jeff quadratic root 2

Percentage Accurate: 72.0% → 90.4%

Time: 5.3s

Alternatives: 13

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
  (if (>= b 0.0)
    (/ (* 2.0 c) (- (- b) t_0))
    (/ (+ (- b) t_0) (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b >= 0.0d0) then
        tmp = (2.0d0 * c) / (-b - t_0)
    else
        tmp = (-b + t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b >= 0.0:
		tmp = (2.0 * c) / (-b - t_0)
	else:
		tmp = (-b + t_0) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
	else
		tmp = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (2.0 * c) / (-b - t_0);
	else
		tmp = (-b + t_0) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 72.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
  (if (>= b 0.0)
    (/ (* 2.0 c) (- (- b) t_0))
    (/ (+ (- b) t_0) (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b >= 0.0d0) then
        tmp = (2.0d0 * c) / (-b - t_0)
    else
        tmp = (-b + t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - t_0);
	} else {
		tmp = (-b + t_0) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b >= 0.0:
		tmp = (2.0 * c) / (-b - t_0)
	else:
		tmp = (-b + t_0) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_0));
	else
		tmp = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (2.0 * c) / (-b - t_0);
	else
		tmp = (-b + t_0) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 90.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot b}{2 \cdot \left(a + a\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (sqrt (- (* b b) (* (* a 4.0) c)))))
  (if (<= b -1e+110)
    (if (>= b 0.0)
      (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))
      (/ (* -4.0 b) (* 2.0 (+ a a))))
    (if (<= b 1.1e+105)
      (if (>= b 0.0) (/ (* -2.0 c) (+ b t_0)) (/ (- t_0 b) (+ a a)))
      (if (>= b 0.0)
        (/ (+ c c) (- (+ b b)))
        (/ (- (sqrt (* (* -4.0 a) c)) b) (+ a a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((a * 4.0) * c)));
	double tmp_1;
	if (b <= -1e+110) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - sqrt(((b * b) - ((4.0 * a) * c))));
		} else {
			tmp_2 = (-4.0 * b) / (2.0 * (a + a));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.1e+105) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-2.0 * c) / (b + t_0);
		} else {
			tmp_3 = (t_0 - b) / (a + a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	}
	return tmp_1;
}

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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) - ((a * 4.0d0) * c)))
    if (b <= (-1d+110)) then
        if (b >= 0.0d0) then
            tmp_2 = (2.0d0 * c) / (-b - sqrt(((b * b) - ((4.0d0 * a) * c))))
        else
            tmp_2 = ((-4.0d0) * b) / (2.0d0 * (a + a))
        end if
        tmp_1 = tmp_2
    else if (b <= 1.1d+105) then
        if (b >= 0.0d0) then
            tmp_3 = ((-2.0d0) * c) / (b + t_0)
        else
            tmp_3 = (t_0 - b) / (a + a)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (c + c) / -(b + b)
    else
        tmp_1 = (sqrt((((-4.0d0) * a) * c)) - b) / (a + a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((a * 4.0) * c)));
	double tmp_1;
	if (b <= -1e+110) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - Math.sqrt(((b * b) - ((4.0 * a) * c))));
		} else {
			tmp_2 = (-4.0 * b) / (2.0 * (a + a));
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.1e+105) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-2.0 * c) / (b + t_0);
		} else {
			tmp_3 = (t_0 - b) / (a + a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = (Math.sqrt(((-4.0 * a) * c)) - b) / (a + a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((a * 4.0) * c)))
	tmp_1 = 0
	if b <= -1e+110:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (2.0 * c) / (-b - math.sqrt(((b * b) - ((4.0 * a) * c))))
		else:
			tmp_2 = (-4.0 * b) / (2.0 * (a + a))
		tmp_1 = tmp_2
	elif b <= 1.1e+105:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-2.0 * c) / (b + t_0)
		else:
			tmp_3 = (t_0 - b) / (a + a)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (c + c) / -(b + b)
	else:
		tmp_1 = (math.sqrt(((-4.0 * a) * c)) - b) / (a + a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c)))
	tmp_1 = 0.0
	if (b <= -1e+110)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))));
		else
			tmp_2 = Float64(Float64(-4.0 * b) / Float64(2.0 * Float64(a + a)));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.1e+105)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-2.0 * c) / Float64(b + t_0));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a + a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c + c) / Float64(-Float64(b + b)));
	else
		tmp_1 = Float64(Float64(sqrt(Float64(Float64(-4.0 * a) * c)) - b) / Float64(a + a));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((a * 4.0) * c)));
	tmp_2 = 0.0;
	if (b <= -1e+110)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (2.0 * c) / (-b - sqrt(((b * b) - ((4.0 * a) * c))));
		else
			tmp_3 = (-4.0 * b) / (2.0 * (a + a));
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.1e+105)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-2.0 * c) / (b + t_0);
		else
			tmp_4 = (t_0 - b) / (a + a);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (c + c) / -(b + b);
	else
		tmp_2 = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1e+110], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * b), $MachinePrecision] / N[(2.0 * N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.1e+105], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c + c), $MachinePrecision] / (-N[(b + b), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1e110

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      3. div-add N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{2 \cdot a} + \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      4. common-denominator N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) \cdot \left(2 \cdot a\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(2 \cdot a\right)}{\left(2 \cdot a\right) \cdot \left(2 \cdot a\right)}\\ \end{array} \]

      5. associate-/r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-b\right) \cdot \left(2 \cdot a\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(2 \cdot a\right)}{2 \cdot a}}{2 \cdot a}\\ \end{array} \]

      6. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-b\right) \cdot \left(2 \cdot a\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(2 \cdot a\right)}{2 \cdot a}}{2 \cdot a}\\ \end{array} \]

   3. Applied rewrites 68.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(a + a\right) \cdot \left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b\right)}{a + a}}{a + a}\\ \end{array} \]

   4. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]

      2. lower-*.f64 61.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]

   6. Applied rewrites 61.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]

      2. mult-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot b\right)}{a + a} \cdot \frac{1}{a + a}\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a + a} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      4. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a + a} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      5. count-2-rev N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot a} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      6. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot 2} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      7. associate-/r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      8. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\frac{1}{a}}{2} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}}\\ \end{array} \]

      9. frac-2neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)}{\mathsf{neg}\left(\left(a + a\right)\right)}}\\ \end{array} \]

      10. frac-2neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(a + a\right)\right)\right)\right)}}\\ \end{array} \]

      11. lift-+.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)}\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(a + a\right)\right)\right)\right)}\\ \end{array} \]

      12. add-flip N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)}\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(a - \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right)}\\ \end{array} \]

      13. sub-negate N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}}{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(a\right)\right) - a\right)\right)}\\ \end{array} \]

      14. sub-negate-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{a - \left(\mathsf{neg}\left(a\right)\right)}}\\ \end{array} \]

      15. add-flip N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{a + a}}\\ \end{array} \]

      16. lift-+.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{a + a}}\\ \end{array} \]

   8. Applied rewrites 61.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(-\left(-\left(a \cdot b\right) \cdot -4\right)\right)}{2 \cdot \left(a + a\right)}\\ \end{array} \]

   9. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot b}{2 \cdot \left(a + a\right)}\\ \end{array} \]
   10. Step-by-step derivation

       1. lower-*.f64 69.5%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot b}{2 \cdot \left(a + a\right)}\\ \end{array} \]

   11. Applied rewrites 69.5%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot b}{2 \cdot \left(a + a\right)}\\ \end{array} \]

   if -1e110 < b < 1.1e105

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   if 1.1e105 < b

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 40.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 54.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 54.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   13. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       2. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       3. frac-2neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       4. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{2}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       5. associate-*r/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       6. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       7. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       8. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       9. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       10. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       11. lift-+.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       12. lower-neg.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{\color{blue}{-2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       13. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       14. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       15. lower-+.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   14. Applied rewrites 54.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c + c}{-\left(b + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{-4}{a \cdot c}}\\ t_1 := \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\\ \mathbf{if}\;b \leq -8.5 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot -4}{\left(4 \cdot a\right) \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq -9.8 \cdot 10^{-234}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + t\_1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-1 \cdot \left(c \cdot t\_0\right)\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (sqrt (/ -4.0 (* a c))))
       (t_1 (sqrt (- (* b b) (* (* a 4.0) c)))))
  (if (<= b -8.5e+154)
    (if (>= b 0.0)
      (* (/ c (+ (sqrt (- (* b b) (* (* c 4.0) a))) b)) -2.0)
      (/ (* (* a b) -4.0) (* (* 4.0 a) a)))
    (if (<= b -9.8e-234)
      (if (>= b 0.0) (/ 2.0 (* a t_0)) (/ (- t_1 b) (+ a a)))
      (if (<= b 1.1e+105)
        (if (>= b 0.0)
          (/ (* -2.0 c) (+ b t_1))
          (* 0.5 (* -1.0 (* c t_0))))
        (if (>= b 0.0)
          (/ (+ c c) (- (+ b b)))
          (/ (- (sqrt (* (* -4.0 a) c)) b) (+ a a))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((-4.0 / (a * c)));
	double t_1 = sqrt(((b * b) - ((a * 4.0) * c)));
	double tmp_1;
	if (b <= -8.5e+154) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / (sqrt(((b * b) - ((c * 4.0) * a))) + b)) * -2.0;
		} else {
			tmp_2 = ((a * b) * -4.0) / ((4.0 * a) * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= -9.8e-234) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = 2.0 / (a * t_0);
		} else {
			tmp_3 = (t_1 - b) / (a + a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.1e+105) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-2.0 * c) / (b + t_1);
		} else {
			tmp_4 = 0.5 * (-1.0 * (c * t_0));
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	}
	return tmp_1;
}

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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = sqrt(((-4.0d0) / (a * c)))
    t_1 = sqrt(((b * b) - ((a * 4.0d0) * c)))
    if (b <= (-8.5d+154)) then
        if (b >= 0.0d0) then
            tmp_2 = (c / (sqrt(((b * b) - ((c * 4.0d0) * a))) + b)) * (-2.0d0)
        else
            tmp_2 = ((a * b) * (-4.0d0)) / ((4.0d0 * a) * a)
        end if
        tmp_1 = tmp_2
    else if (b <= (-9.8d-234)) then
        if (b >= 0.0d0) then
            tmp_3 = 2.0d0 / (a * t_0)
        else
            tmp_3 = (t_1 - b) / (a + a)
        end if
        tmp_1 = tmp_3
    else if (b <= 1.1d+105) then
        if (b >= 0.0d0) then
            tmp_4 = ((-2.0d0) * c) / (b + t_1)
        else
            tmp_4 = 0.5d0 * ((-1.0d0) * (c * t_0))
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (c + c) / -(b + b)
    else
        tmp_1 = (sqrt((((-4.0d0) * a) * c)) - b) / (a + a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((-4.0 / (a * c)));
	double t_1 = Math.sqrt(((b * b) - ((a * 4.0) * c)));
	double tmp_1;
	if (b <= -8.5e+154) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / (Math.sqrt(((b * b) - ((c * 4.0) * a))) + b)) * -2.0;
		} else {
			tmp_2 = ((a * b) * -4.0) / ((4.0 * a) * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= -9.8e-234) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = 2.0 / (a * t_0);
		} else {
			tmp_3 = (t_1 - b) / (a + a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.1e+105) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-2.0 * c) / (b + t_1);
		} else {
			tmp_4 = 0.5 * (-1.0 * (c * t_0));
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = (Math.sqrt(((-4.0 * a) * c)) - b) / (a + a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((-4.0 / (a * c)))
	t_1 = math.sqrt(((b * b) - ((a * 4.0) * c)))
	tmp_1 = 0
	if b <= -8.5e+154:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (c / (math.sqrt(((b * b) - ((c * 4.0) * a))) + b)) * -2.0
		else:
			tmp_2 = ((a * b) * -4.0) / ((4.0 * a) * a)
		tmp_1 = tmp_2
	elif b <= -9.8e-234:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = 2.0 / (a * t_0)
		else:
			tmp_3 = (t_1 - b) / (a + a)
		tmp_1 = tmp_3
	elif b <= 1.1e+105:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (-2.0 * c) / (b + t_1)
		else:
			tmp_4 = 0.5 * (-1.0 * (c * t_0))
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (c + c) / -(b + b)
	else:
		tmp_1 = (math.sqrt(((-4.0 * a) * c)) - b) / (a + a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(-4.0 / Float64(a * c)))
	t_1 = sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c)))
	tmp_1 = 0.0
	if (b <= -8.5e+154)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c / Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(c * 4.0) * a))) + b)) * -2.0);
		else
			tmp_2 = Float64(Float64(Float64(a * b) * -4.0) / Float64(Float64(4.0 * a) * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= -9.8e-234)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(2.0 / Float64(a * t_0));
		else
			tmp_3 = Float64(Float64(t_1 - b) / Float64(a + a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.1e+105)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(-2.0 * c) / Float64(b + t_1));
		else
			tmp_4 = Float64(0.5 * Float64(-1.0 * Float64(c * t_0)));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c + c) / Float64(-Float64(b + b)));
	else
		tmp_1 = Float64(Float64(sqrt(Float64(Float64(-4.0 * a) * c)) - b) / Float64(a + a));
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = sqrt((-4.0 / (a * c)));
	t_1 = sqrt(((b * b) - ((a * 4.0) * c)));
	tmp_2 = 0.0;
	if (b <= -8.5e+154)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (c / (sqrt(((b * b) - ((c * 4.0) * a))) + b)) * -2.0;
		else
			tmp_3 = ((a * b) * -4.0) / ((4.0 * a) * a);
		end
		tmp_2 = tmp_3;
	elseif (b <= -9.8e-234)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = 2.0 / (a * t_0);
		else
			tmp_4 = (t_1 - b) / (a + a);
		end
		tmp_2 = tmp_4;
	elseif (b <= 1.1e+105)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (-2.0 * c) / (b + t_1);
		else
			tmp_5 = 0.5 * (-1.0 * (c * t_0));
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (c + c) / -(b + b);
	else
		tmp_2 = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -8.5e+154], If[GreaterEqual[b, 0.0], N[(N[(c / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(c * 4.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(a * b), $MachinePrecision] * -4.0), $MachinePrecision] / N[(N[(4.0 * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -9.8e-234], If[GreaterEqual[b, 0.0], N[(2.0 / N[(a * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.1e+105], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(b + t$95$1), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(-1.0 * N[(c * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c + c), $MachinePrecision] / (-N[(b + b), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.5000000000000002e154

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      3. div-add N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{2 \cdot a} + \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      4. common-denominator N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) \cdot \left(2 \cdot a\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(2 \cdot a\right)}{\left(2 \cdot a\right) \cdot \left(2 \cdot a\right)}\\ \end{array} \]

      5. associate-/r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-b\right) \cdot \left(2 \cdot a\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(2 \cdot a\right)}{2 \cdot a}}{2 \cdot a}\\ \end{array} \]

      6. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-b\right) \cdot \left(2 \cdot a\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(2 \cdot a\right)}{2 \cdot a}}{2 \cdot a}\\ \end{array} \]

   3. Applied rewrites 68.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(a + a\right) \cdot \left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b\right)}{a + a}}{a + a}\\ \end{array} \]

   4. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]

      2. lower-*.f64 61.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]

   6. Applied rewrites 61.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]

      2. mult-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot b\right)}{a + a} \cdot \frac{1}{a + a}\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a + a} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      4. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a + a} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      5. count-2-rev N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot a} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      6. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot 2} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      7. associate-/r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      8. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\frac{1}{a}}{2} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}}\\ \end{array} \]

      9. frac-2neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)}{\mathsf{neg}\left(\left(a + a\right)\right)}}\\ \end{array} \]

      10. frac-2neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(a + a\right)\right)\right)\right)}}\\ \end{array} \]

      11. lift-+.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)}\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(a + a\right)\right)\right)\right)}\\ \end{array} \]

      12. add-flip N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)}\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(a - \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right)}\\ \end{array} \]

      13. sub-negate N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}}{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(a\right)\right) - a\right)\right)}\\ \end{array} \]

      14. sub-negate-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{a - \left(\mathsf{neg}\left(a\right)\right)}}\\ \end{array} \]

      15. add-flip N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{a + a}}\\ \end{array} \]

      16. lift-+.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{a + a}}\\ \end{array} \]

   8. Applied rewrites 61.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(-\left(-\left(a \cdot b\right) \cdot -4\right)\right)}{2 \cdot \left(a + a\right)}\\ \end{array} \]

   10. Applied rewrites 57.8%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot -4}{\left(4 \cdot a\right) \cdot a}\\ } \end{array}} \]

   if -8.5000000000000002e154 < b < -9.8000000000000001e-234

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   4. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      2. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      3. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{a \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      5. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      6. lower-/.f64 43.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

   6. Applied rewrites 43.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

   7. Taylor expanded in c around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\color{blue}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \color{blue}{\sqrt{\frac{-4}{a \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      4. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      5. lower-*.f64 50.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

   9. Applied rewrites 50.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\color{blue}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

   if -9.8000000000000001e-234 < b < 1.1e105

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   4. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \end{array} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}}\\ \end{array} \]

      3. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot \frac{c}{a}}}\\ \end{array} \]

      4. lower-/.f64 43.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot \color{blue}{\frac{c}{a}}}\\ \end{array} \]

   6. Applied rewrites 43.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \end{array} \]

   7. Taylor expanded in c around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \color{blue}{\left(-1 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\right)}\\ \end{array} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-1 \cdot \color{blue}{\left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)}\right)\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-1 \cdot \left(c \cdot \color{blue}{\sqrt{\frac{-4}{a \cdot c}}}\right)\right)\\ \end{array} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-1 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\right)\\ \end{array} \]

      4. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-1 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\right)\\ \end{array} \]

      5. lower-*.f64 50.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(-1 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\right)\\ \end{array} \]

   9. Applied rewrites 50.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \color{blue}{\left(-1 \cdot \left(c \cdot \sqrt{\frac{-4}{a \cdot c}}\right)\right)}\\ \end{array} \]

   if 1.1e105 < b

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 40.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 54.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 54.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   13. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       2. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       3. frac-2neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       4. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{2}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       5. associate-*r/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       6. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       7. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       8. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       9. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       10. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       11. lift-+.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       12. lower-neg.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{\color{blue}{-2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       13. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       14. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       15. lower-+.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   14. Applied rewrites 54.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c + c}{-\left(b + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\\ \mathbf{if}\;b \leq -8.5 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot -4}{\left(4 \cdot a\right) \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq -2.8 \cdot 10^{-270}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + t\_0}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (sqrt (- (* b b) (* (* a 4.0) c)))))
  (if (<= b -8.5e+154)
    (if (>= b 0.0)
      (* (/ c (+ (sqrt (- (* b b) (* (* c 4.0) a))) b)) -2.0)
      (/ (* (* a b) -4.0) (* (* 4.0 a) a)))
    (if (<= b -2.8e-270)
      (if (>= b 0.0)
        (/ 2.0 (* a (sqrt (/ -4.0 (* a c)))))
        (/ (- t_0 b) (+ a a)))
      (if (<= b 1.1e+105)
        (if (>= b 0.0)
          (/ (* -2.0 c) (+ b t_0))
          (* -0.5 (/ (* c (sqrt (* -4.0 (/ a c)))) a)))
        (if (>= b 0.0)
          (/ (+ c c) (- (+ b b)))
          (/ (- (sqrt (* (* -4.0 a) c)) b) (+ a a))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((a * 4.0) * c)));
	double tmp_1;
	if (b <= -8.5e+154) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / (sqrt(((b * b) - ((c * 4.0) * a))) + b)) * -2.0;
		} else {
			tmp_2 = ((a * b) * -4.0) / ((4.0 * a) * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= -2.8e-270) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = 2.0 / (a * sqrt((-4.0 / (a * c))));
		} else {
			tmp_3 = (t_0 - b) / (a + a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.1e+105) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-2.0 * c) / (b + t_0);
		} else {
			tmp_4 = -0.5 * ((c * sqrt((-4.0 * (a / c)))) / a);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	}
	return tmp_1;
}

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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    real(8) :: tmp_4
    t_0 = sqrt(((b * b) - ((a * 4.0d0) * c)))
    if (b <= (-8.5d+154)) then
        if (b >= 0.0d0) then
            tmp_2 = (c / (sqrt(((b * b) - ((c * 4.0d0) * a))) + b)) * (-2.0d0)
        else
            tmp_2 = ((a * b) * (-4.0d0)) / ((4.0d0 * a) * a)
        end if
        tmp_1 = tmp_2
    else if (b <= (-2.8d-270)) then
        if (b >= 0.0d0) then
            tmp_3 = 2.0d0 / (a * sqrt(((-4.0d0) / (a * c))))
        else
            tmp_3 = (t_0 - b) / (a + a)
        end if
        tmp_1 = tmp_3
    else if (b <= 1.1d+105) then
        if (b >= 0.0d0) then
            tmp_4 = ((-2.0d0) * c) / (b + t_0)
        else
            tmp_4 = (-0.5d0) * ((c * sqrt(((-4.0d0) * (a / c)))) / a)
        end if
        tmp_1 = tmp_4
    else if (b >= 0.0d0) then
        tmp_1 = (c + c) / -(b + b)
    else
        tmp_1 = (sqrt((((-4.0d0) * a) * c)) - b) / (a + a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((a * 4.0) * c)));
	double tmp_1;
	if (b <= -8.5e+154) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / (Math.sqrt(((b * b) - ((c * 4.0) * a))) + b)) * -2.0;
		} else {
			tmp_2 = ((a * b) * -4.0) / ((4.0 * a) * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= -2.8e-270) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = 2.0 / (a * Math.sqrt((-4.0 / (a * c))));
		} else {
			tmp_3 = (t_0 - b) / (a + a);
		}
		tmp_1 = tmp_3;
	} else if (b <= 1.1e+105) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-2.0 * c) / (b + t_0);
		} else {
			tmp_4 = -0.5 * ((c * Math.sqrt((-4.0 * (a / c)))) / a);
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = (Math.sqrt(((-4.0 * a) * c)) - b) / (a + a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((a * 4.0) * c)))
	tmp_1 = 0
	if b <= -8.5e+154:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (c / (math.sqrt(((b * b) - ((c * 4.0) * a))) + b)) * -2.0
		else:
			tmp_2 = ((a * b) * -4.0) / ((4.0 * a) * a)
		tmp_1 = tmp_2
	elif b <= -2.8e-270:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = 2.0 / (a * math.sqrt((-4.0 / (a * c))))
		else:
			tmp_3 = (t_0 - b) / (a + a)
		tmp_1 = tmp_3
	elif b <= 1.1e+105:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = (-2.0 * c) / (b + t_0)
		else:
			tmp_4 = -0.5 * ((c * math.sqrt((-4.0 * (a / c)))) / a)
		tmp_1 = tmp_4
	elif b >= 0.0:
		tmp_1 = (c + c) / -(b + b)
	else:
		tmp_1 = (math.sqrt(((-4.0 * a) * c)) - b) / (a + a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c)))
	tmp_1 = 0.0
	if (b <= -8.5e+154)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c / Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(c * 4.0) * a))) + b)) * -2.0);
		else
			tmp_2 = Float64(Float64(Float64(a * b) * -4.0) / Float64(Float64(4.0 * a) * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= -2.8e-270)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(2.0 / Float64(a * sqrt(Float64(-4.0 / Float64(a * c)))));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a + a));
		end
		tmp_1 = tmp_3;
	elseif (b <= 1.1e+105)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(-2.0 * c) / Float64(b + t_0));
		else
			tmp_4 = Float64(-0.5 * Float64(Float64(c * sqrt(Float64(-4.0 * Float64(a / c)))) / a));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c + c) / Float64(-Float64(b + b)));
	else
		tmp_1 = Float64(Float64(sqrt(Float64(Float64(-4.0 * a) * c)) - b) / Float64(a + a));
	end
	return tmp_1
end

MATLAB

function tmp_6 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((a * 4.0) * c)));
	tmp_2 = 0.0;
	if (b <= -8.5e+154)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (c / (sqrt(((b * b) - ((c * 4.0) * a))) + b)) * -2.0;
		else
			tmp_3 = ((a * b) * -4.0) / ((4.0 * a) * a);
		end
		tmp_2 = tmp_3;
	elseif (b <= -2.8e-270)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = 2.0 / (a * sqrt((-4.0 / (a * c))));
		else
			tmp_4 = (t_0 - b) / (a + a);
		end
		tmp_2 = tmp_4;
	elseif (b <= 1.1e+105)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = (-2.0 * c) / (b + t_0);
		else
			tmp_5 = -0.5 * ((c * sqrt((-4.0 * (a / c)))) / a);
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = (c + c) / -(b + b);
	else
		tmp_2 = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	end
	tmp_6 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -8.5e+154], If[GreaterEqual[b, 0.0], N[(N[(c / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(c * 4.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(a * b), $MachinePrecision] * -4.0), $MachinePrecision] / N[(N[(4.0 * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -2.8e-270], If[GreaterEqual[b, 0.0], N[(2.0 / N[(a * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.1e+105], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(c * N[Sqrt[N[(-4.0 * N[(a / c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c + c), $MachinePrecision] / (-N[(b + b), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.5000000000000002e154

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      3. div-add N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{2 \cdot a} + \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      4. common-denominator N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) \cdot \left(2 \cdot a\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(2 \cdot a\right)}{\left(2 \cdot a\right) \cdot \left(2 \cdot a\right)}\\ \end{array} \]

      5. associate-/r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-b\right) \cdot \left(2 \cdot a\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(2 \cdot a\right)}{2 \cdot a}}{2 \cdot a}\\ \end{array} \]

      6. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-b\right) \cdot \left(2 \cdot a\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(2 \cdot a\right)}{2 \cdot a}}{2 \cdot a}\\ \end{array} \]

   3. Applied rewrites 68.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(a + a\right) \cdot \left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b\right)}{a + a}}{a + a}\\ \end{array} \]

   4. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]

      2. lower-*.f64 61.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]

   6. Applied rewrites 61.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]

      2. mult-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot b\right)}{a + a} \cdot \frac{1}{a + a}\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a + a} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      4. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a + a} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      5. count-2-rev N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot a} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      6. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot 2} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      7. associate-/r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      8. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\frac{1}{a}}{2} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}}\\ \end{array} \]

      9. frac-2neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)}{\mathsf{neg}\left(\left(a + a\right)\right)}}\\ \end{array} \]

      10. frac-2neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(a + a\right)\right)\right)\right)}}\\ \end{array} \]

      11. lift-+.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)}\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(a + a\right)\right)\right)\right)}\\ \end{array} \]

      12. add-flip N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)}\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(a - \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right)}\\ \end{array} \]

      13. sub-negate N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}}{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(a\right)\right) - a\right)\right)}\\ \end{array} \]

      14. sub-negate-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{a - \left(\mathsf{neg}\left(a\right)\right)}}\\ \end{array} \]

      15. add-flip N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{a + a}}\\ \end{array} \]

      16. lift-+.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{a + a}}\\ \end{array} \]

   8. Applied rewrites 61.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(-\left(-\left(a \cdot b\right) \cdot -4\right)\right)}{2 \cdot \left(a + a\right)}\\ \end{array} \]

   10. Applied rewrites 57.8%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot -4}{\left(4 \cdot a\right) \cdot a}\\ } \end{array}} \]

   if -8.5000000000000002e154 < b < -2.7999999999999999e-270

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   4. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      2. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      3. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{a \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      5. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      6. lower-/.f64 43.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

   6. Applied rewrites 43.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

   7. Taylor expanded in c around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\color{blue}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \color{blue}{\sqrt{\frac{-4}{a \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      4. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      5. lower-*.f64 50.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

   9. Applied rewrites 50.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\color{blue}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

   if -2.7999999999999999e-270 < b < 1.1e105

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   4. Taylor expanded in c around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{-1}{2} \cdot \frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}}\\ \end{array} \]

      2. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \color{blue}{\frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}}\\ \end{array} \]

      3. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{\color{blue}{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}}{a}\\ \end{array} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c \cdot \color{blue}{\sqrt{-4 \cdot \frac{a}{c}}}}{a}\\ \end{array} \]

      5. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c \cdot \sqrt{\color{blue}{-4 \cdot \frac{a}{c}}}}{a}\\ \end{array} \]

      6. lower-/.f64 42.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot \sqrt{-4 \cdot \color{blue}{\frac{a}{c}}}}{a}\\ \end{array} \]

   6. Applied rewrites 42.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot \sqrt{-4 \cdot \frac{a}{c}}}{a}\\ \end{array} \]

   if 1.1e105 < b

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 40.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 54.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 54.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   13. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       2. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       3. frac-2neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       4. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{2}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       5. associate-*r/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       6. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       7. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       8. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       9. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       10. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       11. lift-+.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       12. lower-neg.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{\color{blue}{-2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       13. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       14. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       15. lower-+.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   14. Applied rewrites 54.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c + c}{-\left(b + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\\ \mathbf{if}\;b \leq -8.5 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot -4}{\left(4 \cdot a\right) \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+105}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (sqrt (- (* b b) (* (* a 4.0) c)))))
  (if (<= b -8.5e+154)
    (if (>= b 0.0)
      (* (/ c (+ (sqrt (- (* b b) (* (* c 4.0) a))) b)) -2.0)
      (/ (* (* a b) -4.0) (* (* 4.0 a) a)))
    (if (<= b 1.1e+105)
      (if (>= b 0.0) (/ (* -2.0 c) (+ b t_0)) (/ (- t_0 b) (+ a a)))
      (if (>= b 0.0)
        (/ (+ c c) (- (+ b b)))
        (/ (- (sqrt (* (* -4.0 a) c)) b) (+ a a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((a * 4.0) * c)));
	double tmp_1;
	if (b <= -8.5e+154) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / (sqrt(((b * b) - ((c * 4.0) * a))) + b)) * -2.0;
		} else {
			tmp_2 = ((a * b) * -4.0) / ((4.0 * a) * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.1e+105) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-2.0 * c) / (b + t_0);
		} else {
			tmp_3 = (t_0 - b) / (a + a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	}
	return tmp_1;
}

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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = sqrt(((b * b) - ((a * 4.0d0) * c)))
    if (b <= (-8.5d+154)) then
        if (b >= 0.0d0) then
            tmp_2 = (c / (sqrt(((b * b) - ((c * 4.0d0) * a))) + b)) * (-2.0d0)
        else
            tmp_2 = ((a * b) * (-4.0d0)) / ((4.0d0 * a) * a)
        end if
        tmp_1 = tmp_2
    else if (b <= 1.1d+105) then
        if (b >= 0.0d0) then
            tmp_3 = ((-2.0d0) * c) / (b + t_0)
        else
            tmp_3 = (t_0 - b) / (a + a)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (c + c) / -(b + b)
    else
        tmp_1 = (sqrt((((-4.0d0) * a) * c)) - b) / (a + a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((a * 4.0) * c)));
	double tmp_1;
	if (b <= -8.5e+154) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (c / (Math.sqrt(((b * b) - ((c * 4.0) * a))) + b)) * -2.0;
		} else {
			tmp_2 = ((a * b) * -4.0) / ((4.0 * a) * a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.1e+105) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-2.0 * c) / (b + t_0);
		} else {
			tmp_3 = (t_0 - b) / (a + a);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = (Math.sqrt(((-4.0 * a) * c)) - b) / (a + a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((a * 4.0) * c)))
	tmp_1 = 0
	if b <= -8.5e+154:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (c / (math.sqrt(((b * b) - ((c * 4.0) * a))) + b)) * -2.0
		else:
			tmp_2 = ((a * b) * -4.0) / ((4.0 * a) * a)
		tmp_1 = tmp_2
	elif b <= 1.1e+105:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (-2.0 * c) / (b + t_0)
		else:
			tmp_3 = (t_0 - b) / (a + a)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (c + c) / -(b + b)
	else:
		tmp_1 = (math.sqrt(((-4.0 * a) * c)) - b) / (a + a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c)))
	tmp_1 = 0.0
	if (b <= -8.5e+154)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(c / Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(c * 4.0) * a))) + b)) * -2.0);
		else
			tmp_2 = Float64(Float64(Float64(a * b) * -4.0) / Float64(Float64(4.0 * a) * a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.1e+105)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-2.0 * c) / Float64(b + t_0));
		else
			tmp_3 = Float64(Float64(t_0 - b) / Float64(a + a));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c + c) / Float64(-Float64(b + b)));
	else
		tmp_1 = Float64(Float64(sqrt(Float64(Float64(-4.0 * a) * c)) - b) / Float64(a + a));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((a * 4.0) * c)));
	tmp_2 = 0.0;
	if (b <= -8.5e+154)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (c / (sqrt(((b * b) - ((c * 4.0) * a))) + b)) * -2.0;
		else
			tmp_3 = ((a * b) * -4.0) / ((4.0 * a) * a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.1e+105)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (-2.0 * c) / (b + t_0);
		else
			tmp_4 = (t_0 - b) / (a + a);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (c + c) / -(b + b);
	else
		tmp_2 = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -8.5e+154], If[GreaterEqual[b, 0.0], N[(N[(c / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(c * 4.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(a * b), $MachinePrecision] * -4.0), $MachinePrecision] / N[(N[(4.0 * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.1e+105], If[GreaterEqual[b, 0.0], N[(N[(-2.0 * c), $MachinePrecision] / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c + c), $MachinePrecision] / (-N[(b + b), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.5000000000000002e154

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      3. div-add N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{2 \cdot a} + \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      4. common-denominator N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) \cdot \left(2 \cdot a\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(2 \cdot a\right)}{\left(2 \cdot a\right) \cdot \left(2 \cdot a\right)}\\ \end{array} \]

      5. associate-/r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-b\right) \cdot \left(2 \cdot a\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(2 \cdot a\right)}{2 \cdot a}}{2 \cdot a}\\ \end{array} \]

      6. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-b\right) \cdot \left(2 \cdot a\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(2 \cdot a\right)}{2 \cdot a}}{2 \cdot a}\\ \end{array} \]

   3. Applied rewrites 68.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(a + a\right) \cdot \left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b\right)}{a + a}}{a + a}\\ \end{array} \]

   4. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]

      2. lower-*.f64 61.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]

   6. Applied rewrites 61.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]

      2. mult-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot b\right)}{a + a} \cdot \frac{1}{a + a}\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a + a} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      4. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a + a} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      5. count-2-rev N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot a} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      6. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot 2} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      7. associate-/r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      8. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\frac{1}{a}}{2} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}}\\ \end{array} \]

      9. frac-2neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)}{\mathsf{neg}\left(\left(a + a\right)\right)}}\\ \end{array} \]

      10. frac-2neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(a + a\right)\right)\right)\right)}}\\ \end{array} \]

      11. lift-+.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)}\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(a + a\right)\right)\right)\right)}\\ \end{array} \]

      12. add-flip N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)}\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(a - \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right)}\\ \end{array} \]

      13. sub-negate N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}}{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(a\right)\right) - a\right)\right)}\\ \end{array} \]

      14. sub-negate-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{a - \left(\mathsf{neg}\left(a\right)\right)}}\\ \end{array} \]

      15. add-flip N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{a + a}}\\ \end{array} \]

      16. lift-+.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{a + a}}\\ \end{array} \]

   8. Applied rewrites 61.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(-\left(-\left(a \cdot b\right) \cdot -4\right)\right)}{2 \cdot \left(a + a\right)}\\ \end{array} \]

   10. Applied rewrites 57.8%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot -4}{\left(4 \cdot a\right) \cdot a}\\ } \end{array}} \]

   if -8.5000000000000002e154 < b < 1.1e105

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   if 1.1e105 < b

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 40.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 54.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 54.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   13. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       2. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       3. frac-2neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       4. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{2}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       5. associate-*r/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       6. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       7. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       8. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       9. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       10. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       11. lift-+.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       12. lower-neg.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{\color{blue}{-2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       13. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       14. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       15. lower-+.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   14. Applied rewrites 54.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c + c}{-\left(b + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot -4}{\left(4 \cdot a\right) \cdot a}\\ \end{array}\\ \mathbf{if}\;b \leq -8.5 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq -6 \cdot 10^{-309}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0
        (if (>= b 0.0)
          (* (/ c (+ (sqrt (- (* b b) (* (* c 4.0) a))) b)) -2.0)
          (/ (* (* a b) -4.0) (* (* 4.0 a) a)))))
  (if (<= b -8.5e+154)
    t_0
    (if (<= b -6e-309)
      (if (>= b 0.0)
        (/ 2.0 (* a (sqrt (/ -4.0 (* a c)))))
        (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (+ a a)))
      (if (<= b 1.1e+105)
        t_0
        (if (>= b 0.0)
          (/ (+ c c) (- (+ b b)))
          (/ (- (sqrt (* (* -4.0 a) c)) b) (+ a a))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (c / (sqrt(((b * b) - ((c * 4.0) * a))) + b)) * -2.0;
	} else {
		tmp = ((a * b) * -4.0) / ((4.0 * a) * a);
	}
	double t_0 = tmp;
	double tmp_1;
	if (b <= -8.5e+154) {
		tmp_1 = t_0;
	} else if (b <= -6e-309) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 2.0 / (a * sqrt((-4.0 / (a * c))));
		} else {
			tmp_2 = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.1e+105) {
		tmp_1 = t_0;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	}
	return tmp_1;
}

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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b >= 0.0d0) then
        tmp = (c / (sqrt(((b * b) - ((c * 4.0d0) * a))) + b)) * (-2.0d0)
    else
        tmp = ((a * b) * (-4.0d0)) / ((4.0d0 * a) * a)
    end if
    t_0 = tmp
    if (b <= (-8.5d+154)) then
        tmp_1 = t_0
    else if (b <= (-6d-309)) then
        if (b >= 0.0d0) then
            tmp_2 = 2.0d0 / (a * sqrt(((-4.0d0) / (a * c))))
        else
            tmp_2 = (sqrt(((b * b) - ((a * 4.0d0) * c))) - b) / (a + a)
        end if
        tmp_1 = tmp_2
    else if (b <= 1.1d+105) then
        tmp_1 = t_0
    else if (b >= 0.0d0) then
        tmp_1 = (c + c) / -(b + b)
    else
        tmp_1 = (sqrt((((-4.0d0) * a) * c)) - b) / (a + a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (c / (Math.sqrt(((b * b) - ((c * 4.0) * a))) + b)) * -2.0;
	} else {
		tmp = ((a * b) * -4.0) / ((4.0 * a) * a);
	}
	double t_0 = tmp;
	double tmp_1;
	if (b <= -8.5e+154) {
		tmp_1 = t_0;
	} else if (b <= -6e-309) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 2.0 / (a * Math.sqrt((-4.0 / (a * c))));
		} else {
			tmp_2 = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 1.1e+105) {
		tmp_1 = t_0;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = (Math.sqrt(((-4.0 * a) * c)) - b) / (a + a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = (c / (math.sqrt(((b * b) - ((c * 4.0) * a))) + b)) * -2.0
	else:
		tmp = ((a * b) * -4.0) / ((4.0 * a) * a)
	t_0 = tmp
	tmp_1 = 0
	if b <= -8.5e+154:
		tmp_1 = t_0
	elif b <= -6e-309:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = 2.0 / (a * math.sqrt((-4.0 / (a * c))))
		else:
			tmp_2 = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a + a)
		tmp_1 = tmp_2
	elif b <= 1.1e+105:
		tmp_1 = t_0
	elif b >= 0.0:
		tmp_1 = (c + c) / -(b + b)
	else:
		tmp_1 = (math.sqrt(((-4.0 * a) * c)) - b) / (a + a)
	return tmp_1

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(c / Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(c * 4.0) * a))) + b)) * -2.0);
	else
		tmp = Float64(Float64(Float64(a * b) * -4.0) / Float64(Float64(4.0 * a) * a));
	end
	t_0 = tmp
	tmp_1 = 0.0
	if (b <= -8.5e+154)
		tmp_1 = t_0;
	elseif (b <= -6e-309)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(2.0 / Float64(a * sqrt(Float64(-4.0 / Float64(a * c)))));
		else
			tmp_2 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a + a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 1.1e+105)
		tmp_1 = t_0;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c + c) / Float64(-Float64(b + b)));
	else
		tmp_1 = Float64(Float64(sqrt(Float64(Float64(-4.0 * a) * c)) - b) / Float64(a + a));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (c / (sqrt(((b * b) - ((c * 4.0) * a))) + b)) * -2.0;
	else
		tmp = ((a * b) * -4.0) / ((4.0 * a) * a);
	end
	t_0 = tmp;
	tmp_2 = 0.0;
	if (b <= -8.5e+154)
		tmp_2 = t_0;
	elseif (b <= -6e-309)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = 2.0 / (a * sqrt((-4.0 / (a * c))));
		else
			tmp_3 = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a + a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 1.1e+105)
		tmp_2 = t_0;
	elseif (b >= 0.0)
		tmp_2 = (c + c) / -(b + b);
	else
		tmp_2 = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(N[(c / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(c * 4.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(N[(a * b), $MachinePrecision] * -4.0), $MachinePrecision] / N[(N[(4.0 * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]}, If[LessEqual[b, -8.5e+154], t$95$0, If[LessEqual[b, -6e-309], If[GreaterEqual[b, 0.0], N[(2.0 / N[(a * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 1.1e+105], t$95$0, If[GreaterEqual[b, 0.0], N[(N[(c + c), $MachinePrecision] / (-N[(b + b), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.5000000000000002e154 or -6.0000000000000014e-309 < b < 1.1e105

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      3. div-add N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{2 \cdot a} + \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      4. common-denominator N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) \cdot \left(2 \cdot a\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(2 \cdot a\right)}{\left(2 \cdot a\right) \cdot \left(2 \cdot a\right)}\\ \end{array} \]

      5. associate-/r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-b\right) \cdot \left(2 \cdot a\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(2 \cdot a\right)}{2 \cdot a}}{2 \cdot a}\\ \end{array} \]

      6. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-b\right) \cdot \left(2 \cdot a\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(2 \cdot a\right)}{2 \cdot a}}{2 \cdot a}\\ \end{array} \]

   3. Applied rewrites 68.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(a + a\right) \cdot \left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b\right)}{a + a}}{a + a}\\ \end{array} \]

   4. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]

      2. lower-*.f64 61.7%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]

   6. Applied rewrites 61.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(a \cdot b\right)}{a + a}}{a + a}\\ \end{array} \]

      2. mult-flip N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot b\right)}{a + a} \cdot \frac{1}{a + a}\\ \end{array} \]

      3. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a + a} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      4. lift-+.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a + a} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      5. count-2-rev N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot a} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      6. *-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot 2} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      7. associate-/r* N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}\\ \end{array} \]

      8. lift-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\frac{1}{a}}{2} \cdot \frac{-4 \cdot \left(a \cdot b\right)}{a + a}}\\ \end{array} \]

      9. frac-2neg N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)}{\mathsf{neg}\left(\left(a + a\right)\right)}}\\ \end{array} \]

      10. frac-2neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(a + a\right)\right)\right)\right)}}\\ \end{array} \]

      11. lift-+.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)}\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(a + a\right)\right)\right)\right)}\\ \end{array} \]

      12. add-flip N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)}\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(a - \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right)}\\ \end{array} \]

      13. sub-negate N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}}{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(a\right)\right) - a\right)\right)}\\ \end{array} \]

      14. sub-negate-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{a - \left(\mathsf{neg}\left(a\right)\right)}}\\ \end{array} \]

      15. add-flip N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{a + a}}\\ \end{array} \]

      16. lift-+.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{2} \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(-4 \cdot \left(a \cdot b\right)\right)\right)\right)}{a + a}}\\ \end{array} \]

   8. Applied rewrites 61.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(-\left(-\left(a \cdot b\right) \cdot -4\right)\right)}{2 \cdot \left(a + a\right)}\\ \end{array} \]

   10. Applied rewrites 57.8%

       \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot -4}{\left(4 \cdot a\right) \cdot a}\\ } \end{array}} \]

   if -8.5000000000000002e154 < b < -6.0000000000000014e-309

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   4. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      2. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      3. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{a \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      5. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      6. lower-/.f64 43.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

   6. Applied rewrites 43.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

   7. Taylor expanded in c around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\color{blue}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \color{blue}{\sqrt{\frac{-4}{a \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      4. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      5. lower-*.f64 50.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

   9. Applied rewrites 50.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\color{blue}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

   if 1.1e105 < b

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 40.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 54.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 54.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   13. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       2. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       3. frac-2neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       4. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{2}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       5. associate-*r/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       6. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       7. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       8. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       9. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       10. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       11. lift-+.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       12. lower-neg.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{\color{blue}{-2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       13. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       14. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       15. lower-+.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   14. Applied rewrites 54.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c + c}{-\left(b + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq 390000000000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<= b 390000000000.0)
  (if (>= b 0.0)
    (/ 2.0 (* a (sqrt (/ -4.0 (* a c)))))
    (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (+ a a)))
  (if (>= b 0.0)
    (/ (+ c c) (- (+ b b)))
    (/ (- (sqrt (* (* -4.0 a) c)) b) (+ a a)))))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 390000000000.0) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 2.0 / (a * sqrt((-4.0 / (a * c))));
		} else {
			tmp_2 = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	}
	return tmp_1;
}

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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b <= 390000000000.0d0) then
        if (b >= 0.0d0) then
            tmp_2 = 2.0d0 / (a * sqrt(((-4.0d0) / (a * c))))
        else
            tmp_2 = (sqrt(((b * b) - ((a * 4.0d0) * c))) - b) / (a + a)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (c + c) / -(b + b)
    else
        tmp_1 = (sqrt((((-4.0d0) * a) * c)) - b) / (a + a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 390000000000.0) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = 2.0 / (a * Math.sqrt((-4.0 / (a * c))));
		} else {
			tmp_2 = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = (Math.sqrt(((-4.0 * a) * c)) - b) / (a + a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= 390000000000.0:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = 2.0 / (a * math.sqrt((-4.0 / (a * c))))
		else:
			tmp_2 = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a + a)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (c + c) / -(b + b)
	else:
		tmp_1 = (math.sqrt(((-4.0 * a) * c)) - b) / (a + a)
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= 390000000000.0)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(2.0 / Float64(a * sqrt(Float64(-4.0 / Float64(a * c)))));
		else
			tmp_2 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a + a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c + c) / Float64(-Float64(b + b)));
	else
		tmp_1 = Float64(Float64(sqrt(Float64(Float64(-4.0 * a) * c)) - b) / Float64(a + a));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= 390000000000.0)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = 2.0 / (a * sqrt((-4.0 / (a * c))));
		else
			tmp_3 = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a + a);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (c + c) / -(b + b);
	else
		tmp_2 = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 390000000000.0], If[GreaterEqual[b, 0.0], N[(2.0 / N[(a * N[Sqrt[N[(-4.0 / N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c + c), $MachinePrecision] / (-N[(b + b), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.9e11

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2 \cdot c}{b + \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   4. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      2. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      3. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{a \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      5. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      6. lower-/.f64 43.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

   6. Applied rewrites 43.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-2 \cdot \frac{c}{a \cdot \sqrt{-4 \cdot \frac{c}{a}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

   7. Taylor expanded in c around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\color{blue}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \color{blue}{\sqrt{\frac{-4}{a \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      2. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      4. lower-/.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

      5. lower-*.f64 50.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

   9. Applied rewrites 50.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2}{\color{blue}{a \cdot \sqrt{\frac{-4}{a \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a + a}\\ \end{array} \]

   if 3.9e11 < b

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 40.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 54.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 54.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   13. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       2. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       3. frac-2neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       4. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{2}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       5. associate-*r/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       6. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       7. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       8. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       9. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       10. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       11. lift-+.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       12. lower-neg.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{\color{blue}{-2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       13. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       14. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       15. lower-+.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   14. Applied rewrites 54.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c + c}{-\left(b + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 65.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(-4 \cdot a\right) \cdot c\\ t_1 := \sqrt{\left|t\_0\right|}\\ \mathbf{if}\;b \leq 390000000000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t\_1}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{t\_0} - b}{a + a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (* (* -4.0 a) c)) (t_1 (sqrt (fabs t_0))))
  (if (<= b 390000000000.0)
    (if (>= b 0.0)
      (/ (* 2.0 c) (- (- b) t_1))
      (/ (+ (- b) t_1) (* 2.0 a)))
    (if (>= b 0.0)
      (/ (+ c c) (- (+ b b)))
      (/ (- (sqrt t_0) b) (+ a a))))))

C

double code(double a, double b, double c) {
	double t_0 = (-4.0 * a) * c;
	double t_1 = sqrt(fabs(t_0));
	double tmp_1;
	if (b <= 390000000000.0) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - t_1);
		} else {
			tmp_2 = (-b + t_1) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = (sqrt(t_0) - b) / (a + a);
	}
	return tmp_1;
}

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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = ((-4.0d0) * a) * c
    t_1 = sqrt(abs(t_0))
    if (b <= 390000000000.0d0) then
        if (b >= 0.0d0) then
            tmp_2 = (2.0d0 * c) / (-b - t_1)
        else
            tmp_2 = (-b + t_1) / (2.0d0 * a)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (c + c) / -(b + b)
    else
        tmp_1 = (sqrt(t_0) - b) / (a + a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (-4.0 * a) * c;
	double t_1 = Math.sqrt(Math.abs(t_0));
	double tmp_1;
	if (b <= 390000000000.0) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (2.0 * c) / (-b - t_1);
		} else {
			tmp_2 = (-b + t_1) / (2.0 * a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = (Math.sqrt(t_0) - b) / (a + a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (-4.0 * a) * c
	t_1 = math.sqrt(math.fabs(t_0))
	tmp_1 = 0
	if b <= 390000000000.0:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = (2.0 * c) / (-b - t_1)
		else:
			tmp_2 = (-b + t_1) / (2.0 * a)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (c + c) / -(b + b)
	else:
		tmp_1 = (math.sqrt(t_0) - b) / (a + a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-4.0 * a) * c)
	t_1 = sqrt(abs(t_0))
	tmp_1 = 0.0
	if (b <= 390000000000.0)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_1));
		else
			tmp_2 = Float64(Float64(Float64(-b) + t_1) / Float64(2.0 * a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c + c) / Float64(-Float64(b + b)));
	else
		tmp_1 = Float64(Float64(sqrt(t_0) - b) / Float64(a + a));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (-4.0 * a) * c;
	t_1 = sqrt(abs(t_0));
	tmp_2 = 0.0;
	if (b <= 390000000000.0)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = (2.0 * c) / (-b - t_1);
		else
			tmp_3 = (-b + t_1) / (2.0 * a);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (c + c) / -(b + b);
	else
		tmp_2 = (sqrt(t_0) - b) / (a + a);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[Abs[t$95$0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 390000000000.0], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + t$95$1), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c + c), $MachinePrecision] / (-N[(b + b), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.9e11

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 40.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      4. sqr-abs-rev N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\left|\sqrt{-4 \cdot \left(a \cdot c\right)}\right| \cdot \left|\sqrt{-4 \cdot \left(a \cdot c\right)}\right|}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      5. mul-fabs N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\left|\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}\right|}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\left|\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\left|\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      8. rem-square-sqrt N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\left|\color{blue}{-4 \cdot \left(a \cdot c\right)}\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      9. lower-fabs.f64 45.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\left|-4 \cdot \left(a \cdot c\right)\right|}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   9. Applied rewrites 45.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\left|\left(-4 \cdot a\right) \cdot c\right|}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\left|\left(-4 \cdot a\right) \cdot c\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \end{array} \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\left|\left(-4 \cdot a\right) \cdot c\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \end{array} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\left|\left(-4 \cdot a\right) \cdot c\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \end{array} \]

       4. sqr-abs-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\left|\left(-4 \cdot a\right) \cdot c\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left|\sqrt{-4 \cdot \left(a \cdot c\right)}\right| \cdot \left|\sqrt{-4 \cdot \left(a \cdot c\right)}\right|}}{2 \cdot a}\\ \end{array} \]

       5. mul-fabs N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\left|\left(-4 \cdot a\right) \cdot c\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left|\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}\right|}}{2 \cdot a}\\ \end{array} \]

       6. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\left|\left(-4 \cdot a\right) \cdot c\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left|\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}\right|}}{2 \cdot a}\\ \end{array} \]

       7. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\left|\left(-4 \cdot a\right) \cdot c\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left|\sqrt{-4 \cdot \left(a \cdot c\right)} \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}\right|}}{2 \cdot a}\\ \end{array} \]

       8. rem-square-sqrt N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\left|\left(-4 \cdot a\right) \cdot c\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left|-4 \cdot \left(a \cdot c\right)\right|}}{2 \cdot a}\\ \end{array} \]

       9. lower-fabs.f64 50.1%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\left|\left(-4 \cdot a\right) \cdot c\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left|-4 \cdot \left(a \cdot c\right)\right|}}{2 \cdot a}\\ \end{array} \]

   11. Applied rewrites 50.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\left|\left(-4 \cdot a\right) \cdot c\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left|\left(-4 \cdot a\right) \cdot c\right|}}{2 \cdot a}\\ \end{array} \]

   if 3.9e11 < b

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 40.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 54.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 54.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   13. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       2. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       3. frac-2neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       4. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{2}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       5. associate-*r/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       6. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       7. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       8. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       9. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       10. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       11. lift-+.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       12. lower-neg.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{\color{blue}{-2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       13. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       14. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       15. lower-+.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   14. Applied rewrites 54.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c + c}{-\left(b + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 65.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\left|\left(a \cdot c\right) \cdot -4\right|}\\ \mathbf{if}\;b \leq 390000000000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{t\_0 + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (sqrt (fabs (* (* a c) -4.0)))))
  (if (<= b 390000000000.0)
    (if (>= b 0.0) (* c (/ -2.0 (+ t_0 b))) (/ (- t_0 b) (+ a a)))
    (if (>= b 0.0)
      (/ (+ c c) (- (+ b b)))
      (/ (- (sqrt (* (* -4.0 a) c)) b) (+ a a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fabs(((a * c) * -4.0)));
	double tmp_1;
	if (b <= 390000000000.0) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (t_0 + b));
		} else {
			tmp_2 = (t_0 - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	}
	return tmp_1;
}

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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = sqrt(abs(((a * c) * (-4.0d0))))
    if (b <= 390000000000.0d0) then
        if (b >= 0.0d0) then
            tmp_2 = c * ((-2.0d0) / (t_0 + b))
        else
            tmp_2 = (t_0 - b) / (a + a)
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (c + c) / -(b + b)
    else
        tmp_1 = (sqrt((((-4.0d0) * a) * c)) - b) / (a + a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(Math.abs(((a * c) * -4.0)));
	double tmp_1;
	if (b <= 390000000000.0) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (t_0 + b));
		} else {
			tmp_2 = (t_0 - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = (Math.sqrt(((-4.0 * a) * c)) - b) / (a + a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(math.fabs(((a * c) * -4.0)))
	tmp_1 = 0
	if b <= 390000000000.0:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = c * (-2.0 / (t_0 + b))
		else:
			tmp_2 = (t_0 - b) / (a + a)
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (c + c) / -(b + b)
	else:
		tmp_1 = (math.sqrt(((-4.0 * a) * c)) - b) / (a + a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = sqrt(abs(Float64(Float64(a * c) * -4.0)))
	tmp_1 = 0.0
	if (b <= 390000000000.0)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / Float64(t_0 + b)));
		else
			tmp_2 = Float64(Float64(t_0 - b) / Float64(a + a));
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c + c) / Float64(-Float64(b + b)));
	else
		tmp_1 = Float64(Float64(sqrt(Float64(Float64(-4.0 * a) * c)) - b) / Float64(a + a));
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = sqrt(abs(((a * c) * -4.0)));
	tmp_2 = 0.0;
	if (b <= 390000000000.0)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = c * (-2.0 / (t_0 + b));
		else
			tmp_3 = (t_0 - b) / (a + a);
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (c + c) / -(b + b);
	else
		tmp_2 = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[Abs[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 390000000000.0], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(t$95$0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c + c), $MachinePrecision] / (-N[(b + b), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.9e11

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 40.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ } \end{array}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\color{blue}{\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}}} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\color{blue}{\sqrt{\left(-4 \cdot a\right) \cdot c}} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \color{blue}{\sqrt{\left(-4 \cdot a\right) \cdot c}}} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       4. sqr-abs-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\color{blue}{\left|\sqrt{\left(-4 \cdot a\right) \cdot c}\right| \cdot \left|\sqrt{\left(-4 \cdot a\right) \cdot c}\right|}} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       5. mul-fabs N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\color{blue}{\left|\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}\right|}} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       6. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left|\color{blue}{\sqrt{\left(-4 \cdot a\right) \cdot c}} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}\right|} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       7. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left|\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \color{blue}{\sqrt{\left(-4 \cdot a\right) \cdot c}}\right|} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       8. rem-square-sqrt N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left|\color{blue}{\left(-4 \cdot a\right) \cdot c}\right|} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       9. lower-fabs.f64 45.3%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\color{blue}{\left|\left(-4 \cdot a\right) \cdot c\right|}} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   11. Applied rewrites 45.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\color{blue}{\left|\left(a \cdot c\right) \cdot -4\right|}} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   12. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left|\left(a \cdot c\right) \cdot -4\right|} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}} - b}{a + a}\\ \end{array} \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left|\left(a \cdot c\right) \cdot -4\right|} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}} - b}{a + a}\\ \end{array} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left|\left(a \cdot c\right) \cdot -4\right|} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}} - b}{a + a}\\ \end{array} \]

       4. sqr-abs-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left|\left(a \cdot c\right) \cdot -4\right|} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\sqrt{\left(-4 \cdot a\right) \cdot c}\right| \cdot \left|\sqrt{\left(-4 \cdot a\right) \cdot c}\right|} - b}{a + a}\\ \end{array} \]

       5. mul-fabs N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left|\left(a \cdot c\right) \cdot -4\right|} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}\right|} - b}{a + a}\\ \end{array} \]

       6. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left|\left(a \cdot c\right) \cdot -4\right|} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}\right|} - b}{a + a}\\ \end{array} \]

       7. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left|\left(a \cdot c\right) \cdot -4\right|} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}\right|} - b}{a + a}\\ \end{array} \]

       8. rem-square-sqrt N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left|\left(a \cdot c\right) \cdot -4\right|} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\left(-4 \cdot a\right) \cdot c\right|} - b}{a + a}\\ \end{array} \]

       9. lower-fabs.f64 50.1%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left|\left(a \cdot c\right) \cdot -4\right|} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\left(-4 \cdot a\right) \cdot c\right|} - b}{a + a}\\ \end{array} \]

   13. Applied rewrites 50.1%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left|\left(a \cdot c\right) \cdot -4\right|} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\left(a \cdot c\right) \cdot -4\right|} - b}{a + a}\\ \end{array} \]

   if 3.9e11 < b

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 40.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 54.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 54.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   13. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       2. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       3. frac-2neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       4. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{2}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       5. associate-*r/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       6. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       7. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       8. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       9. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       10. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       11. lift-+.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       12. lower-neg.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{\color{blue}{-2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       13. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       14. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       15. lower-+.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   14. Applied rewrites 54.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c + c}{-\left(b + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 64.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(-4 \cdot a\right) \cdot c\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;b \leq -6 \cdot 10^{-224}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|t\_0\right|} - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \leq 390000000000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{t\_1 + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\left(a \cdot c\right) \cdot -4} - b\right)\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 - b}{a + a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (* (* -4.0 a) c)) (t_1 (sqrt t_0)))
  (if (<= b -6e-224)
    (if (>= b 0.0)
      (* c (/ -2.0 (* 2.0 b)))
      (/ (- (sqrt (fabs t_0)) b) (+ a a)))
    (if (<= b 390000000000.0)
      (if (>= b 0.0)
        (* c (/ -2.0 (+ t_1 b)))
        (* (/ 0.5 a) (- (sqrt (* (* a c) -4.0)) b)))
      (if (>= b 0.0) (/ (+ c c) (- (+ b b))) (/ (- t_1 b) (+ a a)))))))

C

double code(double a, double b, double c) {
	double t_0 = (-4.0 * a) * c;
	double t_1 = sqrt(t_0);
	double tmp_1;
	if (b <= -6e-224) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (2.0 * b));
		} else {
			tmp_2 = (sqrt(fabs(t_0)) - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 390000000000.0) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * (-2.0 / (t_1 + b));
		} else {
			tmp_3 = (0.5 / a) * (sqrt(((a * c) * -4.0)) - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = (t_1 - b) / (a + a);
	}
	return tmp_1;
}

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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = ((-4.0d0) * a) * c
    t_1 = sqrt(t_0)
    if (b <= (-6d-224)) then
        if (b >= 0.0d0) then
            tmp_2 = c * ((-2.0d0) / (2.0d0 * b))
        else
            tmp_2 = (sqrt(abs(t_0)) - b) / (a + a)
        end if
        tmp_1 = tmp_2
    else if (b <= 390000000000.0d0) then
        if (b >= 0.0d0) then
            tmp_3 = c * ((-2.0d0) / (t_1 + b))
        else
            tmp_3 = (0.5d0 / a) * (sqrt(((a * c) * (-4.0d0))) - b)
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (c + c) / -(b + b)
    else
        tmp_1 = (t_1 - b) / (a + a)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (-4.0 * a) * c;
	double t_1 = Math.sqrt(t_0);
	double tmp_1;
	if (b <= -6e-224) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (2.0 * b));
		} else {
			tmp_2 = (Math.sqrt(Math.abs(t_0)) - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 390000000000.0) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * (-2.0 / (t_1 + b));
		} else {
			tmp_3 = (0.5 / a) * (Math.sqrt(((a * c) * -4.0)) - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = (t_1 - b) / (a + a);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (-4.0 * a) * c
	t_1 = math.sqrt(t_0)
	tmp_1 = 0
	if b <= -6e-224:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = c * (-2.0 / (2.0 * b))
		else:
			tmp_2 = (math.sqrt(math.fabs(t_0)) - b) / (a + a)
		tmp_1 = tmp_2
	elif b <= 390000000000.0:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = c * (-2.0 / (t_1 + b))
		else:
			tmp_3 = (0.5 / a) * (math.sqrt(((a * c) * -4.0)) - b)
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (c + c) / -(b + b)
	else:
		tmp_1 = (t_1 - b) / (a + a)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-4.0 * a) * c)
	t_1 = sqrt(t_0)
	tmp_1 = 0.0
	if (b <= -6e-224)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / Float64(2.0 * b)));
		else
			tmp_2 = Float64(Float64(sqrt(abs(t_0)) - b) / Float64(a + a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 390000000000.0)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(c * Float64(-2.0 / Float64(t_1 + b)));
		else
			tmp_3 = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(a * c) * -4.0)) - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c + c) / Float64(-Float64(b + b)));
	else
		tmp_1 = Float64(Float64(t_1 - b) / Float64(a + a));
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = (-4.0 * a) * c;
	t_1 = sqrt(t_0);
	tmp_2 = 0.0;
	if (b <= -6e-224)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = c * (-2.0 / (2.0 * b));
		else
			tmp_3 = (sqrt(abs(t_0)) - b) / (a + a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 390000000000.0)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = c * (-2.0 / (t_1 + b));
		else
			tmp_4 = (0.5 / a) * (sqrt(((a * c) * -4.0)) - b);
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (c + c) / -(b + b);
	else
		tmp_2 = (t_1 - b) / (a + a);
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[b, -6e-224], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(2.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[Abs[t$95$0], $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 390000000000.0], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(c + c), $MachinePrecision] / (-N[(b + b), $MachinePrecision])), $MachinePrecision], N[(N[(t$95$1 - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.9999999999999996e-224

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 40.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 54.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 54.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   13. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}} - b}{a + a}\\ \end{array} \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}} - b}{a + a}\\ \end{array} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}} - b}{a + a}\\ \end{array} \]

       4. sqr-abs-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\sqrt{\left(-4 \cdot a\right) \cdot c}\right| \cdot \left|\sqrt{\left(-4 \cdot a\right) \cdot c}\right|} - b}{a + a}\\ \end{array} \]

       5. mul-fabs N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}\right|} - b}{a + a}\\ \end{array} \]

       6. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}\right|} - b}{a + a}\\ \end{array} \]

       7. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}\right|} - b}{a + a}\\ \end{array} \]

       8. rem-square-sqrt N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\left(-4 \cdot a\right) \cdot c\right|} - b}{a + a}\\ \end{array} \]

       9. lower-fabs.f64 59.3%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\left(-4 \cdot a\right) \cdot c\right|} - b}{a + a}\\ \end{array} \]

   14. Applied rewrites 59.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\left(-4 \cdot a\right) \cdot c\right|} - b}{a + a}\\ \end{array} \]

   if -5.9999999999999996e-224 < b < 3.9e11

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 40.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ } \end{array}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       2. mult-flip N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(-4 \cdot a\right) \cdot c} - b\right) \cdot \frac{1}{a + a}\\ \end{array} \]

       3. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a + a} \cdot \left(\sqrt{\left(-4 \cdot a\right) \cdot c} - b\right)\\ \end{array} \]

       4. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a + a} \cdot \left(\sqrt{\left(-4 \cdot a\right) \cdot c} - b\right)\\ \end{array} \]

       5. lift-+.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a + a} \cdot \left(\sqrt{\left(-4 \cdot a\right) \cdot c} - b\right)\\ \end{array} \]

       6. count-2-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot a} \cdot \left(\sqrt{\left(-4 \cdot a\right) \cdot c} - b\right)\\ \end{array} \]

       7. associate-/r* N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{\left(-4 \cdot a\right) \cdot c} - b\right)\\ \end{array} \]

       8. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{\left(-4 \cdot a\right) \cdot c} - b\right)\\ \end{array} \]

       9. lower-/.f64 40.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\left(-4 \cdot a\right) \cdot c} - b\right)\\ \end{array} \]

       10. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{\left(-4 \cdot a\right) \cdot c} - b\right)\\ \end{array} \]

       11. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{\left(-4 \cdot a\right) \cdot c} - b\right)\\ \end{array} \]

       12. associate-*l* N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right)} - b\right)\\ \end{array} \]

       13. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{-4 \cdot \left(a \cdot c\right)} - b\right)\\ \end{array} \]

       14. *-commutative N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{\left(a \cdot c\right) \cdot -4} - b\right)\\ \end{array} \]

       15. lower-*.f64 40.8%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\left(a \cdot c\right) \cdot -4} - b\right)\\ \end{array} \]

   11. Applied rewrites 40.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\left(a \cdot c\right) \cdot -4} - b\right)\\ \end{array} \]

   if 3.9e11 < b

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 40.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 54.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 54.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   13. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       2. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       3. frac-2neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       4. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{2}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       5. associate-*r/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       6. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       7. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       8. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       9. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       10. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       11. lift-+.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       12. lower-neg.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{\color{blue}{-2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       13. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       14. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       15. lower-+.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   14. Applied rewrites 54.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c + c}{-\left(b + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 64.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(-4 \cdot a\right) \cdot c\\ t_1 := \frac{\sqrt{t\_0} - b}{a + a}\\ \mathbf{if}\;b \leq -8.4 \cdot 10^{-137}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|t\_0\right|} - b}{a + a}\\ \end{array}\\ \mathbf{elif}\;b \leq 390000000000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (* (* -4.0 a) c)) (t_1 (/ (- (sqrt t_0) b) (+ a a))))
  (if (<= b -8.4e-137)
    (if (>= b 0.0)
      (* c (/ -2.0 (* 2.0 b)))
      (/ (- (sqrt (fabs t_0)) b) (+ a a)))
    (if (<= b 390000000000.0)
      (if (>= b 0.0) (* c (/ -2.0 (sqrt (- (* 4.0 (* a c)))))) t_1)
      (if (>= b 0.0) (/ (+ c c) (- (+ b b))) t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = (-4.0 * a) * c;
	double t_1 = (sqrt(t_0) - b) / (a + a);
	double tmp_1;
	if (b <= -8.4e-137) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (2.0 * b));
		} else {
			tmp_2 = (sqrt(fabs(t_0)) - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 390000000000.0) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * (-2.0 / sqrt(-(4.0 * (a * c))));
		} else {
			tmp_3 = t_1;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    real(8) :: tmp_3
    t_0 = ((-4.0d0) * a) * c
    t_1 = (sqrt(t_0) - b) / (a + a)
    if (b <= (-8.4d-137)) then
        if (b >= 0.0d0) then
            tmp_2 = c * ((-2.0d0) / (2.0d0 * b))
        else
            tmp_2 = (sqrt(abs(t_0)) - b) / (a + a)
        end if
        tmp_1 = tmp_2
    else if (b <= 390000000000.0d0) then
        if (b >= 0.0d0) then
            tmp_3 = c * ((-2.0d0) / sqrt(-(4.0d0 * (a * c))))
        else
            tmp_3 = t_1
        end if
        tmp_1 = tmp_3
    else if (b >= 0.0d0) then
        tmp_1 = (c + c) / -(b + b)
    else
        tmp_1 = t_1
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (-4.0 * a) * c;
	double t_1 = (Math.sqrt(t_0) - b) / (a + a);
	double tmp_1;
	if (b <= -8.4e-137) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / (2.0 * b));
		} else {
			tmp_2 = (Math.sqrt(Math.abs(t_0)) - b) / (a + a);
		}
		tmp_1 = tmp_2;
	} else if (b <= 390000000000.0) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = c * (-2.0 / Math.sqrt(-(4.0 * (a * c))));
		} else {
			tmp_3 = t_1;
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = t_1;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (-4.0 * a) * c
	t_1 = (math.sqrt(t_0) - b) / (a + a)
	tmp_1 = 0
	if b <= -8.4e-137:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = c * (-2.0 / (2.0 * b))
		else:
			tmp_2 = (math.sqrt(math.fabs(t_0)) - b) / (a + a)
		tmp_1 = tmp_2
	elif b <= 390000000000.0:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = c * (-2.0 / math.sqrt(-(4.0 * (a * c))))
		else:
			tmp_3 = t_1
		tmp_1 = tmp_3
	elif b >= 0.0:
		tmp_1 = (c + c) / -(b + b)
	else:
		tmp_1 = t_1
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-4.0 * a) * c)
	t_1 = Float64(Float64(sqrt(t_0) - b) / Float64(a + a))
	tmp_1 = 0.0
	if (b <= -8.4e-137)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / Float64(2.0 * b)));
		else
			tmp_2 = Float64(Float64(sqrt(abs(t_0)) - b) / Float64(a + a));
		end
		tmp_1 = tmp_2;
	elseif (b <= 390000000000.0)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(c * Float64(-2.0 / sqrt(Float64(-Float64(4.0 * Float64(a * c))))));
		else
			tmp_3 = t_1;
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c + c) / Float64(-Float64(b + b)));
	else
		tmp_1 = t_1;
	end
	return tmp_1
end

MATLAB

function tmp_5 = code(a, b, c)
	t_0 = (-4.0 * a) * c;
	t_1 = (sqrt(t_0) - b) / (a + a);
	tmp_2 = 0.0;
	if (b <= -8.4e-137)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = c * (-2.0 / (2.0 * b));
		else
			tmp_3 = (sqrt(abs(t_0)) - b) / (a + a);
		end
		tmp_2 = tmp_3;
	elseif (b <= 390000000000.0)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = c * (-2.0 / sqrt(-(4.0 * (a * c))));
		else
			tmp_4 = t_1;
		end
		tmp_2 = tmp_4;
	elseif (b >= 0.0)
		tmp_2 = (c + c) / -(b + b);
	else
		tmp_2 = t_1;
	end
	tmp_5 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[t$95$0], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -8.4e-137], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(2.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[Abs[t$95$0], $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 390000000000.0], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[Sqrt[(-N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1], If[GreaterEqual[b, 0.0], N[(N[(c + c), $MachinePrecision] / (-N[(b + b), $MachinePrecision])), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.3999999999999997e-137

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 40.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 54.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 54.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   13. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}} - b}{a + a}\\ \end{array} \]

       2. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}} - b}{a + a}\\ \end{array} \]

       3. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}} - b}{a + a}\\ \end{array} \]

       4. sqr-abs-rev N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\sqrt{\left(-4 \cdot a\right) \cdot c}\right| \cdot \left|\sqrt{\left(-4 \cdot a\right) \cdot c}\right|} - b}{a + a}\\ \end{array} \]

       5. mul-fabs N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}\right|} - b}{a + a}\\ \end{array} \]

       6. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}\right|} - b}{a + a}\\ \end{array} \]

       7. lift-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\sqrt{\left(-4 \cdot a\right) \cdot c} \cdot \sqrt{\left(-4 \cdot a\right) \cdot c}\right|} - b}{a + a}\\ \end{array} \]

       8. rem-square-sqrt N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\left(-4 \cdot a\right) \cdot c\right|} - b}{a + a}\\ \end{array} \]

       9. lower-fabs.f64 59.3%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\left(-4 \cdot a\right) \cdot c\right|} - b}{a + a}\\ \end{array} \]

   14. Applied rewrites 59.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left|\left(-4 \cdot a\right) \cdot c\right|} - b}{a + a}\\ \end{array} \]

   if -8.3999999999999997e-137 < b < 3.9e11

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 40.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-2}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       3. lower-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       4. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       5. lower-*.f64 33.9%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 33.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-2}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   if 3.9e11 < b

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 40.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 54.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 54.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   13. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       2. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       3. frac-2neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       4. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{2}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       5. associate-*r/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       6. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       7. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       8. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       9. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       10. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       11. lift-+.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       12. lower-neg.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{\color{blue}{-2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       13. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       14. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       15. lower-+.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   14. Applied rewrites 54.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c + c}{-\left(b + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 59.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \mathbf{if}\;b \leq 390000000000:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (let* ((t_0 (/ (- (sqrt (* (* -4.0 a) c)) b) (+ a a))))
  (if (<= b 390000000000.0)
    (if (>= b 0.0) (* c (/ -2.0 (sqrt (- (* 4.0 (* a c)))))) t_0)
    (if (>= b 0.0) (/ (+ c c) (- (+ b b))) t_0))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	double tmp_1;
	if (b <= 390000000000.0) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / sqrt(-(4.0 * (a * c))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = (sqrt((((-4.0d0) * a) * c)) - b) / (a + a)
    if (b <= 390000000000.0d0) then
        if (b >= 0.0d0) then
            tmp_2 = c * ((-2.0d0) / sqrt(-(4.0d0 * (a * c))))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (b >= 0.0d0) then
        tmp_1 = (c + c) / -(b + b)
    else
        tmp_1 = t_0
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((-4.0 * a) * c)) - b) / (a + a);
	double tmp_1;
	if (b <= 390000000000.0) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c * (-2.0 / Math.sqrt(-(4.0 * (a * c))));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (b >= 0.0) {
		tmp_1 = (c + c) / -(b + b);
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((-4.0 * a) * c)) - b) / (a + a)
	tmp_1 = 0
	if b <= 390000000000.0:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = c * (-2.0 / math.sqrt(-(4.0 * (a * c))))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif b >= 0.0:
		tmp_1 = (c + c) / -(b + b)
	else:
		tmp_1 = t_0
	return tmp_1

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(-4.0 * a) * c)) - b) / Float64(a + a))
	tmp_1 = 0.0
	if (b <= 390000000000.0)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c * Float64(-2.0 / sqrt(Float64(-Float64(4.0 * Float64(a * c))))));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (b >= 0.0)
		tmp_1 = Float64(Float64(c + c) / Float64(-Float64(b + b)));
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	t_0 = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	tmp_2 = 0.0;
	if (b <= 390000000000.0)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = c * (-2.0 / sqrt(-(4.0 * (a * c))));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (b >= 0.0)
		tmp_2 = (c + c) / -(b + b);
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 390000000000.0], If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[Sqrt[(-N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[GreaterEqual[b, 0.0], N[(N[(c + c), $MachinePrecision] / (-N[(b + b), $MachinePrecision])), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.9e11

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 40.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around 0

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-2}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\mathsf{neg}\left(4 \cdot \left(a \cdot c\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       3. lower-neg.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       4. lower-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       5. lower-*.f64 33.9%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 33.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-2}{\sqrt{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   if 3.9e11 < b

   1. Initial program 72.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 56.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   4. Applied rewrites 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Taylor expanded in a around inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

      2. lower-*.f64 40.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   7. Applied rewrites 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   11. Step-by-step derivation

       1. lower-*.f64 54.6%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   12. Applied rewrites 54.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
   13. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       2. lift-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       3. frac-2neg N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       4. metadata-eval N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{2}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       5. associate-*r/ N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       6. *-commutative N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       7. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       8. lower-/.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       9. lift-*.f64 N/A

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       10. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       11. lift-+.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       12. lower-neg.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{\color{blue}{-2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       13. lift-*.f64 N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       14. count-2-rev N/A

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

       15. lower-+.f64 54.7%

           \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

   14. Applied rewrites 54.7%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c + c}{-\left(b + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 54.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (>= b 0.0)
  (/ (+ c c) (- (+ b b)))
  (/ (- (sqrt (* (* -4.0 a) c)) b) (+ a a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (c + c) / -(b + b);
	} else {
		tmp = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	}
	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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = (c + c) / -(b + b)
    else
        tmp = (sqrt((((-4.0d0) * a) * c)) - b) / (a + a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (c + c) / -(b + b);
	} else {
		tmp = (Math.sqrt(((-4.0 * a) * c)) - b) / (a + a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = (c + c) / -(b + b)
	else:
		tmp = (math.sqrt(((-4.0 * a) * c)) - b) / (a + a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(c + c) / Float64(-Float64(b + b)));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(-4.0 * a) * c)) - b) / Float64(a + a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (c + c) / -(b + b);
	else
		tmp = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[(c + c), $MachinePrecision] / (-N[(b + b), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 72.0%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

2. Taylor expanded in a around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. lower-*.f64 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

4. Applied rewrites 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

5. Taylor expanded in a around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   2. lower-*.f64 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

7. Applied rewrites 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
8. Step-by-step derivation

9. Applied rewrites 40.8%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

10. Taylor expanded in b around inf

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
11. Step-by-step derivation

    1. lower-*.f64 54.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

12. Applied rewrites 54.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
13. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

    2. lift-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{-2}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

    3. frac-2neg N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \color{blue}{\frac{\mathsf{neg}\left(-2\right)}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

    4. metadata-eval N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{\color{blue}{2}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

    5. associate-*r/ N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

    6. *-commutative N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

    7. lift-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

    8. lower-/.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{neg}\left(2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

    9. lift-*.f64 N/A

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

    10. count-2-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

    11. lift-+.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{c + c}}{\mathsf{neg}\left(2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

    12. lower-neg.f64 54.7%

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{\color{blue}{-2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

    13. lift-*.f64 N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

    14. count-2-rev N/A

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

    15. lower-+.f64 54.7%

        \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c + c}{-\left(b + \color{blue}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

14. Applied rewrites 54.7%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c + c}{-\left(b + b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
15. Add Preprocessing

Alternative 13: 54.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (>= b 0.0)
  (* c (/ -2.0 (* 2.0 b)))
  (/ (- (sqrt (* (* -4.0 a) c)) b) (+ a a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = c * (-2.0 / (2.0 * b));
	} else {
		tmp = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	}
	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(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b >= 0.0d0) then
        tmp = c * ((-2.0d0) / (2.0d0 * b))
    else
        tmp = (sqrt((((-4.0d0) * a) * c)) - b) / (a + a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = c * (-2.0 / (2.0 * b));
	} else {
		tmp = (Math.sqrt(((-4.0 * a) * c)) - b) / (a + a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = c * (-2.0 / (2.0 * b))
	else:
		tmp = (math.sqrt(((-4.0 * a) * c)) - b) / (a + a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(c * Float64(-2.0 / Float64(2.0 * b)));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(-4.0 * a) * c)) - b) / Float64(a + a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = c * (-2.0 / (2.0 * b));
	else
		tmp = (sqrt(((-4.0 * a) * c)) - b) / (a + a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(c * N[(-2.0 / N[(2.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 72.0%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

2. Taylor expanded in a around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. lower-*.f64 56.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot \color{blue}{c}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

4. Applied rewrites 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

5. Taylor expanded in a around inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

   2. lower-*.f64 40.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

7. Applied rewrites 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{-4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]
8. Step-by-step derivation

9. Applied rewrites 40.8%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ } \end{array}} \]

10. Taylor expanded in b around inf

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
11. Step-by-step derivation

    1. lower-*.f64 54.6%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{2 \cdot \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]

12. Applied rewrites 54.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{\color{blue}{2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c} - b}{a + a}\\ \end{array} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))