) * abs(z3)) * z2) * z2;
	t_3 = (abs(z1) * z0) * (z0 / t_2);
	tmp = 0.0;
	if (t_2 <= 1e-322)
		tmp = t_1;
	elseif (t_2 <= 1e+287)
		tmp = sqrt((t_3 - z5)) * sqrt((t_3 + z5));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[(N[(z0 / N[Abs[z4], $MachinePrecision]), $MachinePrecision] * N[(N[((-N[Abs[z1], $MachinePrecision]) * z0), $MachinePrecision] / N[(z2 * N[(N[Abs[z3], $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(t$95$0 + z5), $MachinePrecision] * N[(t$95$0 - z5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Abs[z4], $MachinePrecision] * N[Abs[z3], $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision] * z2), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Abs[z1], $MachinePrecision] * z0), $MachinePrecision] * N[(z0 / t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-322], t$95$1, If[LessEqual[t$95$2, 1e+287], N[(N[Sqrt[N[(t$95$3 - z5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$3 + z5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 z4 z3) z2) z2) < 9.8813129168249309e-323 or 1.0000000000000001e287 < (*.f64 (*.f64 (*.f64 z4 z3) z2) z2)

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. sub-flip N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} + \left(\mathsf{neg}\left(z5 \cdot z5\right)\right)}} \]

      3. add-flip N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      7. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      9. remove-double-neg N/A

         \[\leadsto \sqrt{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - \color{blue}{z5 \cdot z5}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - \color{blue}{z5 \cdot z5}} \]

      11. difference-of-squares N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) + z5\right) \cdot \left(\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - z5\right)}} \]

   3. Applied rewrites 67.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{z0 \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\color{blue}{z0 \cdot z1}}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{\frac{z0 \cdot z1}{z2}}{z2 \cdot \left(z3 \cdot z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      8. frac-2neg N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\color{blue}{-\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\color{blue}{\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z0 \cdot z1}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      17. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{\left(z2 \cdot z3\right) \cdot z4}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      18. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      20. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      21. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      22. lower-neg.f64 68.2%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

   5. Applied rewrites 68.2%

      \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)} - z5\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) - z5\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{z0 \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\color{blue}{z0 \cdot z1}}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5\right)} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{\frac{z0 \cdot z1}{z2}}{z2 \cdot \left(z3 \cdot z4\right)}} - z5\right)} \]

      8. frac-2neg N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\color{blue}{-\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\color{blue}{\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z0 \cdot z1}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right)} - z5\right)} \]

      17. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{\left(z2 \cdot z3\right) \cdot z4}\right)} - z5\right)} \]

      18. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} - z5\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} - z5\right)} \]

      20. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} - z5\right)} \]

      21. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} - z5\right)} \]

      22. lower-neg.f64 75.0%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(-z4\right)}} - z5\right)} \]

   7. Applied rewrites 75.0%

      \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} - z5\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{\left(-z0\right) \cdot \left(-\frac{z1 \cdot z0}{z2}\right)}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{\left(-z0\right) \cdot \left(-\frac{z1 \cdot z0}{z2}\right)}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\frac{\left(-z0\right) \cdot \left(-\frac{z1 \cdot z0}{z2}\right)}{\color{blue}{\left(-z4\right) \cdot \left(z3 \cdot z2\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{-z0}{-z4} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{-z0}{-z4} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      8. lift-neg.f64 N/A

         \[\leadsto \sqrt{\left(\frac{\color{blue}{\mathsf{neg}\left(z0\right)}}{-z4} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      9. lift-neg.f64 N/A

         \[\leadsto \sqrt{\left(\frac{\mathsf{neg}\left(z0\right)}{\color{blue}{\mathsf{neg}\left(z4\right)}} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      10. frac-2neg-rev N/A

          \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z4}} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z4}} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      12. lift-neg.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{z1 \cdot z0}{z2}\right)}}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\mathsf{neg}\left(\color{blue}{\frac{z1 \cdot z0}{z2}}\right)}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      14. distribute-neg-frac N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\color{blue}{\frac{\mathsf{neg}\left(z1 \cdot z0\right)}{z2}}}{z3 \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      15. associate-/l/ N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \color{blue}{\frac{\mathsf{neg}\left(z1 \cdot z0\right)}{z2 \cdot \left(z3 \cdot z2\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      16. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \color{blue}{\frac{\mathsf{neg}\left(z1 \cdot z0\right)}{z2 \cdot \left(z3 \cdot z2\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\mathsf{neg}\left(\color{blue}{z1 \cdot z0}\right)}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      18. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\color{blue}{\left(\mathsf{neg}\left(z1\right)\right) \cdot z0}}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\color{blue}{\left(\mathsf{neg}\left(z1\right)\right) \cdot z0}}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      20. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\color{blue}{\left(-z1\right)} \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

      21. lower-*.f64 69.7%

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{\color{blue}{z2 \cdot \left(z3 \cdot z2\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

   9. Applied rewrites 69.7%

      \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} - z5\right)} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} - z5\right)} \]

       3. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\frac{\left(-z0\right) \cdot \left(-\frac{z1 \cdot z0}{z2}\right)}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} - z5\right)} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{\left(-z0\right) \cdot \left(-\frac{z1 \cdot z0}{z2}\right)}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} - z5\right)} \]

       5. *-commutative N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{\left(-z0\right) \cdot \left(-\frac{z1 \cdot z0}{z2}\right)}{\color{blue}{\left(-z4\right) \cdot \left(z3 \cdot z2\right)}} - z5\right)} \]

       6. times-frac N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\frac{-z0}{-z4} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2}} - z5\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\frac{-z0}{-z4} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2}} - z5\right)} \]

       8. lift-neg.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{\color{blue}{\mathsf{neg}\left(z0\right)}}{-z4} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} - z5\right)} \]

       9. lift-neg.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{\mathsf{neg}\left(z0\right)}{\color{blue}{\mathsf{neg}\left(z4\right)}} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} - z5\right)} \]

       10. frac-2neg-rev N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\frac{z0}{z4}} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} - z5\right)} \]

       11. lower-/.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\frac{z0}{z4}} \cdot \frac{-\frac{z1 \cdot z0}{z2}}{z3 \cdot z2} - z5\right)} \]

       12. lift-neg.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{z1 \cdot z0}{z2}\right)}}{z3 \cdot z2} - z5\right)} \]

       13. lift-/.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\mathsf{neg}\left(\color{blue}{\frac{z1 \cdot z0}{z2}}\right)}{z3 \cdot z2} - z5\right)} \]

       14. distribute-neg-frac N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\color{blue}{\frac{\mathsf{neg}\left(z1 \cdot z0\right)}{z2}}}{z3 \cdot z2} - z5\right)} \]

       15. associate-/l/ N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \color{blue}{\frac{\mathsf{neg}\left(z1 \cdot z0\right)}{z2 \cdot \left(z3 \cdot z2\right)}} - z5\right)} \]

       16. lower-/.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \color{blue}{\frac{\mathsf{neg}\left(z1 \cdot z0\right)}{z2 \cdot \left(z3 \cdot z2\right)}} - z5\right)} \]

       17. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\mathsf{neg}\left(\color{blue}{z1 \cdot z0}\right)}{z2 \cdot \left(z3 \cdot z2\right)} - z5\right)} \]

       18. distribute-lft-neg-in N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\color{blue}{\left(\mathsf{neg}\left(z1\right)\right) \cdot z0}}{z2 \cdot \left(z3 \cdot z2\right)} - z5\right)} \]

       19. lower-*.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\color{blue}{\left(\mathsf{neg}\left(z1\right)\right) \cdot z0}}{z2 \cdot \left(z3 \cdot z2\right)} - z5\right)} \]

       20. lower-neg.f64 N/A

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\color{blue}{\left(-z1\right)} \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} - z5\right)} \]

       21. lower-*.f64 71.6%

           \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{\color{blue}{z2 \cdot \left(z3 \cdot z2\right)}} - z5\right)} \]

   11. Applied rewrites 71.6%

       \[\leadsto \sqrt{\left(\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)} + z5\right) \cdot \left(\color{blue}{\frac{z0}{z4} \cdot \frac{\left(-z1\right) \cdot z0}{z2 \cdot \left(z3 \cdot z2\right)}} - z5\right)} \]

   if 9.8813129168249309e-323 < (*.f64 (*.f64 (*.f64 z4 z3) z2) z2) < 1.0000000000000001e287

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      13. lower-*.f64 34.0%

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      16. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      19. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      22. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

   5. Applied rewrites 34.0%

      \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      13. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      16. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      19. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

      22. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

   7. Applied rewrites 40.7%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      3. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      9. lower-/.f64 39.1%

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      14. lower-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      17. lift-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

   9. Applied rewrites 40.4%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) + z5} \]

       3. associate-*l/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} + z5} \]

       5. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       6. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} + z5} \]

       7. associate-/r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       8. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       9. lower-/.f64 41.8%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       10. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} + z5} \]

       12. associate-/l* N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       13. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       14. lower-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} + z5} \]

       16. *-commutative N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

       17. lift-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

   11. Applied rewrites 43.9%

       \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2}} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       3. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       5. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       6. associate-*r/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       7. associate-/l/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       8. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       9. times-frac N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0 \cdot \left(z1 \cdot z0\right)}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       10. *-commutative N/A

           \[\leadsto \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       12. *-commutative N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       13. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z3 \cdot z4\right)} \cdot z2\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       14. *-commutative N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       16. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       17. *-commutative N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       18. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       19. associate-/l* N/A

           \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       20. lower-*.f64 N/A

           \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       21. lower-*.f64 N/A

           \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right)} \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       22. lower-/.f64 37.6%

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \color{blue}{\frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

   13. Applied rewrites 37.6%

       \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2}} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       3. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z3 \cdot z4} + z5} \]

       5. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z3 \cdot z4} + z5} \]

       6. associate-*r/ N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z3 \cdot z4} + z5} \]

       7. associate-/l/ N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

       8. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

       9. times-frac N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z0 \cdot \left(z1 \cdot z0\right)}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

       10. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

       12. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right)}} + z5} \]

       13. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z3 \cdot z4\right)} \cdot z2\right)} + z5} \]

       14. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} + z5} \]

       16. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right)}} + z5} \]

       17. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       18. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       19. associate-/l* N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       20. lower-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       21. lower-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right)} \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + z5} \]

       22. lower-/.f64 38.1%

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\left(z1 \cdot z0\right) \cdot \color{blue}{\frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

   15. Applied rewrites 38.1%

       \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 80.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|z4\right|, \left|z3\right|\right)\\ t_1 := \mathsf{max}\left(\left|z4\right|, \left|z3\right|\right)\\ t_2 := \left(t\_0 \cdot t\_1\right) \cdot z2\\ t_3 := t\_2 \cdot z2\\ t_4 := \frac{\frac{z0}{t\_1 \cdot z2} \cdot z0}{t\_0 \cdot z2}\\ t_5 := \left(-z0\right) \cdot \frac{\frac{\left|z1\right|}{t\_2} \cdot z0}{z2}\\ t_6 := \left(\left|z1\right| \cdot z0\right) \cdot \frac{z0}{t\_3}\\ \mathbf{if}\;t\_3 \leq 10^{-315}:\\ \;\;\;\;\sqrt{\left(t\_5 + z5\right) \cdot \left(t\_5 - z5\right)}\\ \mathbf{elif}\;t\_3 \leq 10^{+269}:\\ \;\;\;\;\sqrt{t\_6 - z5} \cdot \sqrt{t\_6 + z5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left|z1\right| \cdot \left|z1\right|\right) \cdot \left(t\_4 \cdot t\_4\right) - z5 \cdot z5}\\ \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0 (fmin (fabs z4) (fabs z3)))
       (t_1 (fmax (fabs z4) (fabs z3)))
       (t_2 (* (* t_0 t_1) z2))
       (t_3 (* t_2 z2))
       (t_4 (/ (* (/ z0 (* t_1 z2)) z0) (* t_0 z2)))
       (t_5 (* (- z0) (/ (* (/ (fabs z1) t_2) z0) z2)))
       (t_6 (* (* (fabs z1) z0) (/ z0 t_3))))
  (if (<= t_3 1e-315)
    (sqrt (* (+ t_5 z5) (- t_5 z5)))
    (if (<= t_3 1e+269)
      (* (sqrt (- t_6 z5)) (sqrt (+ t_6 z5)))
      (sqrt (- (* (* (fabs z1) (fabs z1)) (* t_4 t_4)) (* z5 z5)))))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmin(fabs(z4), fabs(z3));
	double t_1 = fmax(fabs(z4), fabs(z3));
	double t_2 = (t_0 * t_1) * z2;
	double t_3 = t_2 * z2;
	double t_4 = ((z0 / (t_1 * z2)) * z0) / (t_0 * z2);
	double t_5 = -z0 * (((fabs(z1) / t_2) * z0) / z2);
	double t_6 = (fabs(z1) * z0) * (z0 / t_3);
	double tmp;
	if (t_3 <= 1e-315) {
		tmp = sqrt(((t_5 + z5) * (t_5 - z5)));
	} else if (t_3 <= 1e+269) {
		tmp = sqrt((t_6 - z5)) * sqrt((t_6 + z5));
	} else {
		tmp = sqrt((((fabs(z1) * fabs(z1)) * (t_4 * t_4)) - (z5 * z5)));
	}
	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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_0 = fmin(abs(z4), abs(z3))
    t_1 = fmax(abs(z4), abs(z3))
    t_2 = (t_0 * t_1) * z2
    t_3 = t_2 * z2
    t_4 = ((z0 / (t_1 * z2)) * z0) / (t_0 * z2)
    t_5 = -z0 * (((abs(z1) / t_2) * z0) / z2)
    t_6 = (abs(z1) * z0) * (z0 / t_3)
    if (t_3 <= 1d-315) then
        tmp = sqrt(((t_5 + z5) * (t_5 - z5)))
    else if (t_3 <= 1d+269) then
        tmp = sqrt((t_6 - z5)) * sqrt((t_6 + z5))
    else
        tmp = sqrt((((abs(z1) * abs(z1)) * (t_4 * t_4)) - (z5 * z5)))
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmin(Math.abs(z4), Math.abs(z3));
	double t_1 = fmax(Math.abs(z4), Math.abs(z3));
	double t_2 = (t_0 * t_1) * z2;
	double t_3 = t_2 * z2;
	double t_4 = ((z0 / (t_1 * z2)) * z0) / (t_0 * z2);
	double t_5 = -z0 * (((Math.abs(z1) / t_2) * z0) / z2);
	double t_6 = (Math.abs(z1) * z0) * (z0 / t_3);
	double tmp;
	if (t_3 <= 1e-315) {
		tmp = Math.sqrt(((t_5 + z5) * (t_5 - z5)));
	} else if (t_3 <= 1e+269) {
		tmp = Math.sqrt((t_6 - z5)) * Math.sqrt((t_6 + z5));
	} else {
		tmp = Math.sqrt((((Math.abs(z1) * Math.abs(z1)) * (t_4 * t_4)) - (z5 * z5)));
	}
	return tmp;
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = fmin(math.fabs(z4), math.fabs(z3))
	t_1 = fmax(math.fabs(z4), math.fabs(z3))
	t_2 = (t_0 * t_1) * z2
	t_3 = t_2 * z2
	t_4 = ((z0 / (t_1 * z2)) * z0) / (t_0 * z2)
	t_5 = -z0 * (((math.fabs(z1) / t_2) * z0) / z2)
	t_6 = (math.fabs(z1) * z0) * (z0 / t_3)
	tmp = 0
	if t_3 <= 1e-315:
		tmp = math.sqrt(((t_5 + z5) * (t_5 - z5)))
	elif t_3 <= 1e+269:
		tmp = math.sqrt((t_6 - z5)) * math.sqrt((t_6 + z5))
	else:
		tmp = math.sqrt((((math.fabs(z1) * math.fabs(z1)) * (t_4 * t_4)) - (z5 * z5)))
	return tmp

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = fmin(abs(z4), abs(z3))
	t_1 = fmax(abs(z4), abs(z3))
	t_2 = Float64(Float64(t_0 * t_1) * z2)
	t_3 = Float64(t_2 * z2)
	t_4 = Float64(Float64(Float64(z0 / Float64(t_1 * z2)) * z0) / Float64(t_0 * z2))
	t_5 = Float64(Float64(-z0) * Float64(Float64(Float64(abs(z1) / t_2) * z0) / z2))
	t_6 = Float64(Float64(abs(z1) * z0) * Float64(z0 / t_3))
	tmp = 0.0
	if (t_3 <= 1e-315)
		tmp = sqrt(Float64(Float64(t_5 + z5) * Float64(t_5 - z5)));
	elseif (t_3 <= 1e+269)
		tmp = Float64(sqrt(Float64(t_6 - z5)) * sqrt(Float64(t_6 + z5)));
	else
		tmp = sqrt(Float64(Float64(Float64(abs(z1) * abs(z1)) * Float64(t_4 * t_4)) - Float64(z5 * z5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z4, z3, z2, z5)
	t_0 = min(abs(z4), abs(z3));
	t_1 = max(abs(z4), abs(z3));
	t_2 = (t_0 * t_1) * z2;
	t_3 = t_2 * z2;
	t_4 = ((z0 / (t_1 * z2)) * z0) / (t_0 * z2);
	t_5 = -z0 * (((abs(z1) / t_2) * z0) / z2);
	t_6 = (abs(z1) * z0) * (z0 / t_3);
	tmp = 0.0;
	if (t_3 <= 1e-315)
		tmp = sqrt(((t_5 + z5) * (t_5 - z5)));
	elseif (t_3 <= 1e+269)
		tmp = sqrt((t_6 - z5)) * sqrt((t_6 + z5));
	else
		tmp = sqrt((((abs(z1) * abs(z1)) * (t_4 * t_4)) - (z5 * z5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[Min[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * t$95$1), $MachinePrecision] * z2), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * z2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(z0 / N[(t$95$1 * z2), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / N[(t$95$0 * z2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[((-z0) * N[(N[(N[(N[Abs[z1], $MachinePrecision] / t$95$2), $MachinePrecision] * z0), $MachinePrecision] / z2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[Abs[z1], $MachinePrecision] * z0), $MachinePrecision] * N[(z0 / t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 1e-315], N[Sqrt[N[(N[(t$95$5 + z5), $MachinePrecision] * N[(t$95$5 - z5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+269], N[(N[Sqrt[N[(t$95$6 - z5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$6 + z5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(N[Abs[z1], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision] * N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision] - N[(z5 * z5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 z4 z3) z2) z2) < 9.9999999848168381e-316

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. sub-flip N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} + \left(\mathsf{neg}\left(z5 \cdot z5\right)\right)}} \]

      3. add-flip N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      7. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      9. remove-double-neg N/A

         \[\leadsto \sqrt{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - \color{blue}{z5 \cdot z5}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - \color{blue}{z5 \cdot z5}} \]

      11. difference-of-squares N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) + z5\right) \cdot \left(\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - z5\right)}} \]

   3. Applied rewrites 67.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      4. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \color{blue}{\frac{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}}{z2}}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{\color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}}}{z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}}{z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\color{blue}{z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}}}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      8. lower-/.f64 67.8%

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}}{z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\color{blue}{z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}}}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0}}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      11. lower-*.f64 67.8%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0}}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      14. lower-*.f64 67.8%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      17. lower-*.f64 67.8%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

   5. Applied rewrites 67.8%

      \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)} - z5\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) - z5\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) - z5\right)} \]

      4. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \color{blue}{\frac{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}}{z2}}\right) - z5\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{\color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}}}{z2}\right) - z5\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}}{z2}} - z5\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\color{blue}{z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}}}{z2} - z5\right)} \]

      8. lower-/.f64 72.2%

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}}{z2}} - z5\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\color{blue}{z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}}}{z2} - z5\right)} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0}}{z2} - z5\right)} \]

      11. lower-*.f64 72.2%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0}}{z2} - z5\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0}{z2} - z5\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0}{z2} - z5\right)} \]

      14. lower-*.f64 72.2%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0}{z2} - z5\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0}{z2} - z5\right)} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0}{z2} - z5\right)} \]

      17. lower-*.f64 72.2%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0}{z2} - z5\right)} \]

   7. Applied rewrites 72.2%

      \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0}{z2}} - z5\right)} \]

   if 9.9999999848168381e-316 < (*.f64 (*.f64 (*.f64 z4 z3) z2) z2) < 1e269

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      13. lower-*.f64 34.0%

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      16. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      19. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      22. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

   5. Applied rewrites 34.0%

      \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      13. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      16. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      19. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

      22. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

   7. Applied rewrites 40.7%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      3. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      9. lower-/.f64 39.1%

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      14. lower-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      17. lift-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

   9. Applied rewrites 40.4%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) + z5} \]

       3. associate-*l/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} + z5} \]

       5. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       6. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} + z5} \]

       7. associate-/r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       8. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       9. lower-/.f64 41.8%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       10. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} + z5} \]

       12. associate-/l* N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       13. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       14. lower-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} + z5} \]

       16. *-commutative N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

       17. lift-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

   11. Applied rewrites 43.9%

       \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2}} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       3. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       5. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       6. associate-*r/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       7. associate-/l/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       8. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       9. times-frac N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0 \cdot \left(z1 \cdot z0\right)}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       10. *-commutative N/A

           \[\leadsto \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       12. *-commutative N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       13. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z3 \cdot z4\right)} \cdot z2\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       14. *-commutative N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       16. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       17. *-commutative N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       18. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       19. associate-/l* N/A

           \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       20. lower-*.f64 N/A

           \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       21. lower-*.f64 N/A

           \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right)} \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       22. lower-/.f64 37.6%

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \color{blue}{\frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

   13. Applied rewrites 37.6%

       \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2}} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       3. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z3 \cdot z4} + z5} \]

       5. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z3 \cdot z4} + z5} \]

       6. associate-*r/ N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z3 \cdot z4} + z5} \]

       7. associate-/l/ N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

       8. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

       9. times-frac N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z0 \cdot \left(z1 \cdot z0\right)}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

       10. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

       12. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right)}} + z5} \]

       13. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z3 \cdot z4\right)} \cdot z2\right)} + z5} \]

       14. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} + z5} \]

       16. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right)}} + z5} \]

       17. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       18. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       19. associate-/l* N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       20. lower-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       21. lower-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right)} \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + z5} \]

       22. lower-/.f64 38.1%

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\left(z1 \cdot z0\right) \cdot \color{blue}{\frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

   15. Applied rewrites 38.1%

       \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

   if 1e269 < (*.f64 (*.f64 (*.f64 z4 z3) z2) z2)

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right)} \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z1 \cdot \left(z0 \cdot z0\right)}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      8. associate-/l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \frac{\color{blue}{\left(z1 \cdot z0\right)} \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \frac{\color{blue}{z1 \cdot \left(z0 \cdot z0\right)}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \color{blue}{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} - z5 \cdot z5} \]

      13. swap-sqr N/A

          \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z1\right) \cdot \left(\frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} - z5 \cdot z5} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z1\right) \cdot \left(\frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} - z5 \cdot z5} \]

   3. Applied rewrites 42.5%

      \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right)} - z5 \cdot z5} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)} \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right) \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)} \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(z4 \cdot z2\right)}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(z4 \cdot z2\right)}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      9. lower-*.f64 41.8%

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(z4 \cdot z2\right)}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

   5. Applied rewrites 41.8%

      \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right)\right) - z5 \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)} \cdot z2}\right)\right) - z5 \cdot z5} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)} \cdot z2}\right)\right) - z5 \cdot z5} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(z4 \cdot z2\right)}}\right)\right) - z5 \cdot z5} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(z4 \cdot z2\right)}}\right)\right) - z5 \cdot z5} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      9. lower-*.f64 44.2%

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(z4 \cdot z2\right)}}\right)\right) - z5 \cdot z5} \]

   7. Applied rewrites 44.2%

      \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}}\right)\right) - z5 \cdot z5} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\color{blue}{\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)} \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\color{blue}{\left(\frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)} \cdot z0\right)} \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(\color{blue}{\frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}} \cdot z0\right) \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(\frac{z0}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}} \cdot z0\right) \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      5. associate-/r* N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(\color{blue}{\frac{\frac{z0}{z3 \cdot z2}}{z4 \cdot z2}} \cdot z0\right) \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      6. associate-*l/ N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\color{blue}{\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2}} \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\color{blue}{\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2}} \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\color{blue}{\frac{z0}{z3 \cdot z2} \cdot z0}}{z4 \cdot z2} \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      9. lower-/.f64 43.7%

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\color{blue}{\frac{z0}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

   9. Applied rewrites 43.7%

      \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\color{blue}{\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2}} \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \color{blue}{\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)}\right) - z5 \cdot z5} \]

       2. *-commutative N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \color{blue}{\left(\frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)} \cdot z0\right)}\right) - z5 \cdot z5} \]

       3. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \left(\color{blue}{\frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}} \cdot z0\right)\right) - z5 \cdot z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \left(\frac{z0}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}} \cdot z0\right)\right) - z5 \cdot z5} \]

       5. associate-/r* N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \left(\color{blue}{\frac{\frac{z0}{z3 \cdot z2}}{z4 \cdot z2}} \cdot z0\right)\right) - z5 \cdot z5} \]

       6. associate-*l/ N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \color{blue}{\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2}}\right) - z5 \cdot z5} \]

       7. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \color{blue}{\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2}}\right) - z5 \cdot z5} \]

       8. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \frac{\color{blue}{\frac{z0}{z3 \cdot z2} \cdot z0}}{z4 \cdot z2}\right) - z5 \cdot z5} \]

       9. lower-/.f64 45.2%

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \frac{\color{blue}{\frac{z0}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2}\right) - z5 \cdot z5} \]

   11. Applied rewrites 45.2%

       \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \color{blue}{\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2}}\right) - z5 \cdot z5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 78.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|z4\right|, \left|z3\right|\right)\\ t_1 := \mathsf{min}\left(\left|z4\right|, \left|z3\right|\right)\\ t_2 := \frac{\frac{z0}{t\_0 \cdot z2} \cdot z0}{t\_1 \cdot z2}\\ t_3 := \left(\left(t\_1 \cdot t\_0\right) \cdot z2\right) \cdot z2\\ t_4 := \left(\left|z1\right| \cdot z0\right) \cdot \frac{z0}{t\_3}\\ \mathbf{if}\;t\_3 \leq 10^{+269}:\\ \;\;\;\;\sqrt{t\_4 - z5} \cdot \sqrt{t\_4 + z5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left|z1\right| \cdot \left|z1\right|\right) \cdot \left(t\_2 \cdot t\_2\right) - z5 \cdot z5}\\ \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0 (fmax (fabs z4) (fabs z3)))
       (t_1 (fmin (fabs z4) (fabs z3)))
       (t_2 (/ (* (/ z0 (* t_0 z2)) z0) (* t_1 z2)))
       (t_3 (* (* (* t_1 t_0) z2) z2))
       (t_4 (* (* (fabs z1) z0) (/ z0 t_3))))
  (if (<= t_3 1e+269)
    (* (sqrt (- t_4 z5)) (sqrt (+ t_4 z5)))
    (sqrt (- (* (* (fabs z1) (fabs z1)) (* t_2 t_2)) (* z5 z5))))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmax(fabs(z4), fabs(z3));
	double t_1 = fmin(fabs(z4), fabs(z3));
	double t_2 = ((z0 / (t_0 * z2)) * z0) / (t_1 * z2);
	double t_3 = ((t_1 * t_0) * z2) * z2;
	double t_4 = (fabs(z1) * z0) * (z0 / t_3);
	double tmp;
	if (t_3 <= 1e+269) {
		tmp = sqrt((t_4 - z5)) * sqrt((t_4 + z5));
	} else {
		tmp = sqrt((((fabs(z1) * fabs(z1)) * (t_2 * t_2)) - (z5 * z5)));
	}
	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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = fmax(abs(z4), abs(z3))
    t_1 = fmin(abs(z4), abs(z3))
    t_2 = ((z0 / (t_0 * z2)) * z0) / (t_1 * z2)
    t_3 = ((t_1 * t_0) * z2) * z2
    t_4 = (abs(z1) * z0) * (z0 / t_3)
    if (t_3 <= 1d+269) then
        tmp = sqrt((t_4 - z5)) * sqrt((t_4 + z5))
    else
        tmp = sqrt((((abs(z1) * abs(z1)) * (t_2 * t_2)) - (z5 * z5)))
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmax(Math.abs(z4), Math.abs(z3));
	double t_1 = fmin(Math.abs(z4), Math.abs(z3));
	double t_2 = ((z0 / (t_0 * z2)) * z0) / (t_1 * z2);
	double t_3 = ((t_1 * t_0) * z2) * z2;
	double t_4 = (Math.abs(z1) * z0) * (z0 / t_3);
	double tmp;
	if (t_3 <= 1e+269) {
		tmp = Math.sqrt((t_4 - z5)) * Math.sqrt((t_4 + z5));
	} else {
		tmp = Math.sqrt((((Math.abs(z1) * Math.abs(z1)) * (t_2 * t_2)) - (z5 * z5)));
	}
	return tmp;
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = fmax(math.fabs(z4), math.fabs(z3))
	t_1 = fmin(math.fabs(z4), math.fabs(z3))
	t_2 = ((z0 / (t_0 * z2)) * z0) / (t_1 * z2)
	t_3 = ((t_1 * t_0) * z2) * z2
	t_4 = (math.fabs(z1) * z0) * (z0 / t_3)
	tmp = 0
	if t_3 <= 1e+269:
		tmp = math.sqrt((t_4 - z5)) * math.sqrt((t_4 + z5))
	else:
		tmp = math.sqrt((((math.fabs(z1) * math.fabs(z1)) * (t_2 * t_2)) - (z5 * z5)))
	return tmp

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = fmax(abs(z4), abs(z3))
	t_1 = fmin(abs(z4), abs(z3))
	t_2 = Float64(Float64(Float64(z0 / Float64(t_0 * z2)) * z0) / Float64(t_1 * z2))
	t_3 = Float64(Float64(Float64(t_1 * t_0) * z2) * z2)
	t_4 = Float64(Float64(abs(z1) * z0) * Float64(z0 / t_3))
	tmp = 0.0
	if (t_3 <= 1e+269)
		tmp = Float64(sqrt(Float64(t_4 - z5)) * sqrt(Float64(t_4 + z5)));
	else
		tmp = sqrt(Float64(Float64(Float64(abs(z1) * abs(z1)) * Float64(t_2 * t_2)) - Float64(z5 * z5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z4, z3, z2, z5)
	t_0 = max(abs(z4), abs(z3));
	t_1 = min(abs(z4), abs(z3));
	t_2 = ((z0 / (t_0 * z2)) * z0) / (t_1 * z2);
	t_3 = ((t_1 * t_0) * z2) * z2;
	t_4 = (abs(z1) * z0) * (z0 / t_3);
	tmp = 0.0;
	if (t_3 <= 1e+269)
		tmp = sqrt((t_4 - z5)) * sqrt((t_4 + z5));
	else
		tmp = sqrt((((abs(z1) * abs(z1)) * (t_2 * t_2)) - (z5 * z5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[Max[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z0 / N[(t$95$0 * z2), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / N[(t$95$1 * z2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$1 * t$95$0), $MachinePrecision] * z2), $MachinePrecision] * z2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Abs[z1], $MachinePrecision] * z0), $MachinePrecision] * N[(z0 / t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 1e+269], N[(N[Sqrt[N[(t$95$4 - z5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$4 + z5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(N[Abs[z1], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(z5 * z5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 z4 z3) z2) z2) < 1e269

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      13. lower-*.f64 34.0%

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      16. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      19. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      22. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

   5. Applied rewrites 34.0%

      \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      13. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      16. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      19. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

      22. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

   7. Applied rewrites 40.7%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      3. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      9. lower-/.f64 39.1%

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      14. lower-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      17. lift-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

   9. Applied rewrites 40.4%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) + z5} \]

       3. associate-*l/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} + z5} \]

       5. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       6. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} + z5} \]

       7. associate-/r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       8. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       9. lower-/.f64 41.8%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       10. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} + z5} \]

       12. associate-/l* N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       13. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       14. lower-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} + z5} \]

       16. *-commutative N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

       17. lift-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

   11. Applied rewrites 43.9%

       \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2}} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       3. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       5. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       6. associate-*r/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       7. associate-/l/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       8. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       9. times-frac N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0 \cdot \left(z1 \cdot z0\right)}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       10. *-commutative N/A

           \[\leadsto \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       12. *-commutative N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       13. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z3 \cdot z4\right)} \cdot z2\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       14. *-commutative N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       16. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       17. *-commutative N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       18. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       19. associate-/l* N/A

           \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       20. lower-*.f64 N/A

           \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       21. lower-*.f64 N/A

           \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right)} \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       22. lower-/.f64 37.6%

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \color{blue}{\frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

   13. Applied rewrites 37.6%

       \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2}} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       3. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z3 \cdot z4} + z5} \]

       5. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z3 \cdot z4} + z5} \]

       6. associate-*r/ N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z3 \cdot z4} + z5} \]

       7. associate-/l/ N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

       8. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

       9. times-frac N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z0 \cdot \left(z1 \cdot z0\right)}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

       10. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

       12. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right)}} + z5} \]

       13. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z3 \cdot z4\right)} \cdot z2\right)} + z5} \]

       14. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} + z5} \]

       16. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right)}} + z5} \]

       17. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       18. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       19. associate-/l* N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       20. lower-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       21. lower-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right)} \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + z5} \]

       22. lower-/.f64 38.1%

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\left(z1 \cdot z0\right) \cdot \color{blue}{\frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

   15. Applied rewrites 38.1%

       \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

   if 1e269 < (*.f64 (*.f64 (*.f64 z4 z3) z2) z2)

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right)} \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z1 \cdot \left(z0 \cdot z0\right)}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      8. associate-/l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \frac{\color{blue}{\left(z1 \cdot z0\right)} \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \frac{\color{blue}{z1 \cdot \left(z0 \cdot z0\right)}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \color{blue}{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} - z5 \cdot z5} \]

      13. swap-sqr N/A

          \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z1\right) \cdot \left(\frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} - z5 \cdot z5} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z1\right) \cdot \left(\frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} - z5 \cdot z5} \]

   3. Applied rewrites 42.5%

      \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right)} - z5 \cdot z5} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)} \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right) \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)} \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(z4 \cdot z2\right)}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(z4 \cdot z2\right)}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      9. lower-*.f64 41.8%

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(z4 \cdot z2\right)}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

   5. Applied rewrites 41.8%

      \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right)\right) - z5 \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)} \cdot z2}\right)\right) - z5 \cdot z5} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)} \cdot z2}\right)\right) - z5 \cdot z5} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(z4 \cdot z2\right)}}\right)\right) - z5 \cdot z5} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(z4 \cdot z2\right)}}\right)\right) - z5 \cdot z5} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      9. lower-*.f64 44.2%

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(z4 \cdot z2\right)}}\right)\right) - z5 \cdot z5} \]

   7. Applied rewrites 44.2%

      \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}}\right)\right) - z5 \cdot z5} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\color{blue}{\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)} \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\color{blue}{\left(\frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)} \cdot z0\right)} \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(\color{blue}{\frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}} \cdot z0\right) \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(\frac{z0}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}} \cdot z0\right) \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      5. associate-/r* N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(\color{blue}{\frac{\frac{z0}{z3 \cdot z2}}{z4 \cdot z2}} \cdot z0\right) \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      6. associate-*l/ N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\color{blue}{\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2}} \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\color{blue}{\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2}} \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\color{blue}{\frac{z0}{z3 \cdot z2} \cdot z0}}{z4 \cdot z2} \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      9. lower-/.f64 43.7%

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\color{blue}{\frac{z0}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2} \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

   9. Applied rewrites 43.7%

      \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\color{blue}{\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2}} \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \color{blue}{\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)}\right) - z5 \cdot z5} \]

       2. *-commutative N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \color{blue}{\left(\frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)} \cdot z0\right)}\right) - z5 \cdot z5} \]

       3. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \left(\color{blue}{\frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}} \cdot z0\right)\right) - z5 \cdot z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \left(\frac{z0}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}} \cdot z0\right)\right) - z5 \cdot z5} \]

       5. associate-/r* N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \left(\color{blue}{\frac{\frac{z0}{z3 \cdot z2}}{z4 \cdot z2}} \cdot z0\right)\right) - z5 \cdot z5} \]

       6. associate-*l/ N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \color{blue}{\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2}}\right) - z5 \cdot z5} \]

       7. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \color{blue}{\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2}}\right) - z5 \cdot z5} \]

       8. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \frac{\color{blue}{\frac{z0}{z3 \cdot z2} \cdot z0}}{z4 \cdot z2}\right) - z5 \cdot z5} \]

       9. lower-/.f64 45.2%

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \frac{\color{blue}{\frac{z0}{z3 \cdot z2}} \cdot z0}{z4 \cdot z2}\right) - z5 \cdot z5} \]

   11. Applied rewrites 45.2%

       \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2} \cdot \color{blue}{\frac{\frac{z0}{z3 \cdot z2} \cdot z0}{z4 \cdot z2}}\right) - z5 \cdot z5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 78.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := z0 \cdot \frac{\frac{z0}{\left|z4\right| \cdot z2}}{\left|z3\right| \cdot z2}\\ t_1 := \left(\left(\left|z4\right| \cdot \left|z3\right|\right) \cdot z2\right) \cdot z2\\ t_2 := \left(\left|z1\right| \cdot z0\right) \cdot \frac{z0}{t\_1}\\ \mathbf{if}\;t\_1 \leq 10^{+283}:\\ \;\;\;\;\sqrt{t\_2 - z5} \cdot \sqrt{t\_2 + z5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left|z1\right| \cdot \left|z1\right|\right) \cdot \left(t\_0 \cdot t\_0\right) - z5 \cdot z5}\\ \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0 (* z0 (/ (/ z0 (* (fabs z4) z2)) (* (fabs z3) z2))))
       (t_1 (* (* (* (fabs z4) (fabs z3)) z2) z2))
       (t_2 (* (* (fabs z1) z0) (/ z0 t_1))))
  (if (<= t_1 1e+283)
    (* (sqrt (- t_2 z5)) (sqrt (+ t_2 z5)))
    (sqrt (- (* (* (fabs z1) (fabs z1)) (* t_0 t_0)) (* z5 z5))))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = z0 * ((z0 / (fabs(z4) * z2)) / (fabs(z3) * z2));
	double t_1 = ((fabs(z4) * fabs(z3)) * z2) * z2;
	double t_2 = (fabs(z1) * z0) * (z0 / t_1);
	double tmp;
	if (t_1 <= 1e+283) {
		tmp = sqrt((t_2 - z5)) * sqrt((t_2 + z5));
	} else {
		tmp = sqrt((((fabs(z1) * fabs(z1)) * (t_0 * t_0)) - (z5 * z5)));
	}
	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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = z0 * ((z0 / (abs(z4) * z2)) / (abs(z3) * z2))
    t_1 = ((abs(z4) * abs(z3)) * z2) * z2
    t_2 = (abs(z1) * z0) * (z0 / t_1)
    if (t_1 <= 1d+283) then
        tmp = sqrt((t_2 - z5)) * sqrt((t_2 + z5))
    else
        tmp = sqrt((((abs(z1) * abs(z1)) * (t_0 * t_0)) - (z5 * z5)))
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = z0 * ((z0 / (Math.abs(z4) * z2)) / (Math.abs(z3) * z2));
	double t_1 = ((Math.abs(z4) * Math.abs(z3)) * z2) * z2;
	double t_2 = (Math.abs(z1) * z0) * (z0 / t_1);
	double tmp;
	if (t_1 <= 1e+283) {
		tmp = Math.sqrt((t_2 - z5)) * Math.sqrt((t_2 + z5));
	} else {
		tmp = Math.sqrt((((Math.abs(z1) * Math.abs(z1)) * (t_0 * t_0)) - (z5 * z5)));
	}
	return tmp;
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = z0 * ((z0 / (math.fabs(z4) * z2)) / (math.fabs(z3) * z2))
	t_1 = ((math.fabs(z4) * math.fabs(z3)) * z2) * z2
	t_2 = (math.fabs(z1) * z0) * (z0 / t_1)
	tmp = 0
	if t_1 <= 1e+283:
		tmp = math.sqrt((t_2 - z5)) * math.sqrt((t_2 + z5))
	else:
		tmp = math.sqrt((((math.fabs(z1) * math.fabs(z1)) * (t_0 * t_0)) - (z5 * z5)))
	return tmp

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = Float64(z0 * Float64(Float64(z0 / Float64(abs(z4) * z2)) / Float64(abs(z3) * z2)))
	t_1 = Float64(Float64(Float64(abs(z4) * abs(z3)) * z2) * z2)
	t_2 = Float64(Float64(abs(z1) * z0) * Float64(z0 / t_1))
	tmp = 0.0
	if (t_1 <= 1e+283)
		tmp = Float64(sqrt(Float64(t_2 - z5)) * sqrt(Float64(t_2 + z5)));
	else
		tmp = sqrt(Float64(Float64(Float64(abs(z1) * abs(z1)) * Float64(t_0 * t_0)) - Float64(z5 * z5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z4, z3, z2, z5)
	t_0 = z0 * ((z0 / (abs(z4) * z2)) / (abs(z3) * z2));
	t_1 = ((abs(z4) * abs(z3)) * z2) * z2;
	t_2 = (abs(z1) * z0) * (z0 / t_1);
	tmp = 0.0;
	if (t_1 <= 1e+283)
		tmp = sqrt((t_2 - z5)) * sqrt((t_2 + z5));
	else
		tmp = sqrt((((abs(z1) * abs(z1)) * (t_0 * t_0)) - (z5 * z5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[(z0 * N[(N[(z0 / N[(N[Abs[z4], $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[z3], $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Abs[z4], $MachinePrecision] * N[Abs[z3], $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision] * z2), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Abs[z1], $MachinePrecision] * z0), $MachinePrecision] * N[(z0 / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+283], N[(N[Sqrt[N[(t$95$2 - z5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$2 + z5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(N[Abs[z1], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(z5 * z5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 z4 z3) z2) z2) < 9.9999999999999996e282

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      13. lower-*.f64 34.0%

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      16. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      19. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      22. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

   5. Applied rewrites 34.0%

      \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      13. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      16. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      19. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

      22. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

   7. Applied rewrites 40.7%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      3. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      9. lower-/.f64 39.1%

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      14. lower-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      17. lift-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

   9. Applied rewrites 40.4%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) + z5} \]

       3. associate-*l/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} + z5} \]

       5. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       6. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} + z5} \]

       7. associate-/r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       8. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       9. lower-/.f64 41.8%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       10. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} + z5} \]

       12. associate-/l* N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       13. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       14. lower-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} + z5} \]

       16. *-commutative N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

       17. lift-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

   11. Applied rewrites 43.9%

       \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2}} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       3. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       5. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       6. associate-*r/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       7. associate-/l/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       8. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       9. times-frac N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0 \cdot \left(z1 \cdot z0\right)}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       10. *-commutative N/A

           \[\leadsto \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       12. *-commutative N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       13. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z3 \cdot z4\right)} \cdot z2\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       14. *-commutative N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       16. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       17. *-commutative N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       18. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       19. associate-/l* N/A

           \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       20. lower-*.f64 N/A

           \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       21. lower-*.f64 N/A

           \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right)} \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       22. lower-/.f64 37.6%

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \color{blue}{\frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

   13. Applied rewrites 37.6%

       \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2}} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       3. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z3 \cdot z4} + z5} \]

       5. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z3 \cdot z4} + z5} \]

       6. associate-*r/ N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z3 \cdot z4} + z5} \]

       7. associate-/l/ N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

       8. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

       9. times-frac N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z0 \cdot \left(z1 \cdot z0\right)}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

       10. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

       12. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right)}} + z5} \]

       13. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z3 \cdot z4\right)} \cdot z2\right)} + z5} \]

       14. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} + z5} \]

       16. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right)}} + z5} \]

       17. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       18. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       19. associate-/l* N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       20. lower-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       21. lower-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right)} \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + z5} \]

       22. lower-/.f64 38.1%

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\left(z1 \cdot z0\right) \cdot \color{blue}{\frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

   15. Applied rewrites 38.1%

       \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

   if 9.9999999999999996e282 < (*.f64 (*.f64 (*.f64 z4 z3) z2) z2)

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right)} \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z1 \cdot \left(z0 \cdot z0\right)}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      8. associate-/l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \frac{\color{blue}{\left(z1 \cdot z0\right)} \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \frac{\color{blue}{z1 \cdot \left(z0 \cdot z0\right)}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \color{blue}{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} - z5 \cdot z5} \]

      13. swap-sqr N/A

          \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z1\right) \cdot \left(\frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} - z5 \cdot z5} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z1\right) \cdot \left(\frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} - z5 \cdot z5} \]

   3. Applied rewrites 42.5%

      \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right)} - z5 \cdot z5} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)} \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right) \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)} \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(z4 \cdot z2\right)}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(z4 \cdot z2\right)}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      9. lower-*.f64 41.8%

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(z4 \cdot z2\right)}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

   5. Applied rewrites 41.8%

      \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right)\right) - z5 \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)} \cdot z2}\right)\right) - z5 \cdot z5} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)} \cdot z2}\right)\right) - z5 \cdot z5} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(z4 \cdot z2\right)}}\right)\right) - z5 \cdot z5} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(z4 \cdot z2\right)}}\right)\right) - z5 \cdot z5} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      9. lower-*.f64 44.2%

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(z4 \cdot z2\right)}}\right)\right) - z5 \cdot z5} \]

   7. Applied rewrites 44.2%

      \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}}\right)\right) - z5 \cdot z5} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \color{blue}{\frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z4 \cdot z2\right) \cdot \left(z3 \cdot z2\right)}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      4. associate-/r* N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \color{blue}{\frac{\frac{z0}{z4 \cdot z2}}{z3 \cdot z2}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \color{blue}{\frac{\frac{z0}{z4 \cdot z2}}{z3 \cdot z2}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      6. lower-/.f64 44.2%

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{\color{blue}{\frac{z0}{z4 \cdot z2}}}{z3 \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

   9. Applied rewrites 44.2%

      \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \color{blue}{\frac{\frac{z0}{z4 \cdot z2}}{z3 \cdot z2}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{\frac{z0}{z4 \cdot z2}}{z3 \cdot z2}\right) \cdot \left(z0 \cdot \color{blue}{\frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}}\right)\right) - z5 \cdot z5} \]

       2. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{\frac{z0}{z4 \cdot z2}}{z3 \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}}\right)\right) - z5 \cdot z5} \]

       3. *-commutative N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{\frac{z0}{z4 \cdot z2}}{z3 \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z4 \cdot z2\right) \cdot \left(z3 \cdot z2\right)}}\right)\right) - z5 \cdot z5} \]

       4. associate-/r* N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{\frac{z0}{z4 \cdot z2}}{z3 \cdot z2}\right) \cdot \left(z0 \cdot \color{blue}{\frac{\frac{z0}{z4 \cdot z2}}{z3 \cdot z2}}\right)\right) - z5 \cdot z5} \]

       5. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{\frac{z0}{z4 \cdot z2}}{z3 \cdot z2}\right) \cdot \left(z0 \cdot \color{blue}{\frac{\frac{z0}{z4 \cdot z2}}{z3 \cdot z2}}\right)\right) - z5 \cdot z5} \]

       6. lower-/.f64 45.8%

          \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{\frac{z0}{z4 \cdot z2}}{z3 \cdot z2}\right) \cdot \left(z0 \cdot \frac{\color{blue}{\frac{z0}{z4 \cdot z2}}}{z3 \cdot z2}\right)\right) - z5 \cdot z5} \]

   11. Applied rewrites 45.8%

       \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{\frac{z0}{z4 \cdot z2}}{z3 \cdot z2}\right) \cdot \left(z0 \cdot \color{blue}{\frac{\frac{z0}{z4 \cdot z2}}{z3 \cdot z2}}\right)\right) - z5 \cdot z5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 78.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left(-z0\right) \cdot \left(z0 \cdot \frac{\left|z1\right|}{\left(z2 \cdot \left(\left|z3\right| \cdot z2\right)\right) \cdot \left|z4\right|}\right)\\ t_1 := \left(\left(\left|z4\right| \cdot \left|z3\right|\right) \cdot z2\right) \cdot z2\\ t_2 := \left(\left|z1\right| \cdot z0\right) \cdot \frac{z0}{t\_1}\\ \mathbf{if}\;t\_1 \leq 10^{+287}:\\ \;\;\;\;\sqrt{t\_2 - z5} \cdot \sqrt{t\_2 + z5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t\_0 + z5\right) \cdot \left(t\_0 - z5\right)}\\ \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0
        (*
         (- z0)
         (* z0 (/ (fabs z1) (* (* z2 (* (fabs z3) z2)) (fabs z4))))))
       (t_1 (* (* (* (fabs z4) (fabs z3)) z2) z2))
       (t_2 (* (* (fabs z1) z0) (/ z0 t_1))))
  (if (<= t_1 1e+287)
    (* (sqrt (- t_2 z5)) (sqrt (+ t_2 z5)))
    (sqrt (* (+ t_0 z5) (- t_0 z5))))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = -z0 * (z0 * (fabs(z1) / ((z2 * (fabs(z3) * z2)) * fabs(z4))));
	double t_1 = ((fabs(z4) * fabs(z3)) * z2) * z2;
	double t_2 = (fabs(z1) * z0) * (z0 / t_1);
	double tmp;
	if (t_1 <= 1e+287) {
		tmp = sqrt((t_2 - z5)) * sqrt((t_2 + z5));
	} else {
		tmp = sqrt(((t_0 + z5) * (t_0 - z5)));
	}
	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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = -z0 * (z0 * (abs(z1) / ((z2 * (abs(z3) * z2)) * abs(z4))))
    t_1 = ((abs(z4) * abs(z3)) * z2) * z2
    t_2 = (abs(z1) * z0) * (z0 / t_1)
    if (t_1 <= 1d+287) then
        tmp = sqrt((t_2 - z5)) * sqrt((t_2 + z5))
    else
        tmp = sqrt(((t_0 + z5) * (t_0 - z5)))
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = -z0 * (z0 * (Math.abs(z1) / ((z2 * (Math.abs(z3) * z2)) * Math.abs(z4))));
	double t_1 = ((Math.abs(z4) * Math.abs(z3)) * z2) * z2;
	double t_2 = (Math.abs(z1) * z0) * (z0 / t_1);
	double tmp;
	if (t_1 <= 1e+287) {
		tmp = Math.sqrt((t_2 - z5)) * Math.sqrt((t_2 + z5));
	} else {
		tmp = Math.sqrt(((t_0 + z5) * (t_0 - z5)));
	}
	return tmp;
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = -z0 * (z0 * (math.fabs(z1) / ((z2 * (math.fabs(z3) * z2)) * math.fabs(z4))))
	t_1 = ((math.fabs(z4) * math.fabs(z3)) * z2) * z2
	t_2 = (math.fabs(z1) * z0) * (z0 / t_1)
	tmp = 0
	if t_1 <= 1e+287:
		tmp = math.sqrt((t_2 - z5)) * math.sqrt((t_2 + z5))
	else:
		tmp = math.sqrt(((t_0 + z5) * (t_0 - z5)))
	return tmp

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = Float64(Float64(-z0) * Float64(z0 * Float64(abs(z1) / Float64(Float64(z2 * Float64(abs(z3) * z2)) * abs(z4)))))
	t_1 = Float64(Float64(Float64(abs(z4) * abs(z3)) * z2) * z2)
	t_2 = Float64(Float64(abs(z1) * z0) * Float64(z0 / t_1))
	tmp = 0.0
	if (t_1 <= 1e+287)
		tmp = Float64(sqrt(Float64(t_2 - z5)) * sqrt(Float64(t_2 + z5)));
	else
		tmp = sqrt(Float64(Float64(t_0 + z5) * Float64(t_0 - z5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z4, z3, z2, z5)
	t_0 = -z0 * (z0 * (abs(z1) / ((z2 * (abs(z3) * z2)) * abs(z4))));
	t_1 = ((abs(z4) * abs(z3)) * z2) * z2;
	t_2 = (abs(z1) * z0) * (z0 / t_1);
	tmp = 0.0;
	if (t_1 <= 1e+287)
		tmp = sqrt((t_2 - z5)) * sqrt((t_2 + z5));
	else
		tmp = sqrt(((t_0 + z5) * (t_0 - z5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[((-z0) * N[(z0 * N[(N[Abs[z1], $MachinePrecision] / N[(N[(z2 * N[(N[Abs[z3], $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision] * N[Abs[z4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Abs[z4], $MachinePrecision] * N[Abs[z3], $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision] * z2), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Abs[z1], $MachinePrecision] * z0), $MachinePrecision] * N[(z0 / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+287], N[(N[Sqrt[N[(t$95$2 - z5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$2 + z5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(t$95$0 + z5), $MachinePrecision] * N[(t$95$0 - z5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 z4 z3) z2) z2) < 1.0000000000000001e287

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      13. lower-*.f64 34.0%

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      16. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      19. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

      22. lower-*.f64 34.0%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]

   5. Applied rewrites 34.0%

      \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{\left(z0 \cdot z0\right) \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\left(z0 \cdot z0\right) \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0 \cdot z0}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{\color{blue}{z0 \cdot z0}}{z2} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      8. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\left(\frac{z0}{z2} \cdot z0\right)} \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\color{blue}{\frac{z0}{z2}} \cdot z0\right) \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\left(\frac{z0}{z2} \cdot z0\right) \cdot \color{blue}{\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      13. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(z0 \cdot \frac{z1}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      16. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{z2 \cdot \left(z3 \cdot z4\right)} \cdot z0\right)} + z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} \cdot z0\right) + z5} \]

      18. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      19. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right) \cdot z2}} \cdot z0\right) + z5} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2} \cdot z0\right) + z5} \]

      21. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

      22. lower-*.f64 40.7%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2} \cdot z0\right) + z5} \]

   7. Applied rewrites 40.7%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      3. associate-*l/ N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      9. lower-/.f64 39.1%

         \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      10. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      14. lower-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

      17. lift-*.f64 40.4%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]

   9. Applied rewrites 40.4%

      \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right) + z5} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\left(\frac{z1}{\left(z4 \cdot z3\right) \cdot z2} \cdot z0\right)} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \left(\color{blue}{\frac{z1}{\left(z4 \cdot z3\right) \cdot z2}} \cdot z0\right) + z5} \]

       3. associate-*l/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot z0}}{\left(z4 \cdot z3\right) \cdot z2} + z5} \]

       5. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}} + z5} \]

       6. *-commutative N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z4 \cdot z3\right)}} + z5} \]

       7. associate-/r* N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       8. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       9. lower-/.f64 41.8%

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{\frac{z1 \cdot z0}{z2}}{z4 \cdot z3}} + z5} \]

       10. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z4 \cdot z3} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\frac{\color{blue}{z1 \cdot z0}}{z2}}{z4 \cdot z3} + z5} \]

       12. associate-/l* N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       13. lift-/.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       14. lower-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z4 \cdot z3} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z4 \cdot z3}} + z5} \]

       16. *-commutative N/A

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

       17. lift-*.f64 43.9%

           \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{\color{blue}{z3 \cdot z4}} + z5} \]

   11. Applied rewrites 43.9%

       \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0}{z2}} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       3. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       5. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       6. associate-*r/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z3 \cdot z4} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       7. associate-/l/ N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       8. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       9. times-frac N/A

          \[\leadsto \sqrt{\color{blue}{\frac{z0 \cdot \left(z1 \cdot z0\right)}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       10. *-commutative N/A

           \[\leadsto \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       12. *-commutative N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       13. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z3 \cdot z4\right)} \cdot z2\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       14. *-commutative N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       16. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right)}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       17. *-commutative N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       18. lift-*.f64 N/A

           \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       19. associate-/l* N/A

           \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       20. lower-*.f64 N/A

           \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       21. lower-*.f64 N/A

           \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right)} \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       22. lower-/.f64 37.6%

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \color{blue}{\frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

   13. Applied rewrites 37.6%

       \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]

       2. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z0}{z2}} \cdot \frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4} + z5} \]

       3. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot \frac{z0}{z2}}{z3 \cdot z4}} + z5} \]

       4. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{z1 \cdot \frac{z0}{z2}}}{z3 \cdot z4} + z5} \]

       5. lift-/.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot \color{blue}{\frac{z0}{z2}}}{z3 \cdot z4} + z5} \]

       6. associate-*r/ N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{\color{blue}{\frac{z1 \cdot z0}{z2}}}{z3 \cdot z4} + z5} \]

       7. associate-/l/ N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \color{blue}{\frac{z1 \cdot z0}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

       8. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{z0}{z2} \cdot \frac{z1 \cdot z0}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}} + z5} \]

       9. times-frac N/A

          \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\frac{z0 \cdot \left(z1 \cdot z0\right)}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

       10. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5} \]

       11. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5} \]

       12. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right)}} + z5} \]

       13. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z3 \cdot z4\right)} \cdot z2\right)} + z5} \]

       14. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} + z5} \]

       15. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right)} + z5} \]

       16. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{z2 \cdot \color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right)}} + z5} \]

       17. *-commutative N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       18. lift-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\color{blue}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       19. associate-/l* N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       20. lower-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

       21. lower-*.f64 N/A

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right)} \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + z5} \]

       22. lower-/.f64 38.1%

           \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\left(z1 \cdot z0\right) \cdot \color{blue}{\frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

   15. Applied rewrites 38.1%

       \[\leadsto \sqrt{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} + z5} \]

   if 1.0000000000000001e287 < (*.f64 (*.f64 (*.f64 z4 z3) z2) z2)

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. sub-flip N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} + \left(\mathsf{neg}\left(z5 \cdot z5\right)\right)}} \]

      3. add-flip N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      7. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      9. remove-double-neg N/A

         \[\leadsto \sqrt{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - \color{blue}{z5 \cdot z5}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - \color{blue}{z5 \cdot z5}} \]

      11. difference-of-squares N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) + z5\right) \cdot \left(\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - z5\right)}} \]

   3. Applied rewrites 67.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)}}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right)}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)}}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(z2 \cdot \left(z2 \cdot z3\right)\right) \cdot z4}}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(z2 \cdot \left(z2 \cdot z3\right)\right) \cdot z4}}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(z2 \cdot \left(z2 \cdot z3\right)\right)} \cdot z4}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \color{blue}{\left(z3 \cdot z2\right)}\right) \cdot z4}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      10. lower-*.f64 67.1%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \color{blue}{\left(z3 \cdot z2\right)}\right) \cdot z4}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

   5. Applied rewrites 67.1%

      \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z2\right)\right) \cdot z4}}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z2\right)\right) \cdot z4}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) - z5\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z2\right)\right) \cdot z4}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}}\right) - z5\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z2\right)\right) \cdot z4}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)}}\right) - z5\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z2\right)\right) \cdot z4}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right)}\right) - z5\right)} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z2\right)\right) \cdot z4}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{z2 \cdot \color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)}}\right) - z5\right)} \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z2\right)\right) \cdot z4}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(z2 \cdot \left(z2 \cdot z3\right)\right) \cdot z4}}\right) - z5\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z2\right)\right) \cdot z4}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(z2 \cdot \left(z2 \cdot z3\right)\right) \cdot z4}}\right) - z5\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z2\right)\right) \cdot z4}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(z2 \cdot \left(z2 \cdot z3\right)\right)} \cdot z4}\right) - z5\right)} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z2\right)\right) \cdot z4}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \color{blue}{\left(z3 \cdot z2\right)}\right) \cdot z4}\right) - z5\right)} \]

      10. lower-*.f64 69.1%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z2\right)\right) \cdot z4}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \color{blue}{\left(z3 \cdot z2\right)}\right) \cdot z4}\right) - z5\right)} \]

   7. Applied rewrites 69.1%

      \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z2\right)\right) \cdot z4}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z2\right)\right) \cdot z4}}\right) - z5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 71.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|z4\right|, \left|z3\right|\right)\\ t_1 := \mathsf{max}\left(\left|z4\right|, \left|z3\right|\right)\\ t_2 := \left(\left(t\_0 \cdot t\_1\right) \cdot z2\right) \cdot z2\\ t_3 := \left(z0 \cdot z0\right) \cdot \frac{\left|z1\right|}{\left(z2 \cdot \left(t\_1 \cdot t\_0\right)\right) \cdot z2}\\ t_4 := \left|z1\right| \cdot z0\\ t_5 := t\_4 \cdot z0\\ t_6 := z0 \cdot \frac{t\_4}{\left(-t\_1\right) \cdot \left(\left(z2 \cdot z2\right) \cdot t\_0\right)}\\ \mathbf{if}\;\sqrt{\frac{t\_5 \cdot t\_5}{t\_2 \cdot t\_2} - z5 \cdot z5} \leq \infty:\\ \;\;\;\;\sqrt{t\_3 - z5} \cdot \sqrt{t\_3 + z5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t\_6 + z5\right) \cdot \left(t\_6 - z5\right)}\\ \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0 (fmin (fabs z4) (fabs z3)))
       (t_1 (fmax (fabs z4) (fabs z3)))
       (t_2 (* (* (* t_0 t_1) z2) z2))
       (t_3 (* (* z0 z0) (/ (fabs z1) (* (* z2 (* t_1 t_0)) z2))))
       (t_4 (* (fabs z1) z0))
       (t_5 (* t_4 z0))
       (t_6 (* z0 (/ t_4 (* (- t_1) (* (* z2 z2) t_0))))))
  (if (<= (sqrt (- (/ (* t_5 t_5) (* t_2 t_2)) (* z5 z5))) INFINITY)
    (* (sqrt (- t_3 z5)) (sqrt (+ t_3 z5)))
    (sqrt (* (+ t_6 z5) (- t_6 z5))))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmin(fabs(z4), fabs(z3));
	double t_1 = fmax(fabs(z4), fabs(z3));
	double t_2 = ((t_0 * t_1) * z2) * z2;
	double t_3 = (z0 * z0) * (fabs(z1) / ((z2 * (t_1 * t_0)) * z2));
	double t_4 = fabs(z1) * z0;
	double t_5 = t_4 * z0;
	double t_6 = z0 * (t_4 / (-t_1 * ((z2 * z2) * t_0)));
	double tmp;
	if (sqrt((((t_5 * t_5) / (t_2 * t_2)) - (z5 * z5))) <= ((double) INFINITY)) {
		tmp = sqrt((t_3 - z5)) * sqrt((t_3 + z5));
	} else {
		tmp = sqrt(((t_6 + z5) * (t_6 - z5)));
	}
	return tmp;
}

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmin(Math.abs(z4), Math.abs(z3));
	double t_1 = fmax(Math.abs(z4), Math.abs(z3));
	double t_2 = ((t_0 * t_1) * z2) * z2;
	double t_3 = (z0 * z0) * (Math.abs(z1) / ((z2 * (t_1 * t_0)) * z2));
	double t_4 = Math.abs(z1) * z0;
	double t_5 = t_4 * z0;
	double t_6 = z0 * (t_4 / (-t_1 * ((z2 * z2) * t_0)));
	double tmp;
	if (Math.sqrt((((t_5 * t_5) / (t_2 * t_2)) - (z5 * z5))) <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_3 - z5)) * Math.sqrt((t_3 + z5));
	} else {
		tmp = Math.sqrt(((t_6 + z5) * (t_6 - z5)));
	}
	return tmp;
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = fmin(math.fabs(z4), math.fabs(z3))
	t_1 = fmax(math.fabs(z4), math.fabs(z3))
	t_2 = ((t_0 * t_1) * z2) * z2
	t_3 = (z0 * z0) * (math.fabs(z1) / ((z2 * (t_1 * t_0)) * z2))
	t_4 = math.fabs(z1) * z0
	t_5 = t_4 * z0
	t_6 = z0 * (t_4 / (-t_1 * ((z2 * z2) * t_0)))
	tmp = 0
	if math.sqrt((((t_5 * t_5) / (t_2 * t_2)) - (z5 * z5))) <= math.inf:
		tmp = math.sqrt((t_3 - z5)) * math.sqrt((t_3 + z5))
	else:
		tmp = math.sqrt(((t_6 + z5) * (t_6 - z5)))
	return tmp

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = fmin(abs(z4), abs(z3))
	t_1 = fmax(abs(z4), abs(z3))
	t_2 = Float64(Float64(Float64(t_0 * t_1) * z2) * z2)
	t_3 = Float64(Float64(z0 * z0) * Float64(abs(z1) / Float64(Float64(z2 * Float64(t_1 * t_0)) * z2)))
	t_4 = Float64(abs(z1) * z0)
	t_5 = Float64(t_4 * z0)
	t_6 = Float64(z0 * Float64(t_4 / Float64(Float64(-t_1) * Float64(Float64(z2 * z2) * t_0))))
	tmp = 0.0
	if (sqrt(Float64(Float64(Float64(t_5 * t_5) / Float64(t_2 * t_2)) - Float64(z5 * z5))) <= Inf)
		tmp = Float64(sqrt(Float64(t_3 - z5)) * sqrt(Float64(t_3 + z5)));
	else
		tmp = sqrt(Float64(Float64(t_6 + z5) * Float64(t_6 - z5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z4, z3, z2, z5)
	t_0 = min(abs(z4), abs(z3));
	t_1 = max(abs(z4), abs(z3));
	t_2 = ((t_0 * t_1) * z2) * z2;
	t_3 = (z0 * z0) * (abs(z1) / ((z2 * (t_1 * t_0)) * z2));
	t_4 = abs(z1) * z0;
	t_5 = t_4 * z0;
	t_6 = z0 * (t_4 / (-t_1 * ((z2 * z2) * t_0)));
	tmp = 0.0;
	if (sqrt((((t_5 * t_5) / (t_2 * t_2)) - (z5 * z5))) <= Inf)
		tmp = sqrt((t_3 - z5)) * sqrt((t_3 + z5));
	else
		tmp = sqrt(((t_6 + z5) * (t_6 - z5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[Min[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] * z2), $MachinePrecision] * z2), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z0 * z0), $MachinePrecision] * N[(N[Abs[z1], $MachinePrecision] / N[(N[(z2 * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[z1], $MachinePrecision] * z0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * z0), $MachinePrecision]}, Block[{t$95$6 = N[(z0 * N[(t$95$4 / N[((-t$95$1) * N[(N[(z2 * z2), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(N[(N[(t$95$5 * t$95$5), $MachinePrecision] / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(z5 * z5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], Infinity], N[(N[Sqrt[N[(t$95$3 - z5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$3 + z5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(t$95$6 + z5), $MachinePrecision] * N[(t$95$6 - z5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (-.f64 (/.f64 (*.f64 (*.f64 (*.f64 z1 z0) z0) (*.f64 (*.f64 z1 z0) z0)) (*.f64 (*.f64 (*.f64 (*.f64 z4 z3) z2) z2) (*.f64 (*.f64 (*.f64 z4 z3) z2) z2))) (*.f64 z5 z5))) < +inf.0

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]

   if +inf.0 < (sqrt.f64 (-.f64 (/.f64 (*.f64 (*.f64 (*.f64 z1 z0) z0) (*.f64 (*.f64 z1 z0) z0)) (*.f64 (*.f64 (*.f64 (*.f64 z4 z3) z2) z2) (*.f64 (*.f64 (*.f64 z4 z3) z2) z2))) (*.f64 z5 z5)))

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. sub-flip N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} + \left(\mathsf{neg}\left(z5 \cdot z5\right)\right)}} \]

      3. add-flip N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      7. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      9. remove-double-neg N/A

         \[\leadsto \sqrt{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - \color{blue}{z5 \cdot z5}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - \color{blue}{z5 \cdot z5}} \]

      11. difference-of-squares N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) + z5\right) \cdot \left(\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - z5\right)}} \]

   3. Applied rewrites 67.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{z0 \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\color{blue}{z0 \cdot z1}}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{\frac{z0 \cdot z1}{z2}}{z2 \cdot \left(z3 \cdot z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      8. frac-2neg N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\color{blue}{-\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\color{blue}{\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z0 \cdot z1}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      17. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{\left(z2 \cdot z3\right) \cdot z4}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      18. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      20. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      21. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      22. lower-neg.f64 68.2%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

   5. Applied rewrites 68.2%

      \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)} - z5\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) - z5\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{z0 \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\color{blue}{z0 \cdot z1}}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5\right)} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{\frac{z0 \cdot z1}{z2}}{z2 \cdot \left(z3 \cdot z4\right)}} - z5\right)} \]

      8. frac-2neg N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\color{blue}{-\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\color{blue}{\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z0 \cdot z1}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right)} - z5\right)} \]

      17. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{\left(z2 \cdot z3\right) \cdot z4}\right)} - z5\right)} \]

      18. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} - z5\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} - z5\right)} \]

      20. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} - z5\right)} \]

      21. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} - z5\right)} \]

      22. lower-neg.f64 75.0%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(-z4\right)}} - z5\right)} \]

   7. Applied rewrites 75.0%

      \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} - z5\right)} \]

   9. Applied rewrites 66.4%

      \[\leadsto \sqrt{\left(\color{blue}{z0 \cdot \frac{z1 \cdot z0}{\left(-z3\right) \cdot \left(\left(z2 \cdot z2\right) \cdot z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

   11. Applied rewrites 66.9%

       \[\leadsto \sqrt{\left(z0 \cdot \frac{z1 \cdot z0}{\left(-z3\right) \cdot \left(\left(z2 \cdot z2\right) \cdot z4\right)} + z5\right) \cdot \left(\color{blue}{z0 \cdot \frac{z1 \cdot z0}{\left(-z3\right) \cdot \left(\left(z2 \cdot z2\right) \cdot z4\right)}} - z5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 70.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|z4\right|, \left|z3\right|\right)\\ t_1 := \mathsf{min}\left(\left|z4\right|, \left|z3\right|\right)\\ t_2 := \left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(\left(t\_0 \cdot \left|z2\right|\right) \cdot t\_1\right) \cdot \left|z2\right|}\right)\\ t_3 := z0 \cdot \frac{z1 \cdot z0}{\left(-t\_0\right) \cdot \left(\left(\left|z2\right| \cdot \left|z2\right|\right) \cdot t\_1\right)}\\ \mathbf{if}\;\left|z2\right| \leq 1.22 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\left(t\_3 + z5\right) \cdot \left(t\_3 - z5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t\_2 + z5\right) \cdot \left(t\_2 - z5\right)}\\ \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0 (fmax (fabs z4) (fabs z3)))
       (t_1 (fmin (fabs z4) (fabs z3)))
       (t_2
        (*
         (- z0)
         (* z0 (/ z1 (* (* (* t_0 (fabs z2)) t_1) (fabs z2))))))
       (t_3
        (*
         z0
         (/ (* z1 z0) (* (- t_0) (* (* (fabs z2) (fabs z2)) t_1))))))
  (if (<= (fabs z2) 1.22e+153)
    (sqrt (* (+ t_3 z5) (- t_3 z5)))
    (sqrt (* (+ t_2 z5) (- t_2 z5))))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmax(fabs(z4), fabs(z3));
	double t_1 = fmin(fabs(z4), fabs(z3));
	double t_2 = -z0 * (z0 * (z1 / (((t_0 * fabs(z2)) * t_1) * fabs(z2))));
	double t_3 = z0 * ((z1 * z0) / (-t_0 * ((fabs(z2) * fabs(z2)) * t_1)));
	double tmp;
	if (fabs(z2) <= 1.22e+153) {
		tmp = sqrt(((t_3 + z5) * (t_3 - z5)));
	} else {
		tmp = sqrt(((t_2 + z5) * (t_2 - z5)));
	}
	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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmax(abs(z4), abs(z3))
    t_1 = fmin(abs(z4), abs(z3))
    t_2 = -z0 * (z0 * (z1 / (((t_0 * abs(z2)) * t_1) * abs(z2))))
    t_3 = z0 * ((z1 * z0) / (-t_0 * ((abs(z2) * abs(z2)) * t_1)))
    if (abs(z2) <= 1.22d+153) then
        tmp = sqrt(((t_3 + z5) * (t_3 - z5)))
    else
        tmp = sqrt(((t_2 + z5) * (t_2 - z5)))
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmax(Math.abs(z4), Math.abs(z3));
	double t_1 = fmin(Math.abs(z4), Math.abs(z3));
	double t_2 = -z0 * (z0 * (z1 / (((t_0 * Math.abs(z2)) * t_1) * Math.abs(z2))));
	double t_3 = z0 * ((z1 * z0) / (-t_0 * ((Math.abs(z2) * Math.abs(z2)) * t_1)));
	double tmp;
	if (Math.abs(z2) <= 1.22e+153) {
		tmp = Math.sqrt(((t_3 + z5) * (t_3 - z5)));
	} else {
		tmp = Math.sqrt(((t_2 + z5) * (t_2 - z5)));
	}
	return tmp;
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = fmax(math.fabs(z4), math.fabs(z3))
	t_1 = fmin(math.fabs(z4), math.fabs(z3))
	t_2 = -z0 * (z0 * (z1 / (((t_0 * math.fabs(z2)) * t_1) * math.fabs(z2))))
	t_3 = z0 * ((z1 * z0) / (-t_0 * ((math.fabs(z2) * math.fabs(z2)) * t_1)))
	tmp = 0
	if math.fabs(z2) <= 1.22e+153:
		tmp = math.sqrt(((t_3 + z5) * (t_3 - z5)))
	else:
		tmp = math.sqrt(((t_2 + z5) * (t_2 - z5)))
	return tmp

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = fmax(abs(z4), abs(z3))
	t_1 = fmin(abs(z4), abs(z3))
	t_2 = Float64(Float64(-z0) * Float64(z0 * Float64(z1 / Float64(Float64(Float64(t_0 * abs(z2)) * t_1) * abs(z2)))))
	t_3 = Float64(z0 * Float64(Float64(z1 * z0) / Float64(Float64(-t_0) * Float64(Float64(abs(z2) * abs(z2)) * t_1))))
	tmp = 0.0
	if (abs(z2) <= 1.22e+153)
		tmp = sqrt(Float64(Float64(t_3 + z5) * Float64(t_3 - z5)));
	else
		tmp = sqrt(Float64(Float64(t_2 + z5) * Float64(t_2 - z5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z4, z3, z2, z5)
	t_0 = max(abs(z4), abs(z3));
	t_1 = min(abs(z4), abs(z3));
	t_2 = -z0 * (z0 * (z1 / (((t_0 * abs(z2)) * t_1) * abs(z2))));
	t_3 = z0 * ((z1 * z0) / (-t_0 * ((abs(z2) * abs(z2)) * t_1)));
	tmp = 0.0;
	if (abs(z2) <= 1.22e+153)
		tmp = sqrt(((t_3 + z5) * (t_3 - z5)));
	else
		tmp = sqrt(((t_2 + z5) * (t_2 - z5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[Max[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[((-z0) * N[(z0 * N[(z1 / N[(N[(N[(t$95$0 * N[Abs[z2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z0 * N[(N[(z1 * z0), $MachinePrecision] / N[((-t$95$0) * N[(N[(N[Abs[z2], $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[z2], $MachinePrecision], 1.22e+153], N[Sqrt[N[(N[(t$95$3 + z5), $MachinePrecision] * N[(t$95$3 - z5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(t$95$2 + z5), $MachinePrecision] * N[(t$95$2 - z5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if z2 < 1.2200000000000001e153

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. sub-flip N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} + \left(\mathsf{neg}\left(z5 \cdot z5\right)\right)}} \]

      3. add-flip N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      7. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      9. remove-double-neg N/A

         \[\leadsto \sqrt{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - \color{blue}{z5 \cdot z5}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - \color{blue}{z5 \cdot z5}} \]

      11. difference-of-squares N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) + z5\right) \cdot \left(\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - z5\right)}} \]

   3. Applied rewrites 67.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{z0 \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\color{blue}{z0 \cdot z1}}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{\frac{z0 \cdot z1}{z2}}{z2 \cdot \left(z3 \cdot z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      8. frac-2neg N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\color{blue}{-\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\color{blue}{\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z0 \cdot z1}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      17. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{\left(z2 \cdot z3\right) \cdot z4}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      18. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      20. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      21. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      22. lower-neg.f64 68.2%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

   5. Applied rewrites 68.2%

      \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)} - z5\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) - z5\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{z0 \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\color{blue}{z0 \cdot z1}}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5\right)} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{\frac{z0 \cdot z1}{z2}}{z2 \cdot \left(z3 \cdot z4\right)}} - z5\right)} \]

      8. frac-2neg N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\color{blue}{-\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\color{blue}{\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z0 \cdot z1}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right)} - z5\right)} \]

      17. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{\left(z2 \cdot z3\right) \cdot z4}\right)} - z5\right)} \]

      18. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} - z5\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} - z5\right)} \]

      20. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} - z5\right)} \]

      21. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} - z5\right)} \]

      22. lower-neg.f64 75.0%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(-z4\right)}} - z5\right)} \]

   7. Applied rewrites 75.0%

      \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} - z5\right)} \]

   9. Applied rewrites 66.4%

      \[\leadsto \sqrt{\left(\color{blue}{z0 \cdot \frac{z1 \cdot z0}{\left(-z3\right) \cdot \left(\left(z2 \cdot z2\right) \cdot z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

   11. Applied rewrites 66.9%

       \[\leadsto \sqrt{\left(z0 \cdot \frac{z1 \cdot z0}{\left(-z3\right) \cdot \left(\left(z2 \cdot z2\right) \cdot z4\right)} + z5\right) \cdot \left(\color{blue}{z0 \cdot \frac{z1 \cdot z0}{\left(-z3\right) \cdot \left(\left(z2 \cdot z2\right) \cdot z4\right)}} - z5\right)} \]

   if 1.2200000000000001e153 < z2

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. sub-flip N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} + \left(\mathsf{neg}\left(z5 \cdot z5\right)\right)}} \]

      3. add-flip N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      7. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      9. remove-double-neg N/A

         \[\leadsto \sqrt{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - \color{blue}{z5 \cdot z5}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - \color{blue}{z5 \cdot z5}} \]

      11. difference-of-squares N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) + z5\right) \cdot \left(\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - z5\right)}} \]

   3. Applied rewrites 67.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)} \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right) \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)} \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)} \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z4\right) \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      6. lower-*.f64 68.4%

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z4\right) \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

   5. Applied rewrites 68.4%

      \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(\left(z3 \cdot z2\right) \cdot z4\right)} \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(\left(z3 \cdot z2\right) \cdot z4\right) \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)} \cdot z2}\right) - z5\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(\left(z3 \cdot z2\right) \cdot z4\right) \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right) \cdot z2}\right) - z5\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(\left(z3 \cdot z2\right) \cdot z4\right) \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)} \cdot z2}\right) - z5\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(\left(z3 \cdot z2\right) \cdot z4\right) \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)} \cdot z2}\right) - z5\right)} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(\left(z3 \cdot z2\right) \cdot z4\right) \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z4\right) \cdot z2}\right) - z5\right)} \]

      6. lower-*.f64 70.9%

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(\left(z3 \cdot z2\right) \cdot z4\right) \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z4\right) \cdot z2}\right) - z5\right)} \]

   7. Applied rewrites 70.9%

      \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(\left(z3 \cdot z2\right) \cdot z4\right) \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\color{blue}{\left(\left(z3 \cdot z2\right) \cdot z4\right)} \cdot z2}\right) - z5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 69.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|z4\right|, \left|z3\right|\right)\\ t_1 := \mathsf{max}\left(\left|z4\right|, \left|z3\right|\right)\\ t_2 := z0 \cdot \frac{z1 \cdot z0}{\left(-t\_1\right) \cdot \left(\left(\left|z2\right| \cdot \left|z2\right|\right) \cdot t\_0\right)}\\ t_3 := \frac{z0}{\left(\left|z2\right| \cdot \left(t\_1 \cdot t\_0\right)\right) \cdot \left|z2\right|}\\ \mathbf{if}\;\left|z2\right| \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\left(t\_2 + z5\right) \cdot \left(t\_2 - z5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(z0 \cdot z1\right) \cdot z1\right) \cdot t\_3\right) \cdot \left(z0 \cdot t\_3\right) - z5 \cdot z5}\\ \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0 (fmin (fabs z4) (fabs z3)))
       (t_1 (fmax (fabs z4) (fabs z3)))
       (t_2
        (*
         z0
         (/ (* z1 z0) (* (- t_1) (* (* (fabs z2) (fabs z2)) t_0)))))
       (t_3 (/ z0 (* (* (fabs z2) (* t_1 t_0)) (fabs z2)))))
  (if (<= (fabs z2) 1.35e+154)
    (sqrt (* (+ t_2 z5) (- t_2 z5)))
    (sqrt (- (* (* (* (* z0 z1) z1) t_3) (* z0 t_3)) (* z5 z5))))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmin(fabs(z4), fabs(z3));
	double t_1 = fmax(fabs(z4), fabs(z3));
	double t_2 = z0 * ((z1 * z0) / (-t_1 * ((fabs(z2) * fabs(z2)) * t_0)));
	double t_3 = z0 / ((fabs(z2) * (t_1 * t_0)) * fabs(z2));
	double tmp;
	if (fabs(z2) <= 1.35e+154) {
		tmp = sqrt(((t_2 + z5) * (t_2 - z5)));
	} else {
		tmp = sqrt((((((z0 * z1) * z1) * t_3) * (z0 * t_3)) - (z5 * z5)));
	}
	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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(abs(z4), abs(z3))
    t_1 = fmax(abs(z4), abs(z3))
    t_2 = z0 * ((z1 * z0) / (-t_1 * ((abs(z2) * abs(z2)) * t_0)))
    t_3 = z0 / ((abs(z2) * (t_1 * t_0)) * abs(z2))
    if (abs(z2) <= 1.35d+154) then
        tmp = sqrt(((t_2 + z5) * (t_2 - z5)))
    else
        tmp = sqrt((((((z0 * z1) * z1) * t_3) * (z0 * t_3)) - (z5 * z5)))
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmin(Math.abs(z4), Math.abs(z3));
	double t_1 = fmax(Math.abs(z4), Math.abs(z3));
	double t_2 = z0 * ((z1 * z0) / (-t_1 * ((Math.abs(z2) * Math.abs(z2)) * t_0)));
	double t_3 = z0 / ((Math.abs(z2) * (t_1 * t_0)) * Math.abs(z2));
	double tmp;
	if (Math.abs(z2) <= 1.35e+154) {
		tmp = Math.sqrt(((t_2 + z5) * (t_2 - z5)));
	} else {
		tmp = Math.sqrt((((((z0 * z1) * z1) * t_3) * (z0 * t_3)) - (z5 * z5)));
	}
	return tmp;
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = fmin(math.fabs(z4), math.fabs(z3))
	t_1 = fmax(math.fabs(z4), math.fabs(z3))
	t_2 = z0 * ((z1 * z0) / (-t_1 * ((math.fabs(z2) * math.fabs(z2)) * t_0)))
	t_3 = z0 / ((math.fabs(z2) * (t_1 * t_0)) * math.fabs(z2))
	tmp = 0
	if math.fabs(z2) <= 1.35e+154:
		tmp = math.sqrt(((t_2 + z5) * (t_2 - z5)))
	else:
		tmp = math.sqrt((((((z0 * z1) * z1) * t_3) * (z0 * t_3)) - (z5 * z5)))
	return tmp

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = fmin(abs(z4), abs(z3))
	t_1 = fmax(abs(z4), abs(z3))
	t_2 = Float64(z0 * Float64(Float64(z1 * z0) / Float64(Float64(-t_1) * Float64(Float64(abs(z2) * abs(z2)) * t_0))))
	t_3 = Float64(z0 / Float64(Float64(abs(z2) * Float64(t_1 * t_0)) * abs(z2)))
	tmp = 0.0
	if (abs(z2) <= 1.35e+154)
		tmp = sqrt(Float64(Float64(t_2 + z5) * Float64(t_2 - z5)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(z0 * z1) * z1) * t_3) * Float64(z0 * t_3)) - Float64(z5 * z5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z4, z3, z2, z5)
	t_0 = min(abs(z4), abs(z3));
	t_1 = max(abs(z4), abs(z3));
	t_2 = z0 * ((z1 * z0) / (-t_1 * ((abs(z2) * abs(z2)) * t_0)));
	t_3 = z0 / ((abs(z2) * (t_1 * t_0)) * abs(z2));
	tmp = 0.0;
	if (abs(z2) <= 1.35e+154)
		tmp = sqrt(((t_2 + z5) * (t_2 - z5)));
	else
		tmp = sqrt((((((z0 * z1) * z1) * t_3) * (z0 * t_3)) - (z5 * z5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[Min[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(z0 * N[(N[(z1 * z0), $MachinePrecision] / N[((-t$95$1) * N[(N[(N[Abs[z2], $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z0 / N[(N[(N[Abs[z2], $MachinePrecision] * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[z2], $MachinePrecision], 1.35e+154], N[Sqrt[N[(N[(t$95$2 + z5), $MachinePrecision] * N[(t$95$2 - z5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(z0 * z1), $MachinePrecision] * z1), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(z0 * t$95$3), $MachinePrecision]), $MachinePrecision] - N[(z5 * z5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if z2 < 1.35e154

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. sub-flip N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} + \left(\mathsf{neg}\left(z5 \cdot z5\right)\right)}} \]

      3. add-flip N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      7. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z5 \cdot z5\right)\right)\right)\right)} \]

      9. remove-double-neg N/A

         \[\leadsto \sqrt{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - \color{blue}{z5 \cdot z5}} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - \color{blue}{z5 \cdot z5}} \]

      11. difference-of-squares N/A

          \[\leadsto \sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) + z5\right) \cdot \left(\left(\mathsf{neg}\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)\right) - z5\right)}} \]

   3. Applied rewrites 67.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \left(z0 \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{z0 \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\color{blue}{z0 \cdot z1}}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{\frac{z0 \cdot z1}{z2}}{z2 \cdot \left(z3 \cdot z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      8. frac-2neg N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{\color{blue}{-\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\color{blue}{\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z0 \cdot z1}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      17. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{\left(z2 \cdot z3\right) \cdot z4}\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      18. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      20. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      21. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

      22. lower-neg.f64 68.2%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]

   5. Applied rewrites 68.2%

      \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \color{blue}{\frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) - z5\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\left(z0 \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)} - z5\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \left(z0 \cdot \color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) - z5\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{z0 \cdot z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\color{blue}{z0 \cdot z1}}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} - z5\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{z0 \cdot z1}{\color{blue}{z2 \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5\right)} \]

      7. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{\frac{z0 \cdot z1}{z2}}{z2 \cdot \left(z3 \cdot z4\right)}} - z5\right)} \]

      8. frac-2neg N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{z0 \cdot z1}{z2}\right)}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)}} - z5\right)} \]

      10. lower-neg.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{\color{blue}{-\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\color{blue}{\frac{z0 \cdot z1}{z2}}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z0 \cdot z1}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{\color{blue}{z1 \cdot z0}}{z2}}{\mathsf{neg}\left(z2 \cdot \left(z3 \cdot z4\right)\right)} - z5\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}\right)} - z5\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right)} - z5\right)} \]

      17. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\mathsf{neg}\left(\color{blue}{\left(z2 \cdot z3\right) \cdot z4}\right)} - z5\right)} \]

      18. distribute-rgt-neg-in N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} - z5\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(\mathsf{neg}\left(z4\right)\right)}} - z5\right)} \]

      20. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} - z5\right)} \]

      21. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(\mathsf{neg}\left(z4\right)\right)} - z5\right)} \]

      22. lower-neg.f64 75.0%

          \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(-z4\right)}} - z5\right)} \]

   7. Applied rewrites 75.0%

      \[\leadsto \sqrt{\left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} + z5\right) \cdot \left(\left(-z0\right) \cdot \color{blue}{\frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)}} - z5\right)} \]

   9. Applied rewrites 66.4%

      \[\leadsto \sqrt{\left(\color{blue}{z0 \cdot \frac{z1 \cdot z0}{\left(-z3\right) \cdot \left(\left(z2 \cdot z2\right) \cdot z4\right)}} + z5\right) \cdot \left(\left(-z0\right) \cdot \frac{-\frac{z1 \cdot z0}{z2}}{\left(z3 \cdot z2\right) \cdot \left(-z4\right)} - z5\right)} \]

   11. Applied rewrites 66.9%

       \[\leadsto \sqrt{\left(z0 \cdot \frac{z1 \cdot z0}{\left(-z3\right) \cdot \left(\left(z2 \cdot z2\right) \cdot z4\right)} + z5\right) \cdot \left(\color{blue}{z0 \cdot \frac{z1 \cdot z0}{\left(-z3\right) \cdot \left(\left(z2 \cdot z2\right) \cdot z4\right)}} - z5\right)} \]

   if 1.35e154 < z2

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right)} \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. swap-sqr N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot \left(z1 \cdot z0\right)\right) \cdot \left(z0 \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot \left(z1 \cdot z0\right)\right) \cdot \left(z0 \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      7. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot \left(z1 \cdot z0\right)}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot \left(z1 \cdot z0\right)}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

   3. Applied rewrites 48.3%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\left(z0 \cdot z1\right) \cdot z1\right) \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)} - z5 \cdot z5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 69.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := z0 \cdot \frac{z0}{\left(\left|z3\right| \cdot z2\right) \cdot \left(\left|z4\right| \cdot z2\right)}\\ t_1 := \left(\left(\left|z4\right| \cdot \left|z3\right|\right) \cdot z2\right) \cdot z2\\ t_2 := \left(\frac{z1}{t\_1} \cdot z0\right) \cdot z0\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+195}:\\ \;\;\;\;\sqrt{\left(t\_2 - z5\right) \cdot \left(z5 + t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(z1 \cdot z1\right) \cdot \left(t\_0 \cdot t\_0\right) - z5 \cdot z5}\\ \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0 (* z0 (/ z0 (* (* (fabs z3) z2) (* (fabs z4) z2)))))
       (t_1 (* (* (* (fabs z4) (fabs z3)) z2) z2))
       (t_2 (* (* (/ z1 t_1) z0) z0)))
  (if (<= t_1 5e+195)
    (sqrt (* (- t_2 z5) (+ z5 t_2)))
    (sqrt (- (* (* z1 z1) (* t_0 t_0)) (* z5 z5))))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = z0 * (z0 / ((fabs(z3) * z2) * (fabs(z4) * z2)));
	double t_1 = ((fabs(z4) * fabs(z3)) * z2) * z2;
	double t_2 = ((z1 / t_1) * z0) * z0;
	double tmp;
	if (t_1 <= 5e+195) {
		tmp = sqrt(((t_2 - z5) * (z5 + t_2)));
	} else {
		tmp = sqrt((((z1 * z1) * (t_0 * t_0)) - (z5 * z5)));
	}
	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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = z0 * (z0 / ((abs(z3) * z2) * (abs(z4) * z2)))
    t_1 = ((abs(z4) * abs(z3)) * z2) * z2
    t_2 = ((z1 / t_1) * z0) * z0
    if (t_1 <= 5d+195) then
        tmp = sqrt(((t_2 - z5) * (z5 + t_2)))
    else
        tmp = sqrt((((z1 * z1) * (t_0 * t_0)) - (z5 * z5)))
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = z0 * (z0 / ((Math.abs(z3) * z2) * (Math.abs(z4) * z2)));
	double t_1 = ((Math.abs(z4) * Math.abs(z3)) * z2) * z2;
	double t_2 = ((z1 / t_1) * z0) * z0;
	double tmp;
	if (t_1 <= 5e+195) {
		tmp = Math.sqrt(((t_2 - z5) * (z5 + t_2)));
	} else {
		tmp = Math.sqrt((((z1 * z1) * (t_0 * t_0)) - (z5 * z5)));
	}
	return tmp;
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = z0 * (z0 / ((math.fabs(z3) * z2) * (math.fabs(z4) * z2)))
	t_1 = ((math.fabs(z4) * math.fabs(z3)) * z2) * z2
	t_2 = ((z1 / t_1) * z0) * z0
	tmp = 0
	if t_1 <= 5e+195:
		tmp = math.sqrt(((t_2 - z5) * (z5 + t_2)))
	else:
		tmp = math.sqrt((((z1 * z1) * (t_0 * t_0)) - (z5 * z5)))
	return tmp

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = Float64(z0 * Float64(z0 / Float64(Float64(abs(z3) * z2) * Float64(abs(z4) * z2))))
	t_1 = Float64(Float64(Float64(abs(z4) * abs(z3)) * z2) * z2)
	t_2 = Float64(Float64(Float64(z1 / t_1) * z0) * z0)
	tmp = 0.0
	if (t_1 <= 5e+195)
		tmp = sqrt(Float64(Float64(t_2 - z5) * Float64(z5 + t_2)));
	else
		tmp = sqrt(Float64(Float64(Float64(z1 * z1) * Float64(t_0 * t_0)) - Float64(z5 * z5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z4, z3, z2, z5)
	t_0 = z0 * (z0 / ((abs(z3) * z2) * (abs(z4) * z2)));
	t_1 = ((abs(z4) * abs(z3)) * z2) * z2;
	t_2 = ((z1 / t_1) * z0) * z0;
	tmp = 0.0;
	if (t_1 <= 5e+195)
		tmp = sqrt(((t_2 - z5) * (z5 + t_2)));
	else
		tmp = sqrt((((z1 * z1) * (t_0 * t_0)) - (z5 * z5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[(z0 * N[(z0 / N[(N[(N[Abs[z3], $MachinePrecision] * z2), $MachinePrecision] * N[(N[Abs[z4], $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Abs[z4], $MachinePrecision] * N[Abs[z3], $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision] * z2), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z1 / t$95$1), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+195], N[Sqrt[N[(N[(t$95$2 - z5), $MachinePrecision] * N[(z5 + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(z1 * z1), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(z5 * z5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 z4 z3) z2) z2) < 4.9999999999999998e195

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right)} \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z1 \cdot \left(z0 \cdot z0\right)}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      8. associate-/l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \frac{\color{blue}{\left(z1 \cdot z0\right)} \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \frac{\color{blue}{z1 \cdot \left(z0 \cdot z0\right)}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \color{blue}{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} - z5 \cdot z5} \]

      13. swap-sqr N/A

          \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z1\right) \cdot \left(\frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} - z5 \cdot z5} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z1\right) \cdot \left(\frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} - z5 \cdot z5} \]

   3. Applied rewrites 42.5%

      \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right)} - z5 \cdot z5} \]

   5. Applied rewrites 67.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\frac{z1}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot \left(z5 + \left(\frac{z1}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot z0\right) \cdot z0\right)}} \]

   if 4.9999999999999998e195 < (*.f64 (*.f64 (*.f64 z4 z3) z2) z2)

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right)} \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z1 \cdot \left(z0 \cdot z0\right)}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      8. associate-/l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \frac{\color{blue}{\left(z1 \cdot z0\right)} \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \frac{\color{blue}{z1 \cdot \left(z0 \cdot z0\right)}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \color{blue}{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} - z5 \cdot z5} \]

      13. swap-sqr N/A

          \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z1\right) \cdot \left(\frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} - z5 \cdot z5} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z1\right) \cdot \left(\frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} - z5 \cdot z5} \]

   3. Applied rewrites 42.5%

      \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right)} - z5 \cdot z5} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)} \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right) \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)} \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(z4 \cdot z2\right)}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(z4 \cdot z2\right)}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      9. lower-*.f64 41.8%

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(z4 \cdot z2\right)}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

   5. Applied rewrites 41.8%

      \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right) - z5 \cdot z5} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}}\right)\right) - z5 \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)} \cdot z2}\right)\right) - z5 \cdot z5} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right) \cdot z2}\right)\right) - z5 \cdot z5} \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)} \cdot z2}\right)\right) - z5 \cdot z5} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(z4 \cdot z2\right)}}\right)\right) - z5 \cdot z5} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z2 \cdot z3\right) \cdot \left(z4 \cdot z2\right)}}\right)\right) - z5 \cdot z5} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right)} \cdot \left(z4 \cdot z2\right)}\right)\right) - z5 \cdot z5} \]

      9. lower-*.f64 44.2%

         \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(z4 \cdot z2\right)}}\right)\right) - z5 \cdot z5} \]

   7. Applied rewrites 44.2%

      \[\leadsto \sqrt{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}\right) \cdot \left(z0 \cdot \frac{z0}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}}\right)\right) - z5 \cdot z5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 68.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|z4\right|, \left|z3\right|\right)\\ t_1 := \mathsf{max}\left(\left|z4\right|, \left|z3\right|\right)\\ t_2 := \left(\frac{\left|z1\right|}{\left(\left(t\_0 \cdot t\_1\right) \cdot z2\right) \cdot z2} \cdot z0\right) \cdot z0\\ \mathbf{if}\;t\_0 \leq 8.8 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{\left(t\_2 - \left|z5\right|\right) \cdot \left(\left|z5\right| + t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{z0 \cdot \frac{\left|z1\right|}{\left(t\_1 \cdot z2\right) \cdot z2}}{t\_0} \cdot z0 - \left|z5\right|\right) \cdot \left|z5\right|}\\ \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0 (fmin (fabs z4) (fabs z3)))
       (t_1 (fmax (fabs z4) (fabs z3)))
       (t_2 (* (* (/ (fabs z1) (* (* (* t_0 t_1) z2) z2)) z0) z0)))
  (if (<= t_0 8.8e+27)
    (sqrt (* (- t_2 (fabs z5)) (+ (fabs z5) t_2)))
    (sqrt
     (*
      (-
       (* (/ (* z0 (/ (fabs z1) (* (* t_1 z2) z2))) t_0) z0)
       (fabs z5))
      (fabs z5))))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmin(fabs(z4), fabs(z3));
	double t_1 = fmax(fabs(z4), fabs(z3));
	double t_2 = ((fabs(z1) / (((t_0 * t_1) * z2) * z2)) * z0) * z0;
	double tmp;
	if (t_0 <= 8.8e+27) {
		tmp = sqrt(((t_2 - fabs(z5)) * (fabs(z5) + t_2)));
	} else {
		tmp = sqrt((((((z0 * (fabs(z1) / ((t_1 * z2) * z2))) / t_0) * z0) - fabs(z5)) * fabs(z5)));
	}
	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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmin(abs(z4), abs(z3))
    t_1 = fmax(abs(z4), abs(z3))
    t_2 = ((abs(z1) / (((t_0 * t_1) * z2) * z2)) * z0) * z0
    if (t_0 <= 8.8d+27) then
        tmp = sqrt(((t_2 - abs(z5)) * (abs(z5) + t_2)))
    else
        tmp = sqrt((((((z0 * (abs(z1) / ((t_1 * z2) * z2))) / t_0) * z0) - abs(z5)) * abs(z5)))
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = fmin(Math.abs(z4), Math.abs(z3));
	double t_1 = fmax(Math.abs(z4), Math.abs(z3));
	double t_2 = ((Math.abs(z1) / (((t_0 * t_1) * z2) * z2)) * z0) * z0;
	double tmp;
	if (t_0 <= 8.8e+27) {
		tmp = Math.sqrt(((t_2 - Math.abs(z5)) * (Math.abs(z5) + t_2)));
	} else {
		tmp = Math.sqrt((((((z0 * (Math.abs(z1) / ((t_1 * z2) * z2))) / t_0) * z0) - Math.abs(z5)) * Math.abs(z5)));
	}
	return tmp;
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = fmin(math.fabs(z4), math.fabs(z3))
	t_1 = fmax(math.fabs(z4), math.fabs(z3))
	t_2 = ((math.fabs(z1) / (((t_0 * t_1) * z2) * z2)) * z0) * z0
	tmp = 0
	if t_0 <= 8.8e+27:
		tmp = math.sqrt(((t_2 - math.fabs(z5)) * (math.fabs(z5) + t_2)))
	else:
		tmp = math.sqrt((((((z0 * (math.fabs(z1) / ((t_1 * z2) * z2))) / t_0) * z0) - math.fabs(z5)) * math.fabs(z5)))
	return tmp

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = fmin(abs(z4), abs(z3))
	t_1 = fmax(abs(z4), abs(z3))
	t_2 = Float64(Float64(Float64(abs(z1) / Float64(Float64(Float64(t_0 * t_1) * z2) * z2)) * z0) * z0)
	tmp = 0.0
	if (t_0 <= 8.8e+27)
		tmp = sqrt(Float64(Float64(t_2 - abs(z5)) * Float64(abs(z5) + t_2)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(z0 * Float64(abs(z1) / Float64(Float64(t_1 * z2) * z2))) / t_0) * z0) - abs(z5)) * abs(z5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z4, z3, z2, z5)
	t_0 = min(abs(z4), abs(z3));
	t_1 = max(abs(z4), abs(z3));
	t_2 = ((abs(z1) / (((t_0 * t_1) * z2) * z2)) * z0) * z0;
	tmp = 0.0;
	if (t_0 <= 8.8e+27)
		tmp = sqrt(((t_2 - abs(z5)) * (abs(z5) + t_2)));
	else
		tmp = sqrt((((((z0 * (abs(z1) / ((t_1 * z2) * z2))) / t_0) * z0) - abs(z5)) * abs(z5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[Min[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Abs[z1], $MachinePrecision] / N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] * z2), $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision]}, If[LessEqual[t$95$0, 8.8e+27], N[Sqrt[N[(N[(t$95$2 - N[Abs[z5], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[z5], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(z0 * N[(N[Abs[z1], $MachinePrecision] / N[(N[(t$95$1 * z2), $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * z0), $MachinePrecision] - N[Abs[z5], $MachinePrecision]), $MachinePrecision] * N[Abs[z5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z4 < 8.7999999999999995e27

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(z1 \cdot z0\right)} \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\frac{\color{blue}{z1 \cdot \left(z0 \cdot z0\right)}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      8. associate-/l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \frac{\color{blue}{\left(z1 \cdot z0\right) \cdot z0}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \frac{\color{blue}{\left(z1 \cdot z0\right)} \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \frac{\color{blue}{z1 \cdot \left(z0 \cdot z0\right)}}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - z5 \cdot z5} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right) \cdot \color{blue}{\left(z1 \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} - z5 \cdot z5} \]

      13. swap-sqr N/A

          \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z1\right) \cdot \left(\frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} - z5 \cdot z5} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z1\right) \cdot \left(\frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{z0 \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} - z5 \cdot z5} \]

   3. Applied rewrites 42.5%

      \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z1\right) \cdot \left(\left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right) \cdot \left(z0 \cdot \frac{z0}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}\right)\right)} - z5 \cdot z5} \]

   5. Applied rewrites 67.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\frac{z1}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot \left(z5 + \left(\frac{z1}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot z0\right) \cdot z0\right)}} \]

   if 8.7999999999999995e27 < z4

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{z5}} \]
   5. Step-by-step derivation

   6. Applied rewrites 14.8%

      \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{z5}} \]

   8. Applied rewrites 30.9%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2} \cdot z0\right)} \cdot z0 - z5\right) \cdot z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      3. associate-*l/ N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{z1 \cdot z0}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2}} \cdot z0 - z5\right) \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{z1 \cdot z0}{\color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2}} \cdot z0 - z5\right) \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{z1 \cdot z0}{\color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right)} \cdot z2} \cdot z0 - z5\right) \cdot z5} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\left(\frac{z1 \cdot z0}{\color{blue}{\left(z3 \cdot z4\right) \cdot \left(z2 \cdot z2\right)}} \cdot z0 - z5\right) \cdot z5} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\left(\frac{\color{blue}{z0 \cdot z1}}{\left(z3 \cdot z4\right) \cdot \left(z2 \cdot z2\right)} \cdot z0 - z5\right) \cdot z5} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{z0 \cdot z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot \left(z2 \cdot z2\right)} \cdot z0 - z5\right) \cdot z5} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\left(\frac{z0 \cdot z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot \left(z2 \cdot z2\right)} \cdot z0 - z5\right) \cdot z5} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\frac{z0 \cdot z1}{\color{blue}{z4 \cdot \left(z3 \cdot \left(z2 \cdot z2\right)\right)}} \cdot z0 - z5\right) \cdot z5} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\left(\frac{z0 \cdot z1}{z4 \cdot \color{blue}{\left(\left(z3 \cdot z2\right) \cdot z2\right)}} \cdot z0 - z5\right) \cdot z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0 \cdot z1}{z4 \cdot \left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z2\right)} \cdot z0 - z5\right) \cdot z5} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0 \cdot z1}{z4 \cdot \color{blue}{\left(\left(z3 \cdot z2\right) \cdot z2\right)}} \cdot z0 - z5\right) \cdot z5} \]

      14. *-commutative N/A

          \[\leadsto \sqrt{\left(\frac{z0 \cdot z1}{\color{blue}{\left(\left(z3 \cdot z2\right) \cdot z2\right) \cdot z4}} \cdot z0 - z5\right) \cdot z5} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0 \cdot z1}{\color{blue}{\left(\left(z3 \cdot z2\right) \cdot z2\right) \cdot z4}} \cdot z0 - z5\right) \cdot z5} \]

      16. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\color{blue}{\left(z0 \cdot \frac{z1}{\left(\left(z3 \cdot z2\right) \cdot z2\right) \cdot z4}\right)} \cdot z0 - z5\right) \cdot z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(z0 \cdot \frac{z1}{\color{blue}{\left(\left(z3 \cdot z2\right) \cdot z2\right) \cdot z4}}\right) \cdot z0 - z5\right) \cdot z5} \]

      18. associate-/r* N/A

          \[\leadsto \sqrt{\left(\left(z0 \cdot \color{blue}{\frac{\frac{z1}{\left(z3 \cdot z2\right) \cdot z2}}{z4}}\right) \cdot z0 - z5\right) \cdot z5} \]

      19. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\color{blue}{\frac{z0 \cdot \frac{z1}{\left(z3 \cdot z2\right) \cdot z2}}{z4}} \cdot z0 - z5\right) \cdot z5} \]

      20. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\frac{z0 \cdot \frac{z1}{\left(z3 \cdot z2\right) \cdot z2}}{z4}} \cdot z0 - z5\right) \cdot z5} \]

      21. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{\color{blue}{z0 \cdot \frac{z1}{\left(z3 \cdot z2\right) \cdot z2}}}{z4} \cdot z0 - z5\right) \cdot z5} \]

      22. lower-/.f64 31.7%

          \[\leadsto \sqrt{\left(\frac{z0 \cdot \color{blue}{\frac{z1}{\left(z3 \cdot z2\right) \cdot z2}}}{z4} \cdot z0 - z5\right) \cdot z5} \]

   10. Applied rewrites 31.7%

       \[\leadsto \sqrt{\left(\color{blue}{\frac{z0 \cdot \frac{z1}{\left(z3 \cdot z2\right) \cdot z2}}{z4}} \cdot z0 - z5\right) \cdot z5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 66.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left(z0 \cdot z0\right) \cdot \frac{\left|z1\right|}{\left(\left|z2\right| \cdot \left(\left|z3\right| \cdot \left|z4\right|\right)\right) \cdot \left|z2\right|}\\ \mathbf{if}\;\left|z2\right| \leq 3.5 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{\left(\frac{z0 \cdot \frac{\left|z1\right|}{\left(\left|z3\right| \cdot \left|z2\right|\right) \cdot \left|z2\right|}}{\left|z4\right|} \cdot z0 - \left|z5\right|\right) \cdot \left|z5\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t\_0 - \left|z5\right|\right) \cdot \left(t\_0 + \left|z5\right|\right)}\\ \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0
        (*
         (* z0 z0)
         (/
          (fabs z1)
          (* (* (fabs z2) (* (fabs z3) (fabs z4))) (fabs z2))))))
  (if (<= (fabs z2) 3.5e-33)
    (sqrt
     (*
      (-
       (*
        (/
         (* z0 (/ (fabs z1) (* (* (fabs z3) (fabs z2)) (fabs z2))))
         (fabs z4))
        z0)
       (fabs z5))
      (fabs z5)))
    (sqrt (* (- t_0 (fabs z5)) (+ t_0 (fabs z5)))))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = (z0 * z0) * (fabs(z1) / ((fabs(z2) * (fabs(z3) * fabs(z4))) * fabs(z2)));
	double tmp;
	if (fabs(z2) <= 3.5e-33) {
		tmp = sqrt((((((z0 * (fabs(z1) / ((fabs(z3) * fabs(z2)) * fabs(z2)))) / fabs(z4)) * z0) - fabs(z5)) * fabs(z5)));
	} else {
		tmp = sqrt(((t_0 - fabs(z5)) * (t_0 + fabs(z5))));
	}
	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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z0 * z0) * (abs(z1) / ((abs(z2) * (abs(z3) * abs(z4))) * abs(z2)))
    if (abs(z2) <= 3.5d-33) then
        tmp = sqrt((((((z0 * (abs(z1) / ((abs(z3) * abs(z2)) * abs(z2)))) / abs(z4)) * z0) - abs(z5)) * abs(z5)))
    else
        tmp = sqrt(((t_0 - abs(z5)) * (t_0 + abs(z5))))
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = (z0 * z0) * (Math.abs(z1) / ((Math.abs(z2) * (Math.abs(z3) * Math.abs(z4))) * Math.abs(z2)));
	double tmp;
	if (Math.abs(z2) <= 3.5e-33) {
		tmp = Math.sqrt((((((z0 * (Math.abs(z1) / ((Math.abs(z3) * Math.abs(z2)) * Math.abs(z2)))) / Math.abs(z4)) * z0) - Math.abs(z5)) * Math.abs(z5)));
	} else {
		tmp = Math.sqrt(((t_0 - Math.abs(z5)) * (t_0 + Math.abs(z5))));
	}
	return tmp;
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = (z0 * z0) * (math.fabs(z1) / ((math.fabs(z2) * (math.fabs(z3) * math.fabs(z4))) * math.fabs(z2)))
	tmp = 0
	if math.fabs(z2) <= 3.5e-33:
		tmp = math.sqrt((((((z0 * (math.fabs(z1) / ((math.fabs(z3) * math.fabs(z2)) * math.fabs(z2)))) / math.fabs(z4)) * z0) - math.fabs(z5)) * math.fabs(z5)))
	else:
		tmp = math.sqrt(((t_0 - math.fabs(z5)) * (t_0 + math.fabs(z5))))
	return tmp

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = Float64(Float64(z0 * z0) * Float64(abs(z1) / Float64(Float64(abs(z2) * Float64(abs(z3) * abs(z4))) * abs(z2))))
	tmp = 0.0
	if (abs(z2) <= 3.5e-33)
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(z0 * Float64(abs(z1) / Float64(Float64(abs(z3) * abs(z2)) * abs(z2)))) / abs(z4)) * z0) - abs(z5)) * abs(z5)));
	else
		tmp = sqrt(Float64(Float64(t_0 - abs(z5)) * Float64(t_0 + abs(z5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z4, z3, z2, z5)
	t_0 = (z0 * z0) * (abs(z1) / ((abs(z2) * (abs(z3) * abs(z4))) * abs(z2)));
	tmp = 0.0;
	if (abs(z2) <= 3.5e-33)
		tmp = sqrt((((((z0 * (abs(z1) / ((abs(z3) * abs(z2)) * abs(z2)))) / abs(z4)) * z0) - abs(z5)) * abs(z5)));
	else
		tmp = sqrt(((t_0 - abs(z5)) * (t_0 + abs(z5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[(N[(z0 * z0), $MachinePrecision] * N[(N[Abs[z1], $MachinePrecision] / N[(N[(N[Abs[z2], $MachinePrecision] * N[(N[Abs[z3], $MachinePrecision] * N[Abs[z4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[z2], $MachinePrecision], 3.5e-33], N[Sqrt[N[(N[(N[(N[(N[(z0 * N[(N[Abs[z1], $MachinePrecision] / N[(N[(N[Abs[z3], $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision] * N[Abs[z2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[z4], $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - N[Abs[z5], $MachinePrecision]), $MachinePrecision] * N[Abs[z5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(t$95$0 - N[Abs[z5], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Abs[z5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z2 < 3.4999999999999999e-33

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{z5}} \]
   5. Step-by-step derivation

   6. Applied rewrites 14.8%

      \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{z5}} \]

   8. Applied rewrites 30.9%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2} \cdot z0\right)} \cdot z0 - z5\right) \cdot z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      3. associate-*l/ N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{z1 \cdot z0}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2}} \cdot z0 - z5\right) \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{z1 \cdot z0}{\color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2}} \cdot z0 - z5\right) \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{z1 \cdot z0}{\color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right)} \cdot z2} \cdot z0 - z5\right) \cdot z5} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\left(\frac{z1 \cdot z0}{\color{blue}{\left(z3 \cdot z4\right) \cdot \left(z2 \cdot z2\right)}} \cdot z0 - z5\right) \cdot z5} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\left(\frac{\color{blue}{z0 \cdot z1}}{\left(z3 \cdot z4\right) \cdot \left(z2 \cdot z2\right)} \cdot z0 - z5\right) \cdot z5} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{z0 \cdot z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot \left(z2 \cdot z2\right)} \cdot z0 - z5\right) \cdot z5} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\left(\frac{z0 \cdot z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot \left(z2 \cdot z2\right)} \cdot z0 - z5\right) \cdot z5} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\frac{z0 \cdot z1}{\color{blue}{z4 \cdot \left(z3 \cdot \left(z2 \cdot z2\right)\right)}} \cdot z0 - z5\right) \cdot z5} \]

      11. associate-*l* N/A

          \[\leadsto \sqrt{\left(\frac{z0 \cdot z1}{z4 \cdot \color{blue}{\left(\left(z3 \cdot z2\right) \cdot z2\right)}} \cdot z0 - z5\right) \cdot z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0 \cdot z1}{z4 \cdot \left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z2\right)} \cdot z0 - z5\right) \cdot z5} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0 \cdot z1}{z4 \cdot \color{blue}{\left(\left(z3 \cdot z2\right) \cdot z2\right)}} \cdot z0 - z5\right) \cdot z5} \]

      14. *-commutative N/A

          \[\leadsto \sqrt{\left(\frac{z0 \cdot z1}{\color{blue}{\left(\left(z3 \cdot z2\right) \cdot z2\right) \cdot z4}} \cdot z0 - z5\right) \cdot z5} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{z0 \cdot z1}{\color{blue}{\left(\left(z3 \cdot z2\right) \cdot z2\right) \cdot z4}} \cdot z0 - z5\right) \cdot z5} \]

      16. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\color{blue}{\left(z0 \cdot \frac{z1}{\left(\left(z3 \cdot z2\right) \cdot z2\right) \cdot z4}\right)} \cdot z0 - z5\right) \cdot z5} \]

      17. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(z0 \cdot \frac{z1}{\color{blue}{\left(\left(z3 \cdot z2\right) \cdot z2\right) \cdot z4}}\right) \cdot z0 - z5\right) \cdot z5} \]

      18. associate-/r* N/A

          \[\leadsto \sqrt{\left(\left(z0 \cdot \color{blue}{\frac{\frac{z1}{\left(z3 \cdot z2\right) \cdot z2}}{z4}}\right) \cdot z0 - z5\right) \cdot z5} \]

      19. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\color{blue}{\frac{z0 \cdot \frac{z1}{\left(z3 \cdot z2\right) \cdot z2}}{z4}} \cdot z0 - z5\right) \cdot z5} \]

      20. lower-/.f64 N/A

          \[\leadsto \sqrt{\left(\color{blue}{\frac{z0 \cdot \frac{z1}{\left(z3 \cdot z2\right) \cdot z2}}{z4}} \cdot z0 - z5\right) \cdot z5} \]

      21. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{\color{blue}{z0 \cdot \frac{z1}{\left(z3 \cdot z2\right) \cdot z2}}}{z4} \cdot z0 - z5\right) \cdot z5} \]

      22. lower-/.f64 31.7%

          \[\leadsto \sqrt{\left(\frac{z0 \cdot \color{blue}{\frac{z1}{\left(z3 \cdot z2\right) \cdot z2}}}{z4} \cdot z0 - z5\right) \cdot z5} \]

   10. Applied rewrites 31.7%

       \[\leadsto \sqrt{\left(\color{blue}{\frac{z0 \cdot \frac{z1}{\left(z3 \cdot z2\right) \cdot z2}}{z4}} \cdot z0 - z5\right) \cdot z5} \]

   if 3.4999999999999999e-33 < z2

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      5. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      7. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      8. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 64.2%

      \[\leadsto \sqrt{\color{blue}{\left(\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 66.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := z0 \cdot \left|z1\right|\\ t_1 := \left(\left(\left|z4\right| \cdot \left|z3\right|\right) \cdot z2\right) \cdot z2\\ t_2 := \left|z3\right| \cdot z2\\ t_3 := t\_2 \cdot \left|z4\right|\\ t_4 := \left|z5\right| \cdot \left|z5\right|\\ t_5 := \left(\left|z1\right| \cdot z0\right) \cdot z0\\ \mathbf{if}\;\frac{t\_5 \cdot t\_5}{t\_1 \cdot t\_1} - t\_4 \leq \infty:\\ \;\;\;\;\sqrt{t\_0 \cdot \left(\left(z0 \cdot z0\right) \cdot \frac{t\_0}{\left(\left(t\_3 \cdot z2\right) \cdot z2\right) \cdot t\_3}\right) - t\_4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{\frac{\left|z1\right|}{t\_2}}{\left|z4\right| \cdot z2} \cdot z0\right) \cdot z0 - \left|z5\right|\right) \cdot \left|z5\right|}\\ \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0 (* z0 (fabs z1)))
       (t_1 (* (* (* (fabs z4) (fabs z3)) z2) z2))
       (t_2 (* (fabs z3) z2))
       (t_3 (* t_2 (fabs z4)))
       (t_4 (* (fabs z5) (fabs z5)))
       (t_5 (* (* (fabs z1) z0) z0)))
  (if (<= (- (/ (* t_5 t_5) (* t_1 t_1)) t_4) INFINITY)
    (sqrt
     (- (* t_0 (* (* z0 z0) (/ t_0 (* (* (* t_3 z2) z2) t_3)))) t_4))
    (sqrt
     (*
      (-
       (* (* (/ (/ (fabs z1) t_2) (* (fabs z4) z2)) z0) z0)
       (fabs z5))
      (fabs z5))))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = z0 * fabs(z1);
	double t_1 = ((fabs(z4) * fabs(z3)) * z2) * z2;
	double t_2 = fabs(z3) * z2;
	double t_3 = t_2 * fabs(z4);
	double t_4 = fabs(z5) * fabs(z5);
	double t_5 = (fabs(z1) * z0) * z0;
	double tmp;
	if ((((t_5 * t_5) / (t_1 * t_1)) - t_4) <= ((double) INFINITY)) {
		tmp = sqrt(((t_0 * ((z0 * z0) * (t_0 / (((t_3 * z2) * z2) * t_3)))) - t_4));
	} else {
		tmp = sqrt(((((((fabs(z1) / t_2) / (fabs(z4) * z2)) * z0) * z0) - fabs(z5)) * fabs(z5)));
	}
	return tmp;
}

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = z0 * Math.abs(z1);
	double t_1 = ((Math.abs(z4) * Math.abs(z3)) * z2) * z2;
	double t_2 = Math.abs(z3) * z2;
	double t_3 = t_2 * Math.abs(z4);
	double t_4 = Math.abs(z5) * Math.abs(z5);
	double t_5 = (Math.abs(z1) * z0) * z0;
	double tmp;
	if ((((t_5 * t_5) / (t_1 * t_1)) - t_4) <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((t_0 * ((z0 * z0) * (t_0 / (((t_3 * z2) * z2) * t_3)))) - t_4));
	} else {
		tmp = Math.sqrt(((((((Math.abs(z1) / t_2) / (Math.abs(z4) * z2)) * z0) * z0) - Math.abs(z5)) * Math.abs(z5)));
	}
	return tmp;
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = z0 * math.fabs(z1)
	t_1 = ((math.fabs(z4) * math.fabs(z3)) * z2) * z2
	t_2 = math.fabs(z3) * z2
	t_3 = t_2 * math.fabs(z4)
	t_4 = math.fabs(z5) * math.fabs(z5)
	t_5 = (math.fabs(z1) * z0) * z0
	tmp = 0
	if (((t_5 * t_5) / (t_1 * t_1)) - t_4) <= math.inf:
		tmp = math.sqrt(((t_0 * ((z0 * z0) * (t_0 / (((t_3 * z2) * z2) * t_3)))) - t_4))
	else:
		tmp = math.sqrt(((((((math.fabs(z1) / t_2) / (math.fabs(z4) * z2)) * z0) * z0) - math.fabs(z5)) * math.fabs(z5)))
	return tmp

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = Float64(z0 * abs(z1))
	t_1 = Float64(Float64(Float64(abs(z4) * abs(z3)) * z2) * z2)
	t_2 = Float64(abs(z3) * z2)
	t_3 = Float64(t_2 * abs(z4))
	t_4 = Float64(abs(z5) * abs(z5))
	t_5 = Float64(Float64(abs(z1) * z0) * z0)
	tmp = 0.0
	if (Float64(Float64(Float64(t_5 * t_5) / Float64(t_1 * t_1)) - t_4) <= Inf)
		tmp = sqrt(Float64(Float64(t_0 * Float64(Float64(z0 * z0) * Float64(t_0 / Float64(Float64(Float64(t_3 * z2) * z2) * t_3)))) - t_4));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(Float64(abs(z1) / t_2) / Float64(abs(z4) * z2)) * z0) * z0) - abs(z5)) * abs(z5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z4, z3, z2, z5)
	t_0 = z0 * abs(z1);
	t_1 = ((abs(z4) * abs(z3)) * z2) * z2;
	t_2 = abs(z3) * z2;
	t_3 = t_2 * abs(z4);
	t_4 = abs(z5) * abs(z5);
	t_5 = (abs(z1) * z0) * z0;
	tmp = 0.0;
	if ((((t_5 * t_5) / (t_1 * t_1)) - t_4) <= Inf)
		tmp = sqrt(((t_0 * ((z0 * z0) * (t_0 / (((t_3 * z2) * z2) * t_3)))) - t_4));
	else
		tmp = sqrt(((((((abs(z1) / t_2) / (abs(z4) * z2)) * z0) * z0) - abs(z5)) * abs(z5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[(z0 * N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Abs[z4], $MachinePrecision] * N[Abs[z3], $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision] * z2), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[z3], $MachinePrecision] * z2), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Abs[z4], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[z5], $MachinePrecision] * N[Abs[z5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Abs[z1], $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$5 * t$95$5), $MachinePrecision] / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision], Infinity], N[Sqrt[N[(N[(t$95$0 * N[(N[(z0 * z0), $MachinePrecision] * N[(t$95$0 / N[(N[(N[(t$95$3 * z2), $MachinePrecision] * z2), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(N[(N[Abs[z1], $MachinePrecision] / t$95$2), $MachinePrecision] / N[(N[Abs[z4], $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision] - N[Abs[z5], $MachinePrecision]), $MachinePrecision] * N[Abs[z5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (*.f64 (*.f64 (*.f64 z1 z0) z0) (*.f64 (*.f64 z1 z0) z0)) (*.f64 (*.f64 (*.f64 (*.f64 z4 z3) z2) z2) (*.f64 (*.f64 (*.f64 z4 z3) z2) z2))) (*.f64 z5 z5)) < +inf.0

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      2. mult-flip N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)\right) \cdot \frac{1}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)\right)} \cdot \frac{1}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right)} \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)\right) \cdot \frac{1}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot \left(z0 \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)\right)\right)} \cdot \frac{1}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \left(\left(z0 \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)\right) \cdot \frac{1}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}\right)} - z5 \cdot z5} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \left(\left(z0 \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)\right) \cdot \frac{1}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}\right)} - z5 \cdot z5} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right)} \cdot \left(\left(z0 \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)\right) \cdot \frac{1}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}\right) - z5 \cdot z5} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 \cdot z1\right)} \cdot \left(\left(z0 \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)\right) \cdot \frac{1}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}\right) - z5 \cdot z5} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(z0 \cdot z1\right)} \cdot \left(\left(z0 \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)\right) \cdot \frac{1}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}\right) - z5 \cdot z5} \]

      11. mult-flip-rev N/A

          \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \color{blue}{\frac{z0 \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \frac{z0 \cdot \color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \frac{z0 \cdot \color{blue}{\left(z0 \cdot \left(z1 \cdot z0\right)\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      14. associate-*r* N/A

          \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \frac{\color{blue}{\left(z0 \cdot z0\right) \cdot \left(z1 \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      15. associate-/l* N/A

          \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \color{blue}{\left(\left(z0 \cdot z0\right) \cdot \frac{z1 \cdot z0}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}\right)} - z5 \cdot z5} \]

   3. Applied rewrites 45.2%

      \[\leadsto \sqrt{\color{blue}{\left(z0 \cdot z1\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \frac{z0 \cdot z1}{\left(\left(\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2\right) \cdot z2\right) \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}\right)} - z5 \cdot z5} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \frac{z0 \cdot z1}{\left(\left(\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)} \cdot z2\right) \cdot z2\right) \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}\right) - z5 \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \frac{z0 \cdot z1}{\left(\left(\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right) \cdot z2\right) \cdot z2\right) \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}\right) - z5 \cdot z5} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \frac{z0 \cdot z1}{\left(\left(\color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)} \cdot z2\right) \cdot z2\right) \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}\right) - z5 \cdot z5} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \frac{z0 \cdot z1}{\left(\left(\color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)} \cdot z2\right) \cdot z2\right) \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}\right) - z5 \cdot z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \frac{z0 \cdot z1}{\left(\left(\left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z4\right) \cdot z2\right) \cdot z2\right) \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}\right) - z5 \cdot z5} \]

      6. lower-*.f64 44.8%

         \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \frac{z0 \cdot z1}{\left(\left(\left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z4\right) \cdot z2\right) \cdot z2\right) \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}\right) - z5 \cdot z5} \]

   5. Applied rewrites 44.8%

      \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \frac{z0 \cdot z1}{\left(\left(\color{blue}{\left(\left(z3 \cdot z2\right) \cdot z4\right)} \cdot z2\right) \cdot z2\right) \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}\right) - z5 \cdot z5} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \frac{z0 \cdot z1}{\left(\left(\left(\left(z3 \cdot z2\right) \cdot z4\right) \cdot z2\right) \cdot z2\right) \cdot \color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)}}\right) - z5 \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \frac{z0 \cdot z1}{\left(\left(\left(\left(z3 \cdot z2\right) \cdot z4\right) \cdot z2\right) \cdot z2\right) \cdot \left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right)}\right) - z5 \cdot z5} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \frac{z0 \cdot z1}{\left(\left(\left(\left(z3 \cdot z2\right) \cdot z4\right) \cdot z2\right) \cdot z2\right) \cdot \color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)}}\right) - z5 \cdot z5} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \frac{z0 \cdot z1}{\left(\left(\left(\left(z3 \cdot z2\right) \cdot z4\right) \cdot z2\right) \cdot z2\right) \cdot \color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)}}\right) - z5 \cdot z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \frac{z0 \cdot z1}{\left(\left(\left(\left(z3 \cdot z2\right) \cdot z4\right) \cdot z2\right) \cdot z2\right) \cdot \left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z4\right)}\right) - z5 \cdot z5} \]

      6. lower-*.f64 47.2%

         \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \frac{z0 \cdot z1}{\left(\left(\left(\left(z3 \cdot z2\right) \cdot z4\right) \cdot z2\right) \cdot z2\right) \cdot \left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z4\right)}\right) - z5 \cdot z5} \]

   7. Applied rewrites 47.2%

      \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \frac{z0 \cdot z1}{\left(\left(\left(\left(z3 \cdot z2\right) \cdot z4\right) \cdot z2\right) \cdot z2\right) \cdot \color{blue}{\left(\left(z3 \cdot z2\right) \cdot z4\right)}}\right) - z5 \cdot z5} \]

   if +inf.0 < (-.f64 (/.f64 (*.f64 (*.f64 (*.f64 z1 z0) z0) (*.f64 (*.f64 z1 z0) z0)) (*.f64 (*.f64 (*.f64 (*.f64 z4 z3) z2) z2) (*.f64 (*.f64 (*.f64 z4 z3) z2) z2))) (*.f64 z5 z5))

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{z5}} \]
   5. Step-by-step derivation

   6. Applied rewrites 14.8%

      \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{z5}} \]

   8. Applied rewrites 30.9%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      3. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{\frac{z1}{\left(z3 \cdot z4\right) \cdot z2}}{z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      13. associate-/r* N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)} \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      16. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)} \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      17. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z4\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z4\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      19. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      20. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(z2 \cdot z4\right)}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      21. associate-/r* N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{\frac{z1}{z3 \cdot z2}}{z2 \cdot z4}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      22. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{z3 \cdot z2}}{\color{blue}{z4 \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   10. Applied rewrites 32.0%

       \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{\frac{z1}{z3 \cdot z2}}{z4 \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 66.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := z0 \cdot \left|z1\right|\\ t_1 := \left(\left(\left|z4\right| \cdot \left|z3\right|\right) \cdot z2\right) \cdot z2\\ t_2 := z2 \cdot \left(\left|z3\right| \cdot \left|z4\right|\right)\\ t_3 := \left|z5\right| \cdot \left|z5\right|\\ t_4 := \left(\left|z1\right| \cdot z0\right) \cdot z0\\ \mathbf{if}\;\sqrt{\frac{t\_4 \cdot t\_4}{t\_1 \cdot t\_1} - t\_3} \leq \infty:\\ \;\;\;\;\sqrt{t\_0 \cdot \left(\left(z0 \cdot z0\right) \cdot \frac{t\_0}{\left(\left(t\_2 \cdot z2\right) \cdot z2\right) \cdot t\_2}\right) - t\_3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{\frac{\left|z1\right|}{\left|z3\right| \cdot z2}}{\left|z4\right| \cdot z2} \cdot z0\right) \cdot z0 - \left|z5\right|\right) \cdot \left|z5\right|}\\ \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0 (* z0 (fabs z1)))
       (t_1 (* (* (* (fabs z4) (fabs z3)) z2) z2))
       (t_2 (* z2 (* (fabs z3) (fabs z4))))
       (t_3 (* (fabs z5) (fabs z5)))
       (t_4 (* (* (fabs z1) z0) z0)))
  (if (<= (sqrt (- (/ (* t_4 t_4) (* t_1 t_1)) t_3)) INFINITY)
    (sqrt
     (- (* t_0 (* (* z0 z0) (/ t_0 (* (* (* t_2 z2) z2) t_2)))) t_3))
    (sqrt
     (*
      (-
       (*
        (* (/ (/ (fabs z1) (* (fabs z3) z2)) (* (fabs z4) z2)) z0)
        z0)
       (fabs z5))
      (fabs z5))))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = z0 * fabs(z1);
	double t_1 = ((fabs(z4) * fabs(z3)) * z2) * z2;
	double t_2 = z2 * (fabs(z3) * fabs(z4));
	double t_3 = fabs(z5) * fabs(z5);
	double t_4 = (fabs(z1) * z0) * z0;
	double tmp;
	if (sqrt((((t_4 * t_4) / (t_1 * t_1)) - t_3)) <= ((double) INFINITY)) {
		tmp = sqrt(((t_0 * ((z0 * z0) * (t_0 / (((t_2 * z2) * z2) * t_2)))) - t_3));
	} else {
		tmp = sqrt(((((((fabs(z1) / (fabs(z3) * z2)) / (fabs(z4) * z2)) * z0) * z0) - fabs(z5)) * fabs(z5)));
	}
	return tmp;
}

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = z0 * Math.abs(z1);
	double t_1 = ((Math.abs(z4) * Math.abs(z3)) * z2) * z2;
	double t_2 = z2 * (Math.abs(z3) * Math.abs(z4));
	double t_3 = Math.abs(z5) * Math.abs(z5);
	double t_4 = (Math.abs(z1) * z0) * z0;
	double tmp;
	if (Math.sqrt((((t_4 * t_4) / (t_1 * t_1)) - t_3)) <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((t_0 * ((z0 * z0) * (t_0 / (((t_2 * z2) * z2) * t_2)))) - t_3));
	} else {
		tmp = Math.sqrt(((((((Math.abs(z1) / (Math.abs(z3) * z2)) / (Math.abs(z4) * z2)) * z0) * z0) - Math.abs(z5)) * Math.abs(z5)));
	}
	return tmp;
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = z0 * math.fabs(z1)
	t_1 = ((math.fabs(z4) * math.fabs(z3)) * z2) * z2
	t_2 = z2 * (math.fabs(z3) * math.fabs(z4))
	t_3 = math.fabs(z5) * math.fabs(z5)
	t_4 = (math.fabs(z1) * z0) * z0
	tmp = 0
	if math.sqrt((((t_4 * t_4) / (t_1 * t_1)) - t_3)) <= math.inf:
		tmp = math.sqrt(((t_0 * ((z0 * z0) * (t_0 / (((t_2 * z2) * z2) * t_2)))) - t_3))
	else:
		tmp = math.sqrt(((((((math.fabs(z1) / (math.fabs(z3) * z2)) / (math.fabs(z4) * z2)) * z0) * z0) - math.fabs(z5)) * math.fabs(z5)))
	return tmp

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = Float64(z0 * abs(z1))
	t_1 = Float64(Float64(Float64(abs(z4) * abs(z3)) * z2) * z2)
	t_2 = Float64(z2 * Float64(abs(z3) * abs(z4)))
	t_3 = Float64(abs(z5) * abs(z5))
	t_4 = Float64(Float64(abs(z1) * z0) * z0)
	tmp = 0.0
	if (sqrt(Float64(Float64(Float64(t_4 * t_4) / Float64(t_1 * t_1)) - t_3)) <= Inf)
		tmp = sqrt(Float64(Float64(t_0 * Float64(Float64(z0 * z0) * Float64(t_0 / Float64(Float64(Float64(t_2 * z2) * z2) * t_2)))) - t_3));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(Float64(abs(z1) / Float64(abs(z3) * z2)) / Float64(abs(z4) * z2)) * z0) * z0) - abs(z5)) * abs(z5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z4, z3, z2, z5)
	t_0 = z0 * abs(z1);
	t_1 = ((abs(z4) * abs(z3)) * z2) * z2;
	t_2 = z2 * (abs(z3) * abs(z4));
	t_3 = abs(z5) * abs(z5);
	t_4 = (abs(z1) * z0) * z0;
	tmp = 0.0;
	if (sqrt((((t_4 * t_4) / (t_1 * t_1)) - t_3)) <= Inf)
		tmp = sqrt(((t_0 * ((z0 * z0) * (t_0 / (((t_2 * z2) * z2) * t_2)))) - t_3));
	else
		tmp = sqrt(((((((abs(z1) / (abs(z3) * z2)) / (abs(z4) * z2)) * z0) * z0) - abs(z5)) * abs(z5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[(z0 * N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Abs[z4], $MachinePrecision] * N[Abs[z3], $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision] * z2), $MachinePrecision]}, Block[{t$95$2 = N[(z2 * N[(N[Abs[z3], $MachinePrecision] * N[Abs[z4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[z5], $MachinePrecision] * N[Abs[z5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Abs[z1], $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(N[(N[(t$95$4 * t$95$4), $MachinePrecision] / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision], Infinity], N[Sqrt[N[(N[(t$95$0 * N[(N[(z0 * z0), $MachinePrecision] * N[(t$95$0 / N[(N[(N[(t$95$2 * z2), $MachinePrecision] * z2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(N[(N[Abs[z1], $MachinePrecision] / N[(N[Abs[z3], $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[z4], $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision] - N[Abs[z5], $MachinePrecision]), $MachinePrecision] * N[Abs[z5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (-.f64 (/.f64 (*.f64 (*.f64 (*.f64 z1 z0) z0) (*.f64 (*.f64 z1 z0) z0)) (*.f64 (*.f64 (*.f64 (*.f64 z4 z3) z2) z2) (*.f64 (*.f64 (*.f64 z4 z3) z2) z2))) (*.f64 z5 z5))) < +inf.0

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      2. mult-flip N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)\right) \cdot \frac{1}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)\right)} \cdot \frac{1}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right)} \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)\right) \cdot \frac{1}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot \left(z0 \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)\right)\right)} \cdot \frac{1}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \left(\left(z0 \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)\right) \cdot \frac{1}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}\right)} - z5 \cdot z5} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right) \cdot \left(\left(z0 \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)\right) \cdot \frac{1}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}\right)} - z5 \cdot z5} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(z1 \cdot z0\right)} \cdot \left(\left(z0 \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)\right) \cdot \frac{1}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}\right) - z5 \cdot z5} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(z0 \cdot z1\right)} \cdot \left(\left(z0 \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)\right) \cdot \frac{1}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}\right) - z5 \cdot z5} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\left(z0 \cdot z1\right)} \cdot \left(\left(z0 \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)\right) \cdot \frac{1}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}\right) - z5 \cdot z5} \]

      11. mult-flip-rev N/A

          \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \color{blue}{\frac{z0 \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \frac{z0 \cdot \color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      13. *-commutative N/A

          \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \frac{z0 \cdot \color{blue}{\left(z0 \cdot \left(z1 \cdot z0\right)\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      14. associate-*r* N/A

          \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \frac{\color{blue}{\left(z0 \cdot z0\right) \cdot \left(z1 \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      15. associate-/l* N/A

          \[\leadsto \sqrt{\left(z0 \cdot z1\right) \cdot \color{blue}{\left(\left(z0 \cdot z0\right) \cdot \frac{z1 \cdot z0}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}\right)} - z5 \cdot z5} \]

   3. Applied rewrites 45.2%

      \[\leadsto \sqrt{\color{blue}{\left(z0 \cdot z1\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \frac{z0 \cdot z1}{\left(\left(\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2\right) \cdot z2\right) \cdot \left(z2 \cdot \left(z3 \cdot z4\right)\right)}\right)} - z5 \cdot z5} \]

   if +inf.0 < (sqrt.f64 (-.f64 (/.f64 (*.f64 (*.f64 (*.f64 z1 z0) z0) (*.f64 (*.f64 z1 z0) z0)) (*.f64 (*.f64 (*.f64 (*.f64 z4 z3) z2) z2) (*.f64 (*.f64 (*.f64 z4 z3) z2) z2))) (*.f64 z5 z5)))

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{z5}} \]
   5. Step-by-step derivation

   6. Applied rewrites 14.8%

      \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{z5}} \]

   8. Applied rewrites 30.9%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      3. associate-/r* N/A

         \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{\frac{z1}{\left(z3 \cdot z4\right) \cdot z2}}{z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      13. associate-/r* N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      14. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)} \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      15. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      16. associate-*r* N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)} \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      17. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z4\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z4\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      19. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      20. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(z2 \cdot z4\right)}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      21. associate-/r* N/A

          \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{\frac{z1}{z3 \cdot z2}}{z2 \cdot z4}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      22. *-commutative N/A

          \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{z3 \cdot z2}}{\color{blue}{z4 \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   10. Applied rewrites 32.0%

       \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{\frac{z1}{z3 \cdot z2}}{z4 \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 63.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(\left|z4\right| \cdot \left|z3\right|\right) \cdot z2\right) \cdot z2\\ \mathbf{if}\;t\_0 \leq 10^{+269}:\\ \;\;\;\;\sqrt{\left(\left(\left|z1\right| \cdot \frac{z0}{t\_0}\right) \cdot z0 - \left|z5\right|\right) \cdot \left|z5\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{\left|z1\right|}{\left|z3\right| \cdot \left(\left(z2 \cdot z2\right) \cdot \left|z4\right|\right)} \cdot z0\right) \cdot z0 - \left|z5\right|\right) \cdot \left|z5\right|}\\ \end{array} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (let* ((t_0 (* (* (* (fabs z4) (fabs z3)) z2) z2)))
  (if (<= t_0 1e+269)
    (sqrt (* (- (* (* (fabs z1) (/ z0 t_0)) z0) (fabs z5)) (fabs z5)))
    (sqrt
     (*
      (-
       (*
        (* (/ (fabs z1) (* (fabs z3) (* (* z2 z2) (fabs z4)))) z0)
        z0)
       (fabs z5))
      (fabs z5))))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = ((fabs(z4) * fabs(z3)) * z2) * z2;
	double tmp;
	if (t_0 <= 1e+269) {
		tmp = sqrt(((((fabs(z1) * (z0 / t_0)) * z0) - fabs(z5)) * fabs(z5)));
	} else {
		tmp = sqrt((((((fabs(z1) / (fabs(z3) * ((z2 * z2) * fabs(z4)))) * z0) * z0) - fabs(z5)) * fabs(z5)));
	}
	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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((abs(z4) * abs(z3)) * z2) * z2
    if (t_0 <= 1d+269) then
        tmp = sqrt(((((abs(z1) * (z0 / t_0)) * z0) - abs(z5)) * abs(z5)))
    else
        tmp = sqrt((((((abs(z1) / (abs(z3) * ((z2 * z2) * abs(z4)))) * z0) * z0) - abs(z5)) * abs(z5)))
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	double t_0 = ((Math.abs(z4) * Math.abs(z3)) * z2) * z2;
	double tmp;
	if (t_0 <= 1e+269) {
		tmp = Math.sqrt(((((Math.abs(z1) * (z0 / t_0)) * z0) - Math.abs(z5)) * Math.abs(z5)));
	} else {
		tmp = Math.sqrt((((((Math.abs(z1) / (Math.abs(z3) * ((z2 * z2) * Math.abs(z4)))) * z0) * z0) - Math.abs(z5)) * Math.abs(z5)));
	}
	return tmp;
}

Python

def code(z1, z0, z4, z3, z2, z5):
	t_0 = ((math.fabs(z4) * math.fabs(z3)) * z2) * z2
	tmp = 0
	if t_0 <= 1e+269:
		tmp = math.sqrt(((((math.fabs(z1) * (z0 / t_0)) * z0) - math.fabs(z5)) * math.fabs(z5)))
	else:
		tmp = math.sqrt((((((math.fabs(z1) / (math.fabs(z3) * ((z2 * z2) * math.fabs(z4)))) * z0) * z0) - math.fabs(z5)) * math.fabs(z5)))
	return tmp

Julia

function code(z1, z0, z4, z3, z2, z5)
	t_0 = Float64(Float64(Float64(abs(z4) * abs(z3)) * z2) * z2)
	tmp = 0.0
	if (t_0 <= 1e+269)
		tmp = sqrt(Float64(Float64(Float64(Float64(abs(z1) * Float64(z0 / t_0)) * z0) - abs(z5)) * abs(z5)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(abs(z1) / Float64(abs(z3) * Float64(Float64(z2 * z2) * abs(z4)))) * z0) * z0) - abs(z5)) * abs(z5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z4, z3, z2, z5)
	t_0 = ((abs(z4) * abs(z3)) * z2) * z2;
	tmp = 0.0;
	if (t_0 <= 1e+269)
		tmp = sqrt(((((abs(z1) * (z0 / t_0)) * z0) - abs(z5)) * abs(z5)));
	else
		tmp = sqrt((((((abs(z1) / (abs(z3) * ((z2 * z2) * abs(z4)))) * z0) * z0) - abs(z5)) * abs(z5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := Block[{t$95$0 = N[(N[(N[(N[Abs[z4], $MachinePrecision] * N[Abs[z3], $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision] * z2), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+269], N[Sqrt[N[(N[(N[(N[(N[Abs[z1], $MachinePrecision] * N[(z0 / t$95$0), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - N[Abs[z5], $MachinePrecision]), $MachinePrecision] * N[Abs[z5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(N[Abs[z1], $MachinePrecision] / N[(N[Abs[z3], $MachinePrecision] * N[(N[(z2 * z2), $MachinePrecision] * N[Abs[z4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision] - N[Abs[z5], $MachinePrecision]), $MachinePrecision] * N[Abs[z5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 z4 z3) z2) z2) < 1e269

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{z5}} \]
   5. Step-by-step derivation

   6. Applied rewrites 14.8%

      \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{z5}} \]

   8. Applied rewrites 30.9%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2} \cdot z0\right)} \cdot z0 - z5\right) \cdot z5} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      3. associate-*l/ N/A

         \[\leadsto \sqrt{\left(\color{blue}{\frac{z1 \cdot z0}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2}} \cdot z0 - z5\right) \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{z1 \cdot z0}{\left(\color{blue}{\left(z3 \cdot z4\right)} \cdot z2\right) \cdot z2} \cdot z0 - z5\right) \cdot z5} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\frac{z1 \cdot z0}{\left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right) \cdot z2} \cdot z0 - z5\right) \cdot z5} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{z1 \cdot z0}{\left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right) \cdot z2} \cdot z0 - z5\right) \cdot z5} \]

      7. associate-/l* N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(z1 \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} \cdot z0 - z5\right) \cdot z5} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(z1 \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} \cdot z0 - z5\right) \cdot z5} \]

      9. lower-/.f64 30.9%

         \[\leadsto \sqrt{\left(\left(z1 \cdot \color{blue}{\frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}}\right) \cdot z0 - z5\right) \cdot z5} \]

   10. Applied rewrites 30.9%

       \[\leadsto \sqrt{\left(\color{blue}{\left(z1 \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} \cdot z0 - z5\right) \cdot z5} \]

   if 1e269 < (*.f64 (*.f64 (*.f64 z4 z3) z2) z2)

   1. Initial program 43.7%

      \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

      6. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

      8. sqr-neg-rev N/A

         \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

      9. difference-of-squares N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

   3. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]

   4. Taylor expanded in z1 around 0

      \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{z5}} \]
   5. Step-by-step derivation

   6. Applied rewrites 14.8%

      \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{z5}} \]

   8. Applied rewrites 30.9%

      \[\leadsto \color{blue}{\sqrt{\left(\left(\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right)} \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)} \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)} \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z4\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z4\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      8. associate-*l* N/A

         \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(z2 \cdot z4\right)}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      10. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(\left(z3 \cdot z2\right) \cdot z2\right) \cdot z4}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      11. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z2\right) \cdot z4} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      12. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(z3 \cdot \left(z2 \cdot z2\right)\right)} \cdot z4} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      13. associate-*l* N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{z3 \cdot \left(\left(z2 \cdot z2\right) \cdot z4\right)}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      14. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{z3 \cdot \left(\left(z2 \cdot z2\right) \cdot z4\right)}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\frac{z1}{z3 \cdot \color{blue}{\left(\left(z2 \cdot z2\right) \cdot z4\right)}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

      16. lower-*.f64 30.9%

          \[\leadsto \sqrt{\left(\left(\frac{z1}{z3 \cdot \left(\color{blue}{\left(z2 \cdot z2\right)} \cdot z4\right)} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   10. Applied rewrites 30.9%

       \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{z3 \cdot \left(\left(z2 \cdot z2\right) \cdot z4\right)}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 62.7% accurate, 1.3× speedup

Math

\[\sqrt{\left(\left(\frac{\frac{\left|z1\right|}{\left|z3\right| \cdot z2}}{\left|z4\right| \cdot z2} \cdot z0\right) \cdot z0 - \left|z5\right|\right) \cdot \left|z5\right|} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (sqrt
 (*
  (-
   (* (* (/ (/ (fabs z1) (* (fabs z3) z2)) (* (fabs z4) z2)) z0) z0)
   (fabs z5))
  (fabs z5))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	return sqrt(((((((fabs(z1) / (fabs(z3) * z2)) / (fabs(z4) * z2)) * z0) * z0) - fabs(z5)) * fabs(z5)));
}

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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    code = sqrt(((((((abs(z1) / (abs(z3) * z2)) / (abs(z4) * z2)) * z0) * z0) - abs(z5)) * abs(z5)))
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	return Math.sqrt(((((((Math.abs(z1) / (Math.abs(z3) * z2)) / (Math.abs(z4) * z2)) * z0) * z0) - Math.abs(z5)) * Math.abs(z5)));
}

Python

def code(z1, z0, z4, z3, z2, z5):
	return math.sqrt(((((((math.fabs(z1) / (math.fabs(z3) * z2)) / (math.fabs(z4) * z2)) * z0) * z0) - math.fabs(z5)) * math.fabs(z5)))

Julia

function code(z1, z0, z4, z3, z2, z5)
	return sqrt(Float64(Float64(Float64(Float64(Float64(Float64(abs(z1) / Float64(abs(z3) * z2)) / Float64(abs(z4) * z2)) * z0) * z0) - abs(z5)) * abs(z5)))
end

MATLAB

function tmp = code(z1, z0, z4, z3, z2, z5)
	tmp = sqrt(((((((abs(z1) / (abs(z3) * z2)) / (abs(z4) * z2)) * z0) * z0) - abs(z5)) * abs(z5)));
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := N[Sqrt[N[(N[(N[(N[(N[(N[(N[Abs[z1], $MachinePrecision] / N[(N[Abs[z3], $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[z4], $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision] - N[Abs[z5], $MachinePrecision]), $MachinePrecision] * N[Abs[z5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 43.7%

   \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
2. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

   2. lift--.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

   3. lift-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

   5. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

   6. times-frac N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

   7. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

   8. sqr-neg-rev N/A

      \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

   9. difference-of-squares N/A

      \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

3. Applied rewrites 34.4%

   \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]

4. Taylor expanded in z1 around 0

   \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{z5}} \]
5. Step-by-step derivation

6. Applied rewrites 14.8%

   \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{z5}} \]

8. Applied rewrites 30.9%

   \[\leadsto \color{blue}{\sqrt{\left(\left(\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   2. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   3. associate-/r* N/A

      \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{\frac{z1}{\left(z3 \cdot z4\right) \cdot z2}}{z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   6. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   7. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z4 \cdot z3\right) \cdot z2}}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   8. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z4 \cdot z3\right)} \cdot z2}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   9. *-commutative N/A

      \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   10. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{\left(z3 \cdot z4\right)} \cdot z2}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   11. *-commutative N/A

       \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   12. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{\color{blue}{z2 \cdot \left(z3 \cdot z4\right)}}}{z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   13. associate-/r* N/A

       \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   14. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)} \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   15. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   16. associate-*r* N/A

       \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)} \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   17. *-commutative N/A

       \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z4\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   18. lift-*.f64 N/A

       \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z4\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   19. associate-*l* N/A

       \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(z3 \cdot z2\right) \cdot \left(z4 \cdot z2\right)}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   20. *-commutative N/A

       \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(z3 \cdot z2\right) \cdot \color{blue}{\left(z2 \cdot z4\right)}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   21. associate-/r* N/A

       \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{\frac{z1}{z3 \cdot z2}}{z2 \cdot z4}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   22. *-commutative N/A

       \[\leadsto \sqrt{\left(\left(\frac{\frac{z1}{z3 \cdot z2}}{\color{blue}{z4 \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

10. Applied rewrites 32.0%

    \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{\frac{z1}{z3 \cdot z2}}{z4 \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]
11. Add Preprocessing

Alternative 21: 62.4% accurate, 0.3× speedup

Math

\[\sqrt{\left(\left(\frac{\left|z1\right|}{\left(\left(\mathsf{max}\left(\left|z4\right|, \left|z3\right|\right) \cdot z2\right) \cdot \mathsf{min}\left(\left|z4\right|, \left|z3\right|\right)\right) \cdot z2} \cdot z0\right) \cdot z0 - \left|z5\right|\right) \cdot \left|z5\right|} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (sqrt
 (*
  (-
   (*
    (*
     (/
      (fabs z1)
      (*
       (*
        (* (fmax (fabs z4) (fabs z3)) z2)
        (fmin (fabs z4) (fabs z3)))
       z2))
     z0)
    z0)
   (fabs z5))
  (fabs z5))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	return sqrt((((((fabs(z1) / (((fmax(fabs(z4), fabs(z3)) * z2) * fmin(fabs(z4), fabs(z3))) * z2)) * z0) * z0) - fabs(z5)) * fabs(z5)));
}

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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    code = sqrt((((((abs(z1) / (((fmax(abs(z4), abs(z3)) * z2) * fmin(abs(z4), abs(z3))) * z2)) * z0) * z0) - abs(z5)) * abs(z5)))
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	return Math.sqrt((((((Math.abs(z1) / (((fmax(Math.abs(z4), Math.abs(z3)) * z2) * fmin(Math.abs(z4), Math.abs(z3))) * z2)) * z0) * z0) - Math.abs(z5)) * Math.abs(z5)));
}

Python

def code(z1, z0, z4, z3, z2, z5):
	return math.sqrt((((((math.fabs(z1) / (((fmax(math.fabs(z4), math.fabs(z3)) * z2) * fmin(math.fabs(z4), math.fabs(z3))) * z2)) * z0) * z0) - math.fabs(z5)) * math.fabs(z5)))

Julia

function code(z1, z0, z4, z3, z2, z5)
	return sqrt(Float64(Float64(Float64(Float64(Float64(abs(z1) / Float64(Float64(Float64(fmax(abs(z4), abs(z3)) * z2) * fmin(abs(z4), abs(z3))) * z2)) * z0) * z0) - abs(z5)) * abs(z5)))
end

MATLAB

function tmp = code(z1, z0, z4, z3, z2, z5)
	tmp = sqrt((((((abs(z1) / (((max(abs(z4), abs(z3)) * z2) * min(abs(z4), abs(z3))) * z2)) * z0) * z0) - abs(z5)) * abs(z5)));
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := N[Sqrt[N[(N[(N[(N[(N[(N[Abs[z1], $MachinePrecision] / N[(N[(N[(N[Max[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision] * z2), $MachinePrecision] * N[Min[N[Abs[z4], $MachinePrecision], N[Abs[z3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] * z0), $MachinePrecision] - N[Abs[z5], $MachinePrecision]), $MachinePrecision] * N[Abs[z5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 43.7%

   \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
2. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

   2. lift--.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

   3. lift-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

   5. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

   6. times-frac N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

   7. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

   8. sqr-neg-rev N/A

      \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

   9. difference-of-squares N/A

      \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

3. Applied rewrites 34.4%

   \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]

4. Taylor expanded in z1 around 0

   \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{z5}} \]
5. Step-by-step derivation

6. Applied rewrites 14.8%

   \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{z5}} \]

8. Applied rewrites 30.9%

   \[\leadsto \color{blue}{\sqrt{\left(\left(\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(\left(z3 \cdot z4\right) \cdot z2\right)} \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   2. *-commutative N/A

      \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(z2 \cdot \left(z3 \cdot z4\right)\right)} \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   3. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(z2 \cdot \color{blue}{\left(z3 \cdot z4\right)}\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   4. associate-*r* N/A

      \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(\left(z2 \cdot z3\right) \cdot z4\right)} \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z4\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   6. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\left(\frac{z1}{\left(\color{blue}{\left(z3 \cdot z2\right)} \cdot z4\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   7. lower-*.f64 32.0%

      \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(\left(z3 \cdot z2\right) \cdot z4\right)} \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

10. Applied rewrites 32.0%

    \[\leadsto \sqrt{\left(\left(\frac{z1}{\color{blue}{\left(\left(z3 \cdot z2\right) \cdot z4\right)} \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]
11. Add Preprocessing

Alternative 22: 60.9% accurate, 1.4× speedup

Math

\[\sqrt{\left(\left(\left|z1\right| \cdot \frac{z0}{\left(\left(\left|z4\right| \cdot \left|z3\right|\right) \cdot z2\right) \cdot z2}\right) \cdot z0 - \left|z5\right|\right) \cdot \left|z5\right|} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (sqrt
 (*
  (-
   (* (* (fabs z1) (/ z0 (* (* (* (fabs z4) (fabs z3)) z2) z2))) z0)
   (fabs z5))
  (fabs z5))))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	return sqrt(((((fabs(z1) * (z0 / (((fabs(z4) * fabs(z3)) * z2) * z2))) * z0) - fabs(z5)) * fabs(z5)));
}

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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    code = sqrt(((((abs(z1) * (z0 / (((abs(z4) * abs(z3)) * z2) * z2))) * z0) - abs(z5)) * abs(z5)))
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	return Math.sqrt(((((Math.abs(z1) * (z0 / (((Math.abs(z4) * Math.abs(z3)) * z2) * z2))) * z0) - Math.abs(z5)) * Math.abs(z5)));
}

Python

def code(z1, z0, z4, z3, z2, z5):
	return math.sqrt(((((math.fabs(z1) * (z0 / (((math.fabs(z4) * math.fabs(z3)) * z2) * z2))) * z0) - math.fabs(z5)) * math.fabs(z5)))

Julia

function code(z1, z0, z4, z3, z2, z5)
	return sqrt(Float64(Float64(Float64(Float64(abs(z1) * Float64(z0 / Float64(Float64(Float64(abs(z4) * abs(z3)) * z2) * z2))) * z0) - abs(z5)) * abs(z5)))
end

MATLAB

function tmp = code(z1, z0, z4, z3, z2, z5)
	tmp = sqrt(((((abs(z1) * (z0 / (((abs(z4) * abs(z3)) * z2) * z2))) * z0) - abs(z5)) * abs(z5)));
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := N[Sqrt[N[(N[(N[(N[(N[Abs[z1], $MachinePrecision] * N[(z0 / N[(N[(N[(N[Abs[z4], $MachinePrecision] * N[Abs[z3], $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - N[Abs[z5], $MachinePrecision]), $MachinePrecision] * N[Abs[z5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 43.7%

   \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]
2. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

   2. lift--.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5}} \]

   3. lift-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

   5. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\color{blue}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)}} - z5 \cdot z5} \]

   6. times-frac N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}} - z5 \cdot z5} \]

   7. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{z5 \cdot z5}} \]

   8. sqr-neg-rev N/A

      \[\leadsto \sqrt{\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} \cdot \frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \color{blue}{\left(\mathsf{neg}\left(z5\right)\right) \cdot \left(\mathsf{neg}\left(z5\right)\right)}} \]

   9. difference-of-squares N/A

      \[\leadsto \sqrt{\color{blue}{\left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} + \left(\mathsf{neg}\left(z5\right)\right)\right) \cdot \left(\frac{\left(z1 \cdot z0\right) \cdot z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2} - \left(\mathsf{neg}\left(z5\right)\right)\right)}} \]

3. Applied rewrites 34.4%

   \[\leadsto \color{blue}{\sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} + z5}} \]

4. Taylor expanded in z1 around 0

   \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{z5}} \]
5. Step-by-step derivation

6. Applied rewrites 14.8%

   \[\leadsto \sqrt{\left(z0 \cdot z0\right) \cdot \frac{z1}{\left(z2 \cdot \left(z3 \cdot z4\right)\right) \cdot z2} - z5} \cdot \sqrt{\color{blue}{z5}} \]

8. Applied rewrites 30.9%

   \[\leadsto \color{blue}{\sqrt{\left(\left(\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2} \cdot z0\right) \cdot z0 - z5\right) \cdot z5}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\color{blue}{\left(\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2} \cdot z0\right)} \cdot z0 - z5\right) \cdot z5} \]

   2. lift-/.f64 N/A

      \[\leadsto \sqrt{\left(\left(\color{blue}{\frac{z1}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2}} \cdot z0\right) \cdot z0 - z5\right) \cdot z5} \]

   3. associate-*l/ N/A

      \[\leadsto \sqrt{\left(\color{blue}{\frac{z1 \cdot z0}{\left(\left(z3 \cdot z4\right) \cdot z2\right) \cdot z2}} \cdot z0 - z5\right) \cdot z5} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\frac{z1 \cdot z0}{\left(\color{blue}{\left(z3 \cdot z4\right)} \cdot z2\right) \cdot z2} \cdot z0 - z5\right) \cdot z5} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\left(\frac{z1 \cdot z0}{\left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right) \cdot z2} \cdot z0 - z5\right) \cdot z5} \]

   6. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\frac{z1 \cdot z0}{\left(\color{blue}{\left(z4 \cdot z3\right)} \cdot z2\right) \cdot z2} \cdot z0 - z5\right) \cdot z5} \]

   7. associate-/l* N/A

      \[\leadsto \sqrt{\left(\color{blue}{\left(z1 \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} \cdot z0 - z5\right) \cdot z5} \]

   8. lower-*.f64 N/A

      \[\leadsto \sqrt{\left(\color{blue}{\left(z1 \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} \cdot z0 - z5\right) \cdot z5} \]

   9. lower-/.f64 30.9%

      \[\leadsto \sqrt{\left(\left(z1 \cdot \color{blue}{\frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}}\right) \cdot z0 - z5\right) \cdot z5} \]

10. Applied rewrites 30.9%

    \[\leadsto \sqrt{\left(\color{blue}{\left(z1 \cdot \frac{z0}{\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2}\right)} \cdot z0 - z5\right) \cdot z5} \]
11. Add Preprocessing

Alternative 23: 18.7% accurate, 5.6× speedup

Math

\[\sqrt{z5 \cdot z5} \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (sqrt (* z5 z5)))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	return sqrt((z5 * z5));
}

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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    code = sqrt((z5 * z5))
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	return Math.sqrt((z5 * z5));
}

Python

def code(z1, z0, z4, z3, z2, z5):
	return math.sqrt((z5 * z5))

Julia

function code(z1, z0, z4, z3, z2, z5)
	return sqrt(Float64(z5 * z5))
end

MATLAB

function tmp = code(z1, z0, z4, z3, z2, z5)
	tmp = sqrt((z5 * z5));
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := N[Sqrt[N[(z5 * z5), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 43.7%

   \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

2. Taylor expanded in z1 around 0

   \[\leadsto \color{blue}{\sqrt{\mathsf{neg}\left({z5}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left({z5}^{2}\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto \sqrt{-{z5}^{2}} \]

   3. lower-pow.f64 0.7%

      \[\leadsto \sqrt{-{z5}^{2}} \]

4. Applied rewrites 0.7%

   \[\leadsto \color{blue}{\sqrt{-{z5}^{2}}} \]
5. Step-by-step derivation

   1. rem-square-sqrt N/A

      \[\leadsto \sqrt{\sqrt{-{z5}^{2}} \cdot \sqrt{-{z5}^{2}}} \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{\sqrt{-{z5}^{2}} \cdot \sqrt{-{z5}^{2}}} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{\sqrt{-{z5}^{2}} \cdot \sqrt{-{z5}^{2}}} \]

   4. sqr-neg-rev N/A

      \[\leadsto \sqrt{\left(\mathsf{neg}\left(\sqrt{-{z5}^{2}}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{-{z5}^{2}}\right)\right)} \]

   5. sqr-abs N/A

      \[\leadsto \sqrt{\left|\mathsf{neg}\left(\sqrt{-{z5}^{2}}\right)\right| \cdot \left|\mathsf{neg}\left(\sqrt{-{z5}^{2}}\right)\right|} \]

   6. mul-fabs N/A

      \[\leadsto \sqrt{\left|\left(\mathsf{neg}\left(\sqrt{-{z5}^{2}}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{-{z5}^{2}}\right)\right)\right|} \]

   7. sqr-neg-rev N/A

      \[\leadsto \sqrt{\left|\sqrt{-{z5}^{2}} \cdot \sqrt{-{z5}^{2}}\right|} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{\left|\sqrt{-{z5}^{2}} \cdot \sqrt{-{z5}^{2}}\right|} \]

   9. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{\left|\sqrt{-{z5}^{2}} \cdot \sqrt{-{z5}^{2}}\right|} \]

   10. rem-square-sqrt N/A

       \[\leadsto \sqrt{\left|-{z5}^{2}\right|} \]

   11. lift-neg.f64 N/A

       \[\leadsto \sqrt{\left|\mathsf{neg}\left({z5}^{2}\right)\right|} \]

   12. lift-pow.f64 N/A

       \[\leadsto \sqrt{\left|\mathsf{neg}\left({z5}^{2}\right)\right|} \]

   13. pow2 N/A

       \[\leadsto \sqrt{\left|\mathsf{neg}\left(z5 \cdot z5\right)\right|} \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \sqrt{\left|z5 \cdot \left(\mathsf{neg}\left(z5\right)\right)\right|} \]

   15. mul-fabs N/A

       \[\leadsto \sqrt{\left|z5\right| \cdot \left|\mathsf{neg}\left(z5\right)\right|} \]

   16. neg-fabs N/A

       \[\leadsto \sqrt{\left|z5\right| \cdot \left|z5\right|} \]

   17. sqr-abs-rev N/A

       \[\leadsto \sqrt{z5 \cdot z5} \]

   18. lower-*.f64 18.7%

       \[\leadsto \sqrt{z5 \cdot z5} \]

6. Applied rewrites 18.7%

   \[\leadsto \sqrt{z5 \cdot z5} \]
7. Add Preprocessing

Alternative 24: 4.2% accurate, 30.0× speedup

Math

\[\left|z5\right| \]

FPCore

(FPCore (z1 z0 z4 z3 z2 z5)
  :precision binary64
  (fabs z5))

C

double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	return fabs(z5);
}

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(z1, z0, z4, z3, z2, z5)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z4
    real(8), intent (in) :: z3
    real(8), intent (in) :: z2
    real(8), intent (in) :: z5
    code = abs(z5)
end function

Java

public static double code(double z1, double z0, double z4, double z3, double z2, double z5) {
	return Math.abs(z5);
}

Python

def code(z1, z0, z4, z3, z2, z5):
	return math.fabs(z5)

Julia

function code(z1, z0, z4, z3, z2, z5)
	return abs(z5)
end

MATLAB

function tmp = code(z1, z0, z4, z3, z2, z5)
	tmp = abs(z5);
end

Wolfram

code[z1_, z0_, z4_, z3_, z2_, z5_] := N[Abs[z5], $MachinePrecision]

Derivation

1. Initial program 43.7%

   \[\sqrt{\frac{\left(\left(z1 \cdot z0\right) \cdot z0\right) \cdot \left(\left(z1 \cdot z0\right) \cdot z0\right)}{\left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right) \cdot \left(\left(\left(z4 \cdot z3\right) \cdot z2\right) \cdot z2\right)} - z5 \cdot z5} \]

2. Taylor expanded in z1 around 0

   \[\leadsto \color{blue}{\sqrt{\mathsf{neg}\left({z5}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left({z5}^{2}\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto \sqrt{-{z5}^{2}} \]

   3. lower-pow.f64 0.7%

      \[\leadsto \sqrt{-{z5}^{2}} \]

4. Applied rewrites 0.7%

   \[\leadsto \color{blue}{\sqrt{-{z5}^{2}}} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{-{z5}^{2}} \]

   2. sqrt-fabs-rev N/A

      \[\leadsto \left|\sqrt{-{z5}^{2}}\right| \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \left|\sqrt{-{z5}^{2}}\right| \]

   4. rem-sqrt-square-rev N/A

      \[\leadsto \sqrt{\sqrt{-{z5}^{2}} \cdot \sqrt{-{z5}^{2}}} \]

   5. sqr-neg-rev N/A

      \[\leadsto \sqrt{\left(\mathsf{neg}\left(\sqrt{-{z5}^{2}}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{-{z5}^{2}}\right)\right)} \]

   6. sqr-abs N/A

      \[\leadsto \sqrt{\left|\mathsf{neg}\left(\sqrt{-{z5}^{2}}\right)\right| \cdot \left|\mathsf{neg}\left(\sqrt{-{z5}^{2}}\right)\right|} \]

   7. mul-fabs N/A

      \[\leadsto \sqrt{\left|\left(\mathsf{neg}\left(\sqrt{-{z5}^{2}}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{-{z5}^{2}}\right)\right)\right|} \]

   8. sqr-neg-rev N/A

      \[\leadsto \sqrt{\left|\sqrt{-{z5}^{2}} \cdot \sqrt{-{z5}^{2}}\right|} \]

   9. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{\left|\sqrt{-{z5}^{2}} \cdot \sqrt{-{z5}^{2}}\right|} \]

   10. lift-sqrt.f64 N/A

       \[\leadsto \sqrt{\left|\sqrt{-{z5}^{2}} \cdot \sqrt{-{z5}^{2}}\right|} \]

   11. rem-square-sqrt N/A

       \[\leadsto \sqrt{\left|-{z5}^{2}\right|} \]

   12. lift-neg.f64 N/A

       \[\leadsto \sqrt{\left|\mathsf{neg}\left({z5}^{2}\right)\right|} \]

   13. lift-pow.f64 N/A

       \[\leadsto \sqrt{\left|\mathsf{neg}\left({z5}^{2}\right)\right|} \]

   14. pow2 N/A

       \[\leadsto \sqrt{\left|\mathsf{neg}\left(z5 \cdot z5\right)\right|} \]

   15. distribute-rgt-neg-in N/A

       \[\leadsto \sqrt{\left|z5 \cdot \left(\mathsf{neg}\left(z5\right)\right)\right|} \]

   16. mul-fabs N/A

       \[\leadsto \sqrt{\left|z5\right| \cdot \left|\mathsf{neg}\left(z5\right)\right|} \]

   17. neg-fabs N/A

       \[\leadsto \sqrt{\left|z5\right| \cdot \left|z5\right|} \]

   18. rem-sqrt-square-rev N/A

       \[\leadsto \left|\left|z5\right|\right| \]

   19. fabs-fabs N/A

       \[\leadsto \left|z5\right| \]

   20. lower-fabs.f64 4.2%

       \[\leadsto \left|z5\right| \]

6. Applied rewrites 4.2%

   \[\leadsto \left|z5\right| \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(FPCore (z1 z0 z4 z3 z2 z5)
  :name "(sqrt (- (/ (* (* (* z1 z0) z0) (* (* z1 z0) z0)) (* (* (* (* z4 z3) z2) z2) (* (* (* z4 z3) z2) z2))) (* z5 z5)))"
  :precision binary64
  (sqrt (- (/ (* (* (* z1 z0) z0) (* (* z1 z0) z0)) (* (* (* (* z4 z3) z2) z2) (* (* (* z4 z3) z2) z2))) (* z5 z5))))